GCE (Advanced Level) Examination

G.C.E. (Advanced Level) Examination - 2011 onwards

Structure of Question Papers and Prototype Questions

Volume 1

01 - Physics02 - Chemistry07 - Mathematics 08 - Agriculture 09 - Biology 10 - Combined Mathematics11 - Higher Mathematics

CurriculumAssessment

&Evaluation

Teaching LearningN E T S

Research and Development BranchNational Evaluation and Testing ServiceDepartment of Examinations

Structure of Question Papers and Prototype Questions for G.C.E. (Advanced Level) Examination - 2011 onwards

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All rights reserved

G.C.E. (Advanced Level) Examination - 2011 onwards Structure of Question Papers and Prototype Questions Volume 1

Department of Examinations First print - 2010

ISBN 978-955-668-028-7

Research and Development BranchNational Evaluation and Testing ServiceDepartment of Examinations

Financial AidWorld Bank Education Sector Development Framework and Programme

Education Sector Development Grant (ESDG)

Printing : Department of Examinations


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Message of the Commissioner General of Examinations

Various methodologies are applied to measure the knowledge and comprehension achieved from any course of studies. Out of them the most widespread method used today is the written examination mode. Hence the fact that it builds up a relationship between the process that enhances the efficiency of the Learning Teaching process and the summative evaluation methodology and determines the achivement levels are taken into account.

The syllabi amended according to the new educational reforms of 2007 were introduced to Grade 12 in 2009. Those pupils will appear for the G.C.E. (A.L) examination for the first time in 2011. Accordingly, this is a document containing the structure of the question paper and prototype questions for the use of the pupils appearing for the G.C.E. (A.L) examinations scheduled to be held in and after the year 2011 in terms of the amended syllabi.

With the revision of the subject content of the G.C.E. (A.L) syllabi, it was necessary to effect changes in the structure of question papers to suit these changes. Accordingly, an attempt has been made to collect the subject groups and to maintain the identity of the structures of question papers within each group. The structures were decided on by academic committees and prototype questions too were framed by these committees.

At a time when the confidence and attention on educational measurement and evaluation is increasing it is important to raise awareness in all the parties concerned on how this evaluation process is performed at G.C.E.(A.L) examination. The G.C.E.(A.L) examination is mainly an achievement examination aimed at certification. However, since pupils are selected for universities on the results of this examination, by the University Grants Commission and other national as well as international higher educational institutions, this examination also embodies features of a selection examination. Therefore, by using the structures of question papers and the prototype questions contained in this document, the pupils will get a clear idea about the type of evaluation they will have to face at this examination. As such this will be of immense help for principals and teachers of schools, those who provide guidance in this regard and the school community, in preparing the pupils for the examination.

Part I of this Instruction Manual contains general information about the examination, Part II contains the structure and nature of the question papers while Part III contains prototype questions on each subject.

I wish to express my gratitude to the Director General and staff of the National Institute of Education the Commissioner General of Educational Publications and his staff, the Controlling Examiners of all the subjects, the resource persons, all officers of the Department of Examinations Sri Lanka, for the co-operation extended in drafting the structures of the question papers and the prototype questions contained in this manual, the staff of the World Bank, Educational Sector Development Grant for providing financial assistance for printing this manual to the staff and the Head of Printing Division of the Department of Examinations for the dedication in printing and all who contributed for the successful completion of this task.

Anura Edirisinghe Commissioner General of Examinations2010.06.22Research and Development BranchDepartment of Examinations


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Guidance : Mr. Anura Edirisinghe Commissioner General of Examinations

Direction and Organization : Mrs. Gayathri Abeygunasekara Commissioner of Examinations (Research and Development)

Mr. H.J.M.C.A. Jayasundara Commissioner of Examinations (School Examinations Organization)

Mr. B. Sanath Pujitha Commissioner of Examinations (Evaluation - School Examinations)

Subject Co-ordination and Editing : Mr. J.A.J.R. Jayakody Assistant Commissioner of Examinations

Mrs. Manomi Senevirathne Assistant Commissioner of Examinations

Mr. E. Kulasekara Deputy Commissioner of Examinations

Mrs. T. Balasinkhgm Assistant Commissioner of Examinations

Mrs. Thamara Vidanapathirana Assistant Commissioner of Examinations

Computer typesetting : Mrs. K.P.D. Anusha Maduwanthi Dissanayake Data Entry Operator

Cover page : Mr. Asanka Sameera Deepal Data Entry Operator


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Table of Contents Page No.

Part I

General information about the Examination ----------------- 1

Part II

Structure of question papers and nature ----------------- 6

Part III Prototype questions --------------------------------------------- 7 (01) Physics ---------------------------------------------------- 9 (02) Chemistry ----------------------------------------------- 29

(07) Mathematics -------------------------------------------- 47 (08) Agricultural Science ----------------------------------- 93 (09) Biology ------------------------------------------------- 110 (10) Combined Mathematics ------------------------------ 122 (11) Higher Mathematics ---------------------------------- 169 Annex 01 (Sinhala) ------------------------------------------ 184 Annex 01 (Tamil) -------------------------------------------- 199


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PART I General Certificate of Examination (Advanced Level)

General Information about the Examination

1. Introduction

The G.C.E. (Advanced Level) Examination is the final certification examination of senior secondary education in Sri Lanka. Even though this is conducted mainly as a certifying examination since selection for Universities, other Higher Educational Institutions and Colleges of Education are made on the basis of the results of this examination, this is used also as a selection examination.

Moreover the results of the examination are also regarded a basic qualification to obtain middle level employment. The G.C.E. (Advanced Level) Examination is held under the subject streams, Biological Sciences, Physical sciences, Commerce and Arts. This is held in all three media Sinhala, Tamil and English.

2. Application for Examination

2.1 School Candidates

It is essential that a candidate has completed 80% of attendance out of the time advanced level classes were held at the time of applying for the examination.

The principal of each school should submit applications with information about the name of the school candidates who have completed the requirements to apply for the examination, the subjects offered and the medium. Here strict attention should be paid to fill the name of candidates correctly as indicated in the birth cetiticate and correctly enter the subjects, subject numbers and the medium.

2.2 Private Candidates Applicants who have not appeared as school candidates previously but appearing for the

first time as a private candidate and those candidates who wish to appear by changing even one subject offerd as a previous sitting, should have registered in the respectire Provincial Department of Education to fulfill the requirements of assessment and projects implemented as an alternative to the School Based Assesment programme. They must join the school they are assigned at registration complete the assessment and projects, and a copy of the certificate issued by the principal of such school to that effect should be sent to the Provincial Department of Education, and another copy should be sent to the Department of Examinations. Applicants who are over 21 years of age are exempt from this requirement. The applicants who are below 21 years of age, and not registered are not eligible to apply for the examination.

At the time applications are called from private candidates through a newspaper advertisement published by the Department of Examinations the applicants who wish to appear for the examination, should send to the Department of Examinations before the stipulated date mentioned in the newspaper advertisement, a duly perfected application with the receipt obtained on payment of the required examination fees by reqistered post.


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3. Selection of Subjects

This examination will be held from 2011 according to the ciruler (annex 1) No 2009/16 dated 18.05.2009 issued by the Ministry of Education regarding G.C.E. (A.L) subject combinations and the subject combination for university entrance.

The new syllabi for G.C.E. (A.L) were introduced for Grade 12 in 2009 and G.C.E. (A.L) examination based on these syllabi will be held for the first time in 2011. According to the provisions contained in the circular No. 2009/16 mentioned above candidates should select the subjects relevant to the streams of Biological Sciences, Physical Sciences, Commerce and Arts.

Candidates should appear for three main subjects for the G.C.E. (A.L) Examination and in addition the pupils who wish to apply for university entrance should pass in the "Common General Examination" paper too. Even though not considered for university entrance, pupils have the option to sit the "General English" paper too.

3.1 Subjects approved for G.C.E .(A.L) Examination

The conditions regarding the subject combination to be selcted by pupils for all the steams are mentioned in circular No. 2009/16 embodied in annex 1.

The subjects approved for G.C.E. (A.L) Examination and the subject numbers are given below. When applying for the examination the relevant subject numbers should be used.

Subject Subject Numbers

Physics 01 Chemistry 02Mathematics 07Agricultural Science 08Biology 09Combined Mathematics 10Higher Mathematics 11Civil Technology 14Mechanical Technology 15Electrical, Electronics and Information Technology 16Food Technology 17Agriculture Technology 18Bio Resource Technology 19Information & Communication Technology 20Economics 21Geography 22Political Science 23Logic and Scientific Method 24History of Sri Lanka 25History of India 25 AHistory of Europe 25 BHistory of Modern 25 CHome Economics 28Communication & Media Studies 29Business Statistics 31Business Studies 32Accountancy 33


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Buddhism 41Hinduism 42Christianity 43Islam 44Buddhist Civilization 45Hindu Civilization 46Islam Civilization 47Greek and Roman Civilization 48Christian Civilization 49Art 51Dancing (Indigenous) 52Dancing (Bharatha) 53Oriental Music 54Carnatic Music 55Western Music 56Drama and Theatre (Sinhala) 57Drama and Theatre (Tamil) 58Drama and Theatre (English) 59Sinhala 71Tamil 72English 73Pali 74Sanskrit 75Arabic 78Malay 79French 81German 82Russian 83Hindi 84Chinese 86Japanese 87

For the G.C.E. (A.L) Examination three main subjects out of the above subjects in terms of circular No. 2009/16 issued by the Ministry of Education can be selected. Besides these main subjects candidates must sit the following two subjects.

* Common General Examination (12) For admission to a university in Sri Lanka as an internal student it is essential to obtain a minimum mark as determined. Obtaining the qualification on one occasion can be applied for admission to university entrance on a subsequent occasion. The marks obtained for this subject will not be used for the calculation of the Z score.

* General English (13) This subject is not a main subject at G.C.E. (A.L) Examination. The marks or the pass obtained for General English will not be used for university admission. However, the result obtained for the subject will be entered separately in the G.C.E. (A.L) certificate.


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4. Deciding on Grades

According to the raw score obtained for each subject grades will be determined in the following mannar.

Mark Range Grade

75 - 10065 - 7450 - 6435 - 4900 - 34

A - Distinction PassB - Very Good Pass C - Credit Pass S - Ordinary Pass F - Fail

5. School Based Assessment

5.1 Objectives

The objective of this exercise is to measure skills, competencies etc. of the pupils studying in grades 12 & 13, that cannot be measured at G.C.E. (A.L) examination, during the teaching – learning process and identifying the weaknesses and strengths of pupils, and if there are weak pupils to implement feedback programmes in respect of such pupils. Under this arrangement assessment is carried out in respect of the subjects learnt in the classroom while two projects carried out by pupils are also assessed.

5.2 How Assessment Proceeds

5.2.1 Assessment carried out in respect of subjects learnt in the classroom (a) In respect of every subject learnt in the classroom, three assessments are conducted.

(for them the assessment modalities introduced should be used) (b) In respect of the subjects will practicals learnt in the classroom (subjects such as

Physics, Chemistry, Biology, Agriculture, Home Economics, Music & Dancing) at least one practical assessment should be done per term, and for the remaining assessment other modalities may be used.

(c) There should be 15 sessions of assessment during the two years at the rate of 09 for the three terms in Grade 12 as three assessments per term, and six during the 1st and 2nd terms in Grade 13. making up a total of 15 assessments.

(d) The average scores of these 15 assessments shall be collected by the Department of Examinations, Sri Lanka at the end of the second term in Grade 13. The competency levels determined according to these scores shall be entered in the result sheet of G.C.E.(A.L) in a separate column in the following manner.

School BasedAssessment marks Level of competency

9, 108

6, 74, 5

1, 2, 3

Excellent Level CompetencyHigh Level CompetencyCredit Level CompetencyNear CompetencyNot reached to Competency Level


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5.2.2 Assessment through Projects

Every pupil studying for G.C.E.(A.L) Examination should complete two projects. One of them should be an individual project while other shall be a Group project.

5.2.2.1 Individual Project

(a) In order to design the individual project a suitable topic should be selected and approved by the School Project Committee.

(b) This project should be completed during the 1st and 2nd terms of Grade 12 and should be under the guidance of the class teacher or a subject teacher

(c) The day to day activities regarding the project should be noted in a field note book and certified by the teacher in charge of the project.

(d) Through the completion of this project, it is expected, that the pupil will be able to display his talents, maintain direct dealings with various institutions and the society and provide opportunities to develop an understanding about working with various types of people.

5.2.2.2 Group Project

(a) A group of 10 pupils should be named by the Project Committee of the School for the Group Project.

(b) This group should be so formed as to represent the three streams of the school, namely, Science, Commerce and Arts

(c) For Group Projects too, the group must submit a suitable topic to the Project Committee and obtain approval for it.

(d) Work on this project should be completed during the 3rd term of Grade 12 and 1st term of Grade 13.

(e) Working with team spirit, working with various institutions and people, and gaining the opportunity to be equipped with an idea about the activities in the field are expected from this project.

5.2.3 Assessment of Project

The Projects are subjected to assessment under 05 criteria and the teacher in charge of the project/project committee, shall study how pupils have performed according to each criteria and award marks.

A total of 20 marks is awarded for a project and the score is converted out of 10, which is transmitted to the Department of Examinations.

The competency level decided on according to the final average marks of these two projects, is also entered in the G.C.E.(A.L) result sheet according to the levels mentioned in 5.2.1 (d)


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Part IIG.C.E.(Advanced Level) Examination - 2011 onwards

Structure of question papers and nature

Subject and Subject Number

Paper I Paper IITi

me

(hou

rs)

Part A Part B

Tim

e (h

ours

)

Part A Part B Part C*

Nat

ure

of q

uest

ions

Num

ber

of o

ptio

ns

Num

ber

of q

uest

ions

Num

ber

of q

uest

ions

to b

e an

swer

ed

* N

atur

e of

que

stio

ns

Num

ber

of q

uest

ions

Num

ber

of q

uest

ions

to b

e an

swer

ed

* N

atur

e of

que

stio

ns

Num

ber

of q

uest

ions

Num

ber

of q

uest

ions

to b

e an

swer

ed

* N

atur

e of

que

stio

ns

Num

ber

of q

uest

ions

Num

ber

of q

uest

ions

to b

e an

swer

ed

* N

atur

e of

que

stio

ns

Num

ber

of q

uest

ions

Num

ber

of q

uest

ions

to b

e an

swer

ed

01. Physics 2 1 5 50 50 3 3 4 4 4 6 4/6

02. Chemistry 2 1 5 50 50 3 3 4 4 4 3 2/3 4 3 2/3

07. Mathematics 3 3, 4 10 10 4 7 5/7 3 4 10 10 4 7 5/7

08. Agricultural Science 2 1 5 50 50 3 3 4 4 4 6 4/6

09. Biology 2 1 5 50 50 3 3 4 4 4 6 4/6

10. Combined Mathamatics 3 3, 4 10 10 4 7 5/7 3 4 10 10 4 7 5/7

11. Higher Mathematics 3 3, 4 10 10 4 7 5/7 3 4 10 10 4 7 5/7

* Nature of questions 1. Multiple Choice 2. Other objective 3. Structured 4. Semi Structured 5. Essay 6. Practical


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Part IIIPrototype Questions

The G.C.E.(A.L) Examination is the final cetifying examination held at the end of secondary education. Since pupils are selected for universities and institutions such as other Higher Education Institutions and Colleges of Education too, this embodies the features of a selection examination.

Accordingly when framing question papers for G.C.E(A.L) examination, extra attention should be paid to the features an achievement test should have and since pupils are selected to universities and other Higher Education institutions too, on the results of this examination, attention has also been paid to this aspect.

Accordingly, for the purpose of evaluating pupil achievement assay type tests and objective type tests are used mainly in the written examination system. The subjectivity of the examiner affects an essay type answer. In the case of an objective type of question, since there is only one correct answer it is completely devoid of subjectivity. More attention of the present examination is paid to the type of questions aimed at providing short structured answers, that are half way between these two extremes. Through a structured question the answer that should be provided is strictly controlled within certain limits laid down in the question itself. Accordingly since consistency among examiners in awarding marks can be assured, objective and structured type of questions receive prominence in the examination sphere.

Accordingly in the drafting of question papers in the G.C.E. (A.L) examination, essay type questions are used, only when necessary and more attention will be paid to framing objective type of questions. Moreover in drafting question papers of G.C.E.(A.L) examination attention has already been paid to framing questions of the type that measures advanced mental skills such as comprehension, application, analysis, synthesis and evaluation without merely depending on memory. Questions are framed in relation to practical events as much as possible and they are purported to measure the skills such as understanding something clearly, applying the principles they have learnt to other similar situations, solving problems, argument, presenting new suggestions / plans, comparisons, proper presenting of the language, and expressing ideas clearly.

In this Part III along with the structure of question papers and the method of awarding marks, prototype questions have been included, but they are not model question papers. As such, when drafting question papers depending on the number of parts included in questions, the method of assigning weightage to questions may change as appropriate, according to the subject matter used as the basis for framing questions. In addition to the prototype questions mentioned here, the Commissioner General of Examinations reserves the right to include, depending on the circumstances in the question papers for G.C.E. (A.L) examination, other types of questions too used for objective and essay type tests and any other type of questions including all types of questions mentioned on page 08.


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Objective Tests

Supply Type Selection Type

Matching Type Alternative Type MCQ Type

Question TypeIncomplete Statement TypeNegative TypeCombined Response TypeBest Answer TypeMultiple Response TypeCommon Response TypeSubstitution TypeDouble Statement TypeAssertion & Reason Type

Essay Tests

Essay Type Questions Structured Essay Type Questions

Structuring by using subsections

Structuring by information provided

Structuring by function

Open Response Type Controlled Response Type


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01 - PhysicsStructure of the Question Paper

Paper I - Time : 02 hours. 50 multiple choice questions with 5 options. All questions should be answered. Each question carries 02 marks. Total 100 marks.

Paper II - Time : 03 hours. This paper consists of two parts as Structured Essay and Essay. Part A - Four structured essay type questions. All questions should be answered. 10 marks for each question - altogether 40 marks. Part B - Six essay type questions. Four questions should be answered. Each question carries 15 marks - altogether 60 marks. Total marks for Paper II = 100

Calculation of the final mark : Paper I = 100 Paper II = 100 Final mark = 200 ÷ 2 = 100

Paper IImportant :* Answer all questions.* Select the correct or the most appropriate answer. (A multiple choice answer sheet would be

provided at the examination.) (g = 10 N kg−1)

1. The quantity having dimensions equal to the dimensions of Planck’s constant is, (1) energy. (2) power. (3) angular frequency. (4) torque. (5) angular momentum.

2. When the absolute temperature of an ideal gas is doubled, the root mean square speed of its atoms will increase by a factor of, (1) √2 (2) √3 (3) 2 (4) 3 (5) 4 3. The change of frequency of a sound wave due to Doppler effect depends on (A) the distance of the source from the observer (B) the velocity of sound in air (C) the relative volocity of the source and the observer Of the above statements, (1) only (A) (2) only (B) (3) only (C) (4) only (B) and (C) (5) all (A), (B) and (C)

4. A block of ice of mass 1.5 kg, at 0 °C is heated at a steady rate of 50 W. If the specific latent heat of fusion of ice is 3.0 × 105 J kg−1, the time in hours, for the ice to melt completly is, (1) 150.00 (2) 100.00 (3) 3.75 (4) 3.50 (5) 2.50

5. Consider the following statements made about characteristics of light produced by a laser. (A) It is electromagnetic (B) It is coherent (C) It is monochromatic Of the above statements, (1) only (A) is true. (2) only (B) is true. (3) only (A) and (B) are true. (4) only (B) and (C) are true. (5) all (A), (B) and (C) are true.


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6. The range of length scale in metres dealt in nano-technology is (1) 0.1 − 10. (2) 10 − 1000. (3) 10−4 − 10−2. (4) 10−9 − 10−7. (5) 10−15 − 10−13.

7. No net resultant force acts on a given object. Consider the following statements made about the object. (A) It may be at rest. (B) It may be moving with uniform velocity. (C) It may be moving along a circular path. Of the above statements, (1) only (A) is true. (2) only (C) is true. (3) only (A) and (B) are true. (4) only (A) and (C) are true. (5) all (A), (B) and (C) are true.

8. An immersion heater boils a certain quantity of water in time t1. Another immersion heater boils the same quantity of water in time t2. If both heaters are used together, they will boil the same quantity of water in time, (Neglect the heat capacities of the containers and heat loss to the souroundings.)

(1) (t1 + t2) (2) t1 + t2 (3) (4) (5)12

t1 t2t1 + t2

√ t1 t2

t1 + t2

t1 t2

9. The surface energy of a liquid film on a ring of area 0.15 m2 is, (Surface tension of the liquid is 5 N m-1) (1) 0.75 J (2) 1.50 J (3) 2.25 J (4) 3.00 J (5) 6.00 J 10. If each of the resistances shown in the network is R,

A BR

R RR

R the equivalent resistance between terminals A and B will be, (1) R (2) 2R (3) 3R (4) 4R (5) 5R

11. When a metallic bar is heated from 0 °C to 100 °C, its length increases by 0.05%. The linear expansivity of the metal is,

(1) 5 × 10−3 °C−1 (2) 5 × 10−4 °C−1 (3) 5 × 10−5 °C−1 (4) 5 × 10−6 °C−1 (5) 5 × 10−7 °C−1

12. A block is in static equilibrium on an inclined surface. Which of the following represents the correct free body force diagram?

(1) (2) (3)

(4) (5)

W

R

F

W

R F

W

R F

W

R F

W

R F

13. The total magnifying power of a compound microscope is 15. The magnification of objective is 5. If

the final image is formed at the least distance of distinct vision, the focal length of the eye piece is,

(1) 25 cm (2) 25 cm 2 (3) 25 cm

3 (4) 25 cm10 (5) 25 cm

14


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14. An object starting from rest moves first 50 m at constant acceleration, next 200 m with uniform velocity and finally comes to rest after travelling a further 30 m with constant deceleration. The total time taken is 30 seconds. The maximum velocity attained by the object is,

(1) 12.0 m s−1 (2) 8.7 m s−1 (3) 6.7 m s−1 (4) 6.0 m s−1 (5) 2.7 m s−1

15. The equation of motion of a particle executing simple harmonic motion is given by a = − 4x, where a is the acceleration and x is the displacement. What is the length of a simple pendulum that oscillates with the same frequency as that of the simple harmonic motion?

(1) 3.0 m (2) 2.5 m (3) 2.0 m (4) 1.5 m (5) 0.4 m

1 Ω4 V

I 1 Ω

1 Ω1 Ω1 Ω

1 Ω1 Ω

16. Figure shows a network of seven resistors, each of resistance 1 Ω, connected to a 4 V battery of negligible internal resistance. The current I in the circuit is,

(1) 0.5 A (2) 1.5 A (3) 2.0 A (4) 3.0 A (5) 3.5 A

17. A particle is moving from a point X to a point Y in a gravitatinal field. If there are no other forces acting on the particle, which of the following statement/s is/are correct about the change in the gravitational potential energy?

(A) It depends on the path between X and Y. (B) It is equal to the change in the kinetic energy. (C) It does not depend on the mass of the particle. (1) (A) only (2) (B) only (3) (A) and (B) only (4) (B) and (C) only (5) all (A), (B) and (C) 18. A stone dropped from rest reaches the ground in 8 seconds. The distance travelled by the stone in the last second is, (1) 320 m (2) 245 m (3) 160 m (4) 75 m (5) 70 m

19. A charge +Q is located at point X in an isolated space as shown in figure. Which one of the following statements is incorrect?

(1) The magnitude of the electric field intensity at Y and at Z is equal.

X

+Q

Y

Z

r

r

(2) The potentials at Y and at Z are equal. (3) The potential at Y is positive. (4) No work is done in taking a charge from Y to Z. (5) The electric field at Z acts along ZX.

Fig (1) Fig (2)

airdd2d2

20. An air filled parallel plate capacitor shown in Fig (1) has a capacitance of C. When it is half filled with a dielectric of dielectric constant k = 2 as shown in Fig (2), its capacitance becomes

(1) 3C (2) 53 C (3) 3

2 C

(4) 43 C (5) C

θA

C

B

75°21. A monochromatic ray of light falls on a transparent glass slab of refractive index √2 held in air as shown in the figure. The minimum value of the angle (θ) of incidence for the ray to be totally internally reflected from surface AC is,

(1) 90° (2) 60° (3) 45° (4) 30° (5) 0°


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22. A wooden block of mass m having an internal cavity is attached to the bottom of a water vessel by a weightless string as shown in the figure. The density of water and wood are d and ρ respectively (d > ρ ). If the tension in the string is 12 mg, the volume of the cavity will be,

(1) 0 (2) d - ρ d m( ) (3) m

ρ

(4) 3m2d (5) m( )3

2d1ρ

23. 250 J of work is done on the gas during an adiabatic compression of 5 moles of a perfect gas. The change in the internal energy will be,

(1) 50 J (2) -150 J (3) 250 J (4) -250 J (5) 750 J

A B C

V

x0

24. The graph shows the variation of an electric potential (V) with distance (x). The variation of the corresponding electric field intensity (E) with x is best represented by

(1) (2) (3) (4) (5)

A B C

E

x A B C

E

xA B C

E

xA B C

E

xA B C

E

x

90N

10 kg 60°

25. A force of 90 N is applied on a block placed on a horizontal plane as

shown in the figure. The coefficients of static and dynamic friction between the block and plane are 0.4 and 0.3 respectively. The magnitude of the frictional force exerted by the plane on the block is,

(1) 0 (2) 30 N (3) 40 N (4) 45 N (5) 78 N

26. A small fish at 0.4 m below the surface of a lake is viewed through a convex lens of focal length 3 m. The lens is kept at 0.2 m above the water surface such that the fish lies on the main axis of the lens. The position of the fish seen by the observer is, (The refractive index of water is 4/3.)

(1) virtual and 0.6 m below the lens. (2) real and 0.6 m below the lens. (3) virtual and 0.6 m above the lens. (4) real and 0.43 m below the lens. (5) virtual and 3 m below the lens.

2m mX Y27. Figure shows two charged spheres, X and Y, of masses 2m and m

respectively, which are just prevented from falling under gravity by the application of a potential difference between the two parallel metal plates. If the plates are moved closer together,

(1) X and Y will both remain stationary. (2) X and Y will both begin to move upwards with the same acceleration. (3) X and Y will both begin to move downwards with the same acceleration. (4) X will begin to accelerate upwards faster than Y. (5) X will begin to accelerate downwards faster than Y.


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BA

C

F28. Which of the following gives the correct logic values for A, B and C

respectively for F to be 1? (1) 0, 0, 0 (2) 0, 1, 0 (3) 1, 0, 0 (4) 1, 0, 1 (5) 1, 1, 0

29. You are provided with necessary instruments to find resonance lengths for air columns closed at one end and opened at both ends, using 250 Hz, 300 Hz, 400 Hz and 500 Hz frequency generators. If the velocity of sound in air is 330 m s−1 the shortest resonance length will be,

(1) 8.25 cm (2) 16.5 cm (3) 33 cm (4) 66 cm (5) 132 cm

AB

7 kg

3 kg

30°

30. Two objects A and B touching each other as shown in the figure are released from rest on a smooth inclined plane. The force exerted by B on A is,

(1) 0 (2) 15 N (3) 35 N (4) 50 N (5) 100 N

31. A tiny sphere of mass m made of a material with density d is dropped in a tall jar filled with glycerine of density ρ. When the sphere acquires terminal velocity, the magnitude of the viscous force acting on the sphere is,

(1) mgd ρ (2) mgρ

d (3) ρ d

mg 1 -( ) (4) d ρ

mg 1 +( ) (5) mg

A

hB32. A U-tube contains a liquid of density ρ as shown in the diagram. The

radius of B is twice that of A, which is r. If the liquid level in limb A is raised to height h, the increase in gravitational potential energy of the liquid will be,

(1) 0 (2) 3 h2 π r2ρ g 8 (3) 5 h2 π r2ρ g

(4) 5 h2 π r2ρ g 8 (5) 5 h2 π r2ρ g

4

33. l, M, T, ρ and E denote the length, mass, tension, density and Youngs modulus respectively of a stretched string.

The ratio, Speed of transverse wavesSpeed of longitudinal waves

for the string is given by

(1) TρlEM

(2) TρEM√ (3) EM

Tρ√ (4) TρlEM√ (5)

EMTρl√

34. In a potentiometer experiment the galvanometer shows no deflection when a cell is connected across 60 cm of the potentiometer wire. If the cell is shunted by a resistance of 6 Ω, a balance point is obtained at a length of 50 cm of the wire. The internal resistance of the cell is,

(1) 0.5 Ω (2) 0.6 Ω (3) 1.2 Ω (4) 1.4 Ω (5) 1.5 Ω

35. Water from a faucet of a tap emerges vertically downwards with an initial speed of 1.0 m s−1. The cross-sectional area of the tap is 10−4 m2. Assume that the pressure is constant throughout the stream of water and that the flow is steady. The cross - sectional area of the stream 0.15 m below the tap is,

(1) 5 × 10−5 m2 (2) 2.5 × 10−5 m2 (3) 5.8 × 10−5 m2 (4) 6.7 × 10−4 m2 (5) 1 × 10−4 m2

36. A mass m is placed on a freely rotating table at a distance x from the centre. The maximum frictional force between the mass and the table is mg. The table’s angular velocity is then steadily increased. The angular velocity of the table at which the mass will begin to slide is given by,

(1) � g2x (2) � g

x (3) � m g 2x (4) � m g x

2 (5) � g 3m x


- 14 -

37. The focal length of a concave lens is 25 cm and the focal length of a convex lens is 30 cm. They are placed in contact to make a combined lens. Which of the following defect of vision can be corrected by using a lens having a focal length equal to the focal length of the combination?

(1) Inability to see objects clearly beyond 150 cm (2) Inability to see objects clearly closer than 150 cm (3) Inability to see objects clearly beyond 30 cm (4) Inability to see objects clearly closer than 30 cm (5) Inability to see objects clearly closer than 28 cm

38. The decay chain that leads from 23892

U to 20682

Pb consists of a series of alpha decays and beta decays. The number of alpha particles emitted in such a decay is,

(1) 4 (2) 5 (3) 6 (4) 8 (5) 10

Fig (1)

39. Fig (1) shows a disturbance at a given moment of a transverse wave travelling along a string to the right. A small ribbon is tied to the string as shown. Which of the following best represents the ribbon’s displacement (y) with time t.

(1)

t

y (2)

t

y (3)

t

y

(4)

t

y (5)

t

y

40. When a source of sound is moving in air with the speed faster than the speed of sound, the shape of the wave front produced would be a,

(1) plane (2) sphere (3) cylinder (4) cone (5) parabola

m41. Suppose that a magnetic monopole exist and such a monopole (m)

starting from a long distance at time t = 0 and moving with a constant speed as shown in the figure is passing through a ring made of superconducting material at t = t0. Which of the following graphs correctly represents the variation of induced current (I) in the ring?

(1)

t0t

I (2)

t0t

I (3)

t0t

I

(4)

t0t

I

(5)

t0t

I


- 15 -

NNorth poleof a strong

magnet

N

S42. A magnetised symmetric top can be levitated in a magnetic field as shown in

the figure. It is essential to spin the top about its symmetric vertical axis in order to maintain the system under stable equilibrium, because the angular momentum of the top,

(1) makes the magnetization of the top stronger. (2) makes the magnetic field of the north pole of the magnet more stronger. (3) makes the repulsive force stronger. (4) makes the magnetic flux density through the top larger. (5) preserves the orientation of the top without flipping it.

P Q

S R

II

B

d 43. A slab PQRS of thickness d and charge density n is placed

horizontally in a magnetic field of flux density B which is in the downward direction as shown in the figure to study the Hall effect. A current I is passing along the slab in the direction shown. If the carriers are negatively charged, and the charge of the carriers is q, the direction of the Hall voltage and its magnitude would be,

(1) from side RS to side PQ, and E/B. (2) from side RS to side PQ, and BI/ndq.

(3) from side PQ to side RS, and E/B. (4) from side PQ to side RS, and BI/ndq.

(5) from side PS to side QR, and BI/ndq.

44. Consider the following statements made about an n - channel junction field effect transistor. (A) The transistor always operates with gate - source, voltage (VGS) positive. (B) The drain current through the transistor decreases when VGS is reduced. (C) The width of the depletion region controls the current through the transistor. Of the statement above, (1) only (A) is true. (2) only (B) is true. (3) only (A) and (C) are true. (4) only (B) and (C) are true. (5) all (A), (B) and (C) are true.

45. Which of the following represents the correct truth table for an S - R flip - flop S QR Q shown in figure?

(Qold represents the logic level of the Q output before application of the input and Qnew represents its value after the application of the input.)

(1) S R Qnew

0 0 00 1 01 0 01 1 Qold

(2) S R Qnew

0 0 Qold

0 1 01 0 11 1 ambiguous

(3) S R Qnew

0 0 Qold

0 1 01 0 11 1 Qold

(4) S R Qnew

0 0 01 0 00 1 11 1 ambiguous

(5) S R Qnew

0 0 Qold

0 1 11 0 01 1 ambiguous


- 16 -

46. X - rays are produced in an X - ray tube by a target potential of 62.0 kV. If an electron looses one half of its incoming energy after one collision the minimum wavelength of the X - ray photon produced is (hc = 1240 eVnm)

(1) 4 × 10−2 nm (2) 6 × 10−2 nm (3) 8 × 10−2 nm (4) 4 × 10−3 nm (5) 8 × 10−3 nm

47. An atom of mass m travels opposite to a laser beam whose wavelength is λ with a constant velocity. The change in speed of the atom, if it absorbs a single photon from the laser beam is,

(1) 4 hλm (2) 2 h

λm (3) h

λm (4) h

2λm (5) h4λm

ba

48. Figure shows a uniform magnetic field of flux density B confined to a cylindrical volume of radius a. B is decreasing in magnitude at a constant rate R. The instantaneous acceleration experienced by an electron placed at a distance b is,

(e = charge of the electron, m = mass of the electron)

(1) eRb 2m

, to the right. (2) eRb 2m

, to the left.

(3) eRb2am , to the right. (4) eRb

2am, to the left.

(5) 0.

49. A proton (p) is projected along the axis of a uniform positively charged ring p

+ ++ +

+ ++

+ as shown in the figure. Consider the following velocity (v) - time (t) graphs.

0 t

v

0

(A)

0 t

v

0

(B)

0 t

v

0

(C) If the gravitational forces are neglected which of the above graph/s best represents the variation of

the velocity (v) of the proton with time t? (1) (A) (2) (B) (3) (C) (4) (A) and (B) (5) all (A), (B) and (C)

c ba

50. A sphere of radius a carries a uniform charge per unit volume ρ. A spherical cavity of radius b is created in the sphere as shown in the figure. If c is the distance between the center of the sphere and the center of the cavity, the magnitude of the electric field intensity at all points within the cavity is,

(1) 0 (2) ρ b3ε0

(3) ρ a3ε0

(4) ρ c2

3 bε0 (5) ρ c

3ε0

* * *


- 17 -

01 - PhysicsPaper II

* Answer all questions of part A* Answer four questions only of part B

Part A - Structured Essay(g = 10 N kg−1)

1. In order to determine the density of glass in a laboratory, you are provided with a glass cube of side about 3 cm and mass of about 60 g.

(a) (i) In order to determine the length of a side (ℓ) of the cube with an accurency better then 1%, what laboratory instrument would you use?

(ii) Write down an expression for the volume of the glass cube in terms of ℓ.

(b) In order to find the mass of the cube using the principle of moments, you are provided with the following items.

• metre rule • knife edge • 20 g, 50 g and 100 g weights, pieces of strings

(i) In order to determine the mass of the cube with highest accuracy, which weight would you select out of the weights given above? Give reason for your selection.

weight : reason :

(ii) Firstly, the metre rule has to be placed on the knife edge. How would you find out the position of the metre rule, which has to be placed on the knife edge?

(iii) Draw a labeled diagram of the setup that you would use to find out the mass. Use only the items given above.


- 18 -

(iv) Let mass of the glass cube and the mass of the selected weight be m and M respectively. Mark the length measurements (ℓ1, ℓ2) that have to be obtained on the diagram, drawn above in (b) (iii) and write down an expression relating m, M, ℓ1 and ℓ2.

(v) Using expressions written in (a) (ii) and (b) (iv) above, write down an expression for the density (dg) of glass.

(c) (i) If a beaker of water is provided, the density of glass (dg) could be determined without calculating the volume of the cube. Keeping the distance from the knife edge to the glass cube unchanged, what is the extra measurement (ℓ3) that you have to obtain in order to determine dg.

(ii) Obtain an expression for dg, in terms of ℓ1, ℓ2, ℓ3 and density of water dw

.

waterthin glass tube

mercury

2. A volume of air saturated with water vapour is trapped in a thin uniform glass tube by a mercury column as shown in the figure. You are asked to determine the saturated vapour pressure (P) of water at room temperature using the above apparatus by plotting a suitable graph.

â& How do you know that the trapped air volume is saturated with water vapour?

^b& Let h1 be the length of the mercury column in cm, H be the atmospheric pressure in cm of mercury and P be the saturated vapour pressure of water at room temperature, in cm of mercury.

Consider the following two positions (1 and 2) of the tube.

Position 1

h1 h1

Position 2 Write down expressions for the pressure in mercury cm of air in the two positions. Position 1 : Pressure of air = Position 2 : Pressure of air =


- 19 -

^c&

lh

θ

L

h /

By varying the inclination (θ) of the tube from 0° to 90° the pressure of the air trapped in the tube can be changed.

(i) Write down an expression for the pressure in mercury cm of the trapped air in the position shown above.

Pressure of air =

(ii) Applying Boyle's law to the trapped air write down an expression relating h, H, P and ℓ. (ℓ = length of the air column in cm)

îii& The saturated vapour pressure (P) of water has to be determined by plotting a graph between two appropriate variables. Rearrange the expression given in (c) (ii) above in terms of H, P, ℓ, h / and L.

îv& What quantities should you extract from the graph to determine P?

^v& In order to determine P, H has to be found out. What instrument do you use to measure H?

^d& Is it possible to perform this experiment using a short mercury column? Give reasons for your answer.

3. You are provided with a resonance tube with both end opened, a tall glass jar filled with water, a stand and a 500 Hz tuning fork in order to determine the velocity of sound in air in a laboratory.

â& What other instrument do you need to perform this experiment?


- 20 -

^b& Draw a diagram to illustrate how you would arrange the above apparatus in order to perform the experiment.

^c& (i) If the end correction of the tube is taken into account, at least how many resonance states do you have to take in this experiment? Identify them.

(ii) State clearly how you obtain the relevant resonance states.

^d& What are the measurements you would take in this experiment? (say ℓ1 and ℓ2.)

ê& Taking the end correction of the tube as e, write down expressions in order to find the velocity (ν) of sound in air in terms of the measurements ℓ1, ℓ2 and e.

^f& If the measurements you have taken in (d) above are 16.5 cm and 50.5 cm respectively determine ν and e.

^g& If the length of the tube is 80 cm, would you be able to obtain the next resonance state? Give reasons for your answer.


- 21 -

4. You are asked to use a potentiometer arrangement to measure the internal resistance (r) of a cell. You are provided with the following items.

Potentiometer, 2 V accumulator, resistance box, sliding key, center zero galvanometer, safety resistor with a plug key, a switch

â& Draw a complete circuit diagram for the potentiometer arrangment that you would use in this experiment.

^b& Write down the test that you would perform to check if all the components of the experimental arrangment are properly connected.

^c& By varying the resistance (R) of the resistance box balance length (l) of the potentiometer is measured. Take the balance length to be l0 when R is infinite. Derive the expression that you would use to determine r by plotting a suitable graph.

^d& How do you obtain different data points in order to draw the graph?

ê& It is not advisable to set the resistance of the resistance box to a very low value. What is the reason for this?

^f& Clearly indicating the axes, draw a rough sketch of the graph that you would obtain in this experiment.

0

0


- 22 -

^g& How do you obtain r from the graph?

^h& If the e.m.f. of the cell is slightly greater than 2 V, will you be able to perform this experiment successfully?

Give reasons for your answer.

* *


- 23 -

Part B - Essay(g = 10 N kg−1)

5. Bernoulli’s equation for a fluid flow is given by P + 1 ρ v2 + ρgh = constant 2

P

Av1

B

v D

h2

h1

C

T

where all symbols have their usual meaning. Identify each term in the equation.

A siphon is a device for removing liquids from a container. Figure shows a container P filled with water of density ρ. Tube T (siphon) of uniform cross - section is initially filled with water keeping the end D of the tube closed with the thumb. Then the water is released by removing the thumb.

Assume water to be non - viscous, incompressible and flow of water to be streamlined.

(a) If the speed of flow of water at the lower end (D) of the tube is v, what is the speed of water just inside the entrance (B) of the tube?

(b) If the cross-sectional area of the container is 100 times that of the tube, write down an expression for the speed v1 of the water moving downwards at the free surface of the water in the container in terms of v.

Since v1 << v, take v1 = 0 hereafter.

(c) (i) Considering a streamline starting from A and ending at D, derive an expression for v in terms of an appropriate constant and a parameter given in the diagram.

(ii) If h2 = 45 cm, determine v. (iii) Show that if h2 = 0, siphon action will not work.

(d) (i) Derive an expression for the pressure PC of water at the topmost point (C) of the water column, in terms of ρ, g, h1, h2 and atmospheric pressure P0.

(ii) If a longer tube [the value of (h1 + h2) is large] is used when PC = Pvapour, where Pvapour is the saturated vapour pressure of water at the temperature of water, the siphon action will cease. What is the reason for this?

(iii) If Pvapour = 4.0 kPa at 30°C, determine the maximum possible value that (h1 + h2) must have for the siphon action to work properly ( P0 = 105 Pa, ρ = 103 kg m−3)

Fig (1)

6. Fig (1) shows a properly cut diamond. A diamond sparkles because the intensity of the light coming from is greatly enhanced due to total internal reflection.

As shown in the Fig (2) a monochromatic ray of light is incident on to the top surface, AB, of a diamond held in air, with an angle of incidence i. It then refracts with a refracting angle r and strikes the inclined surface, CD, of the diamond.

Use the following data for your calculations. sin 5° = 0.0870 sin 24° = 0.4000 sin 7.5° = 0.1305 sin 42° = 0.6667 sin 10° = 0.1737 sin 80° = 0.9800 sin 23° = 0.3920


- 24 -

C

A B

Dθ

90° - r

i1r

i

Fig (2)

(a) Determine the critical angle for diamond - air interface. The refractive index of diamond is 2.5.

(b) Using the geometrical diagram shown in Fig (2) find out an expression for the angle of incidence, iʹ, of the ray on the inclined surface CD in terms of θ and r. Here θ is the inclination of the surface CD to the horizontal.

(c) (i) For i = 80°, determine r. (ii) Determine the minimum value of θ (θmin) for this ray of light

to be just totally internally reflected from the surface CD. (iii) Hence show that all light rays incident on the surface AB with i values less than or equal to 80°

will be totally internally reflected from the surface CD. (iv) What would happen if θ < θmin?

(d) (i) For a glass structure with the same geometry with θ = θmin as in Fig (2), determine the maximum angle of incidence, i, that a ray must have so that it will be totally internally reflected from the surface CD. (Refractive index of glass = 1.5)

(ii) Hence, giving reasons deduce that the glass structure does not sparkle in the same way as the diamond.

7. (a) Write down an expression for the excess pressure of a spherical liquid bubble in terms of the surface tension (T) of the liquid and the radius (r) of the bubble.

soap flim

liquid

h

Fig (1)

rubbertube

clip

(b) Vertical U-tube contains a liquid of density ρ. One end of the tube is opened to the atmosphere and a soap flim is made at the other end of the tube as shown in the Fig (1). The pressure inside the limb which contains the soap flim can be varied so that the shape of the flim could be changed.

î& Show that the product of the radius (r) of the flim and the difference (h) of the liquid levels of the limbs of the U-tube is a constant.

îi& If the value of the constant mentioned in (i) above is 1.23 × 10−5 m2, determine the surface tension of soap solution. (Take the density of the liquid in the U-tube to be 800 kg m−3.)

capillary tube

liquid

air

waterh1Fig (2)

(c) Now the soap flim is removed and the respective end of the limb is sealed. Then as shown in the Fig (2) this limb of the U-tube is connected to a vertical capillary tube of internal diameter 0.7 mm which is immersed in water. Now the U-tube acts as a manometer. When air is sent slowly through the open end of the capillary, the difference in the liquid levels of the manometer increased to 9.1 cm initially, then decreased to 4.0 cm and increased to 9.1 cm again.

(density of water = 1000 kg m−3) î& Explain why the difference in the liquid levels of

the manometer vary as described above. îi& Calculate the depth (h1) of the bottom end of the

capillary tube from the water level. îii& Hence, determine the surface tension of water.


- 25 -

8. Read the following passage and answer the questions given below.

A black hole is one of the most interesting and startling predictions of modern gravitational theory, yet the basic idea can be understood on the basis of Newtonian principles. Consider a spherical, nonrotating star with mass M and radius R. The escape velocity (v) for this star at its surface can be written as,

v = 2GM R

.......(1), where G is the universal gravitational constant. In 1783 Rev. John Mitchell, an amateur astronomer noted that if escape velocity (v) would be greater than the speed of light (c) then all light emitted from such a star would be made to return toward it. Mitchell became the first person to suggest the existence of what we now call a black hole. The above expression for v also suggests that a star of mass M will act as a black hole if its radius R is less than or equal to a certain critical radius. An expression for this critical radius could be simply obtained by setting v = c in the above relationship for v. In 1916, Karl Schwarzschild used Einstein’s general theory of relativity to derive an expression for the critical radius Rs, now called the Schwarzschild radius. Eventhough the kinetic energy of light is not 1

2 mc2 and the gravitational potential near a black hole is not simply given by − GM

R , Schwarzschild found the same expression for Rs, as though we had set v = c in equation (1). This happens due to the fact that the two errors mentioned above compensate each other. The surface of the sphere with radius Rs surrounding a black hole is called the ‘event horizon’, since light cannot escape from within that sphere, we can’t see events occurring inside.

Astronomers have found evidence for the existence of a massive black hole at the center of our Milky way galaxy, about 26,000 light years from earth in the direction of the constellation Sagittarius. High resolution images of the galactic center reveal stars moving at high speed about an unseen object. By analyzing these motions, astronomers can infer the period T and the radius r of each star’s orbit. Using these data the mass m of the unseen object can be calculated. This black hole has a mass of 5.4 × 10 36 kg or 2.7 million times the mass of our sun. Research suggest that even larger black holes, in excess of 10 9 solar masses, lie at the centers of other galaxies. Observational and theoretical studies of black holes of all sizes continue to be an important area of research in both physics and astronomy.

(a) (i) What is meant by the term ‘event horizon’? (ii) Why is it defined that way?

(b) (i) Derive the expression for v given in equation (1) (ii) Assuming the density of the material of the star to be constant, show that the escape

velocity is proportional to the radius (R) of the star.

(c) (i) Write down an expression for Rs, in terms of G, M and c. (ii) In obtaining the expression for Rs from Newtonian gravity what are the two compensating

errors that we have used?

(d) (i) A burned out star with a mass equal to three solar masses collapses to form a black hole. Determine the radius of its event horizon. (G = 6.66 × 10 −11 N m2 kg −2, Solar mass = 2.0 × 10 30 kg, c = 3.0 × 10 8 m s−1) (ii) If the radius of the black hole is just equal to the Schwarzschild radius, what is its density? (Take π = 3.) (iii) What would be the radius of the event horizon of the black hole existing at the center of our

galaxy?

(e) Could X - rays or γ- rays escape through an event horizon? Give reasons for your answer.

(f) Derive an expression for the mass (m) of the black hole at the center of our galaxy in terms of a period T of a revolving star around it with a radius r and G.

(g) Calculate the distance from the earth to the black hole at the center of our galaxy in km.


- 26 -

(h) If our sun somehow collapses to form a black hole, what effect would this event have on the orbit of the earth? (The mass of our sun is not large enough to collapse to a black hole.)

9. Answer either part (A) or part (B) only.

R1

R2 V0

R10 V

Fig (1)

(A) (a) Fig (1) shows a circuit consisting of a variable resistor of total resistance R (R = R1 + R2), which can be used to obtain a variable voltage, V0, from a battery of e.m.f 10 V and of negligible internal resistance.

(i) What is the range of voltages that can be expected for V0 from this circuit?

(ii) Draw a rough sketch to show how V0 varies with R2. (b) A student has modified the above circuit as shown below to obtain a smaller voltage range for

V0. The student expects the value of V0 to vary linearly with R2.

R1

V0

R10 V

R2

R3R4

900 Ω

100 Ω

Fig (2)

(i) What is the range of voltages for V0 that can be obtained from the modified circuit?

(ii) What is the advantage of using the circuit shown in Fig (2) over the circuit shown in Fig (1) to obtain smaller voltages?

(iii) If R = 10 kΩ find V0 when the value of R2 is kept at 5 kΩ.

(1) Hence draw a rough sketch to show how V0 varies with R2. (2) Is the variation of V0 with R2 linear? Explain you answer.

(iv) In order to improve the linearity between R2 and V0, a student has decided to use different values for R3 and R4 as 90 kΩ and 10 kΩ respectively.

(1) State the new range for V0 , and calculate V0 when R2 = 5 kΩ. (2) Has the student succeeded in achieving a better linearity? Explain your answer. (3) When using the circuit given under (iv) above, to provide the prescribed voltages to a

load of resistance Ro he recommends that Ro should always be less than 10 kΩ. Would you agree to that? Explain your answer.

4.7 kΩ

I C

RC

VC

I B

I ERE

VE

3.3 kΩ

10 V

4 V

(B) (a) (i) When the applied voltage V across a resistance R is

increased by a small amount ΔV, the current I through the resistance will increase to I + ΔI. Show that, ΔV = ΔIR.

(ii) In the circuit shown, VBE = 0.7 V, and the current gain of the transistor is 100. Assume IE = IC.

Find (1) VE , (2) IE , (3) VC , (4) IB . Show also that the Base-Collector junction of the

transistor is reverse biased under this situation.

(b) Suppose the applied voltage to the base is increased by a small amount ΔVB (i) If the change in VBE is negligible, find the change (ΔVE) in the emitter voltage (VE) due to the

increase of the base voltage. (ii) Hence, write down an expression for the change (ΔIE) in the emitter current. (iii) Write down an expression for the change (ΔVC) in collector voltage. (iv) If this change is an increase, put + sign in front of your expression in (iii), otherwise put a

− sign. (v) Hence write down an expression for the voltage gain of the amplifier. (vi) Use the expression obtained in (v) to deduce a general expression for the voltage gain of

a common emitter amplifier.


- 27 -

10. Answer either part (A) or part (B) only.

(A) As shown in Fig (1) a mixture of air and petrol undergoes as shown in Fig (2) thermodynamic process in one of the cylinders of a petrol engine. The air - petrol mixture is rapidly compressed (e f) as the piston rises. Then the mixture is ignited almost instantaneously causing a great increase in pressure. (f g) Then the piston descends and the mixture rapidly expands (g h). Finally the hot gas is expelled (h e) and the cylinder is ready to start a another cycle.

g

fh

e

P

V Fig (2)

Piston

Fig (1)

(a) Out of thermodynamic processes given below classify each of the processes, (i) e f (ii) f g (iii) g h and (iv) h e

constant pressure, constant volume, isothermal expansion/compression, adiabatic expansion/ compression

1.00 2.0 3.0 4.0

1.0

2.0

3.0

4.0

5.0

6.0

Volume × 10-4 m3

Pres

sure

/ MPa

g

f h

e

Fig (3)

(b) Fig (3) shows the simplified P - V diagram (curves approximated by straight lines) for a cylinder of the engine.

Calculate, (i) the work done by the gas mixture from g to h, (ii) the work done on the gas mixture from e to f, (iii) the net work done (W) by the gas mixture during one cycle efgh.

(c) (i) If Q1 amount of heat is absorbed by the gas mixture from f to g, what is the change in internal energy (ΔUf → g ) of the gas mixture from f to g?

(ii) If Q2 amount of heat is liberated by the gas mixture from h to e, what is the change in internal energy (ΔUh → e ) of the mixture from h to e?


- 28 -

(d) (i) A quantity called the thermal efficiency (e) of the engine is defined by e = WQ1. Write

down an expression for (e) in terms of Q1 and Q2 only. (ii) If Q1 = 3.5 × 103 J, determine (e) as a percentage. (iii) If heat of combustion of petrol is 3.5 × 103 J g−1 what mass of petrol is burned in each

cycle?

(e) If the engine rotates at 50 cycles per second and it has 4 cylinders, calculate the power generated by the engine.

(B) 1. (a) The radius of our sun is 7.0 × 105 km and its surface temperature is 5800 K. Assume that the sun behaves like a black body.

(i) Calculate the total power radiated from the sun. (Stefan constant σ = 6.0 × 10-8 W m-2 K-4; Take 584 = 1.0 × 107)

(b) The temperature of the center of the sun is 2.0 × 107 K. (i) Why is it so essential to have this very high temperature in the sun’s core? (ii) The inner core of the sun emits γ rays. But the surface of the sun radiates mostly IR,

visible and UV light. What happens to this radiation with short wavelengths produced at the core of the sun?

(c) A “blue supergiant” star has a surface temperature of 30,000 K. Assume that the star behaves like a black body.

(i) What is the principal wavelength it radiates? Is this light visible? Use your answer to explain why this star appears to be blue (Wien constant = 3.0 × 10-3 m K)

(ii) If the total power radiated by this star is 105 times that of our sun, determine the radius of this star.

P Q

300 K 6 000 K

R

2. Two large plates (P and Q), one maintained at a temperature of 300 K and the other at a temperature of 6 000 K are placed side by side as shown in the figure. A small plate (R) is placed exactly in between the large plates. What temperature will the small plate acquire on reaching thermal equilibrium?

Assume that the small plate behaves like a black body. (648¼ = 5.045)

* * *


- 29 -

02 - ChemistryStructure of the Question Paper

Paper I - Time : 02 hours 50 multiple choice questions with 5 options. All questions should be answered. Each question carries 02 marks. Total 100 marks.

Paper II - Time : 03 hours This paper consists of three parts as A, B and C. Part A - Four structured essay type questions. All questions should be answered. 100 marks for each question - altogether 400 marks. Part B - Three essay type questions. Two questions should be answered. Each question carries 150 marks - altogether 300 marks. Part C - Three essay type questions. Two questions should be answered. Each question carries 150 marks - altogether 300 marks. Total marks for paper II 1000 ÷ 10 = 100



provided at the examination.)Universal gas constant, R = 8.314 J K−1 mol−1 Avogadro constant, NA = 6.022 × 1023 mol−1

1. Which of the following is the correct set of quantum numbers that deseribe an electon in a 3p orbital?

n_ l_ ml_ ms_

(1) 4 2 -1 0 (2) 3 1 -1 + ½ (3) 2 2 -2 -1 (4) 3 1 2 - ½ (5) 3 1 -2 + ½

2. Which of the following structure best represent the ion S2O32- ?

(1) S:

S::

O::

:O::

:O::

(2) S

S::

:O::

O :O::

::

+

:

(3) S

S::O:

:

: O::

:O::

: + + (4) S

S::O:

:

O::

:O::

: (5) S

S::O:

:

O::

: +:O:

:

:

3. Which of the following statements is not true with regard to the NOCl molecule? (1) The shape of NOCl is angular. (2) The hybridization of the nitrogen atom is sp2. (3) The electron pair spatial arrangment of the central atom in NOCl is angular. (4) In NOCl, a π bond exists between N and O. (5) N - Cl bond is formed by the overlap of a hybrid sp2 orbital of nitrogen and a 3p atomic orbital of

chlorine.


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4. Which of the following statement is true regarding the atoms of the alkali metals in the metallic crystal? (1) In the metallic crystal, they are held together by weak van der Waals forces only. (2) They have atomic numbers which are odd numbers. (3) They do not have electrons in ‘d’ orbitals. (4) They form ions which are strong reducing agents. (5) They form hydrated ions; Li+ being the least hydrated.

5. Which of the following statements is not true regarding the elements from fluorine to iodine in group 17 of the periodic table?

(1) They form diatomic covalent molecules. (2) release energy when gaseous atoms gain electrons. (3) They form hydrides which dissolve in water to form strong acids. (4) chlorine form oxyacids. (5) They form compounds with one another.

6. The first three ionization energies (kJ mol−1) of elements X and Y are as follows.

Element 1st 2nd 3rd

XY

5281095

73402370

118504660

Which of the following could be the identity of X and Y respectively? (1) Be and O (2) Li and Be (3) Na and Mg (4) Li and C (5) B and Ne

7. The name of the following compound according to the IUPAC system of nomenclature is,

C C CH2 CH2 C

=

O

OH

=O

HCH3

=CH CH2

(1) 4-formyl-4-methyl-5-hexenoic acid. (2) 4-methyl-4-formyl-5-hexenoic acid. (3) 4-formyl-4-methyl-5-enehexanoic acid. (4) 4-oxo-4-methyl- 5-hexenoic acid. (5) 4-methyl-4-formyl-5-enehexanoic acid.

8. Consider the hydrogen atoms labelled as p, q, r, s and t in the following compound.

C C CH2 CH2

O

≡H CH2 HC

=

qp r s t Which one of the following pairs represent the two most acidic hydrogen atoms? (1) p and q. (2) p and s. (3) r and s. (4) p and t. (5) s and t.

9. C C CH2 CHO≡CH3

ba c d e

Which is true out of the following statements regarding the above compound? (1) b, c and e carbon atoms are sp hybridized. (2) a, b, c, d and e carbon atoms are in a straight line. (3) The cde angle is about 109.5°. (4) The abc angle is about 120°. (5) Carbon atoms a, b and d are sp3 hybridized.


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10. Some reactions shown by the organic compound A are given below.

A

YZ

X

Br2 / CCl4 Con. H2SO4

HBrNi / H2

CH3 - CH - CH2BrBr

The structures of X, Y and Z respectively are, (1) CH3

CH2 CH2

OH , CH3 CH2

CH2 Br and CH3

CH2 CH3.

(2) CH3 CH2

CH2 OH , CH3

CH CH3

Br and CH3

CH2 CH3.

(3) CH3 CH

CH3

OH , CH3

CH CH3

Br and CH3

CH2 CH3.

(4) CH3 CH

CH3

OH , CH3

CH2 CH2

Br and CH3 CH2

CH3.

(5) CH3 CH

CH3

OH , CH3

CH CH3

Br and CH3

CH = CH2.

11. What are the positions in the following compound that could be expected to react with a hydroxyl ion acting as a base, to form a stable product?

C CCCh

ac

d bBr

H HeHg H

fH

H

H H

H

(1) a and e. (2) a, d and e. (3) b, d and f. (4) e and d. (5) a, c, e and g.

12. A partial structure of a reaction intermediate is given below.

C

C

HC+

The possibility of obtaining a partial structure of this nature exists, (1) only in a nucleophilic substitution reaction. (2) only in an electrophilic addition reaction. (3) only in an aromatic electrophilic substitution reaction. (4) only in nucleophilic substitution and aromatic electrophilic substitution reactions. (5) in nucleophilic substitution, electrophilic addition and aromatic electrophilic substitution reactions.

13. A solution absorbs 90 % of the radiation passing through it. What is the absorbance of the solution? (1) 0.10 (2) 0.90 (3) 1.0 (4) 9.0 (5) none of the above

14. The conductivity of 0.10 mol dm−3 aqueous solution of a strong electrolyte is 1.25 × 104 μS cm−1. What is the most probable value for the conductivity of the solution if it is diluted ten times with water?

(1) 1.05 × 103 μS cm−1 (2) 1.25 × 103 μS cm−1 (3) 1.45 × 103 μS cm−1 (4) 1.25 × 104 μS cm−1 (5) 1.45 × 104 μS cm−1


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15. When equal volumes are mixed which one of the following pairs of solutions gives the highest buffer capacity?

(1) 0.10 mol dm−3 CH3COOH and 0.10 mol dm−3 CH3CO2Na (2) 0.50 mol dm−3 CH3COOH and 0.10 mol dm−3 CH3CO2Na (3) 0.10 mol dm−3 CH3COOH and 0.50 mol dm−3 CH3CO2Na (4) 0.50 mol dm−3 NH4OH and 0.50 mol dm−3 NH4Cl (5) 0.10 mol dm−3 NH4OH and 0.10 mol dm−3 CH3COOH

16. Gibbs energy changes corresponding to the formation of Ag+(aq), Fe2+(aq) and Fe3+(aq) ions at the standard state at 25 °C are given below.

=

Ion ΔG0 / kJ mol-1

Ag+(aq) +77.1

Fe2+(aq) -85.0

Fe3+(aq) -10.7 What is the Gibbs energy change (in kJ) for the overall reaction of the following cell under

standard state and at 25 0C? Pt(s)/Fe3+(aq, 1.00 mol dm-3), Fe2+(aq, 1.00 mol dm-3) Ag+ (aq, 1.00 mol dm-3)/Ag (s)

(1) -151. 4 (2) -18.6 (3) -2.8 (4) 18.6 (5) None of the above.

17. Which of the following statements are correct?

(a) Catalyst is found to be chemically unchanged at the end of a reaction. (b) Catalyst does not change the enthalpy of a reaction. (c) Catalyst does not change the equilibrium position of a reaction. (d) Catalyst makes a reaction to proceed through a path of lower activation energy.

(1) only (a) and (b) (2) only (b) and (c) (3) only (a), (b) and (c) (4) only (a), (b) and (d) (5) all (a), (b), (c) and (d)

18. A solution containing 0.400 g of a diprotic acid requires 40.00 cm3 of 0.100 mol dm-3 NaOH solution for complete neutralization. What is the approximate relative molecular mass of the acid?

(1) 5.00 × 10 (2) 1.00 × 102 (3) 2.00 × 102 (4) 3.00 × 102 (5) 4.00 × 102

19. At 200 0C, the equilibrium constant of the following reaction is 1.6 × 103. 2HBr(g) H2(g) + Br2(g) What is the amount of H2(g) at equilibrium when 1.0 × 10-2 mol of HBr(g) is heated at

200 0C in a closed vessel? (1) 80 × 10-2 mol (2) 1 × 10-2 mol (3) 20 × 10-2 mol 81 81 81 (4) 10 × 10-2 mol (5) 40 × 10-2 mol 81 81


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20. Dolomite (Ca CO3.MgCO3) can be used as a raw material in the manufacture of, (1) Cement (2) Chlorine (3) Detergents

(4) Urea (5) Sodium carbonate

21. An industrial effluent has very high chemical oxygen demand (COD) due to the presence of a detergent, R − COO Na. (Prior to the discharge of the effluent of the public waters the COD value has to be reduced below the limit given in the Sri lanka standard for the discharge of effluent.)

The COD value cannot be reduced by the following process is, (1) Aeration (2) Dilution with a large quantity of pure water (3) Precipitation as (R COO)2 Ca salt. (4) Dissolution of common salt (NaCl) in the effluent (5) Passing the effluent through a column of an adsorbent mateirial

22. Acid rains (hydrogen ions) can react with one of the following substances in the soil leading to an increase of the sodium adsorption ratio (SAR) of the water which flows through the soil system.

SAR = Na+ Concentrations in mmol dm-3

√Ca2+ + Mg2+( )

(1) Dolomite (2) Calcite (3) Feldspars (4) Silica (5) Granite

23. Which one of the following represent the correct order of the solubility of solid metal sulphates in water?

(1) Ba SO4 > SrSO4 > Ca SO4 > MgSO4 (2) Ba SO4 < SrSO4 < Ca SO4 < MgSO4 (3) Ba SO4 ≈ SrSO4 > Ca SO4 ≈ MgSO4 (4) Ba SO4 > SrSO4 > Ca SO4 ≈ MgSO4 (5) Ba SO4 < SrSO4 ≈ Ca SO4 < MgSO4

24. Which of the following statements are not true?(a) Pressure of a real gas sample will always be less than that of an ideal gas sample having the same

amount, temperature and volume.(b) Volume of a real gas will always be less than that for an ideal gas sample having the same amount,

temperature and pressure.(c) Compressibility factor for a real gas depends on pressure.(d) The coherent SI unit of the van der Waals constant b is m3.

(1) Only (a) and (b) (2) Only (b) and (c) (3) Only (a), (b) and (c) (4) Only (a), (b) and (d) (5) Only (a), (c) and (d) 25. Which of the following statement/statements is/are correct?

(a) An endothermic reaction that takes place under constant pressure with a positive entropy change will always be non spontaneous.

(b) Reactions with positive entropy changes will always be spontaneous.(c) Reactions with negative enthalpy change will always occur spontaneously.(d) Exothermic reactions occur under constant pressure and positive entropy change take place

spontaneously. (1) (a) only. (2) (d) only. (3) (a) and (b) only. (4) (c) and (d) only. (5) (a), (b) and (d) only.


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26. Solute X, which was initially present in Phase I, was partitioned between Phase I and Phase II which are immiscible. Which of the following statements are correct regarding amount of X extracted from Phase I to Phase II ?

(a) It depends on the volume of Phase I. (b) It depends on the volume of phase II. (c) It does not depend on the ratio of the volumes of the two phases. (d) It does not depend on the temperature of the experiment. (1) (a) and (b) only. (2) (a) and (d) only. (3) (b) and (c) only. (4) (b) and (d) only. (5) (c) and (d) only.

27. Which one of the following precipitates dissolves on addition of excess ammonia solution? (1) Barium sulphate (2) Iron(III) hydroxide (3) Nickel(II) hydroxide (4) Calcium oxalate (5) Magnesium hydroxide

28. Which one of the following reactions is involved with the production of soap from coconut oil? (1) Condensation (2) Hydrolysis (3) Hydrogenation (4) Oxidation (5) Esterification

29. Which one of the following is not associated with the solvay process? (1) CaCO3 (2) NH4Cl (3) NaCl (4) (NH4)2C2O4 (5) Ca(OH)2

30. The salt NaA was dissolved in 0.10 mol dm-3 solution of hydrochloric acid until the concentration of

NaA became 0.05 mol dm-3. For HA, pKa = 6.0. Which one of the following sketches represent the titration curve when a 25.0 cm3 portion of the above solution was titrated with 0.10 mol dm-3 solution of sodium hydroxide.

(1)

pH

Volume of 0.10 mol dm-3

solution of NaOH / cm3

02468101214

5 10 15 20 25 30 35 40

(2)

pH



02468101214

5 10 15 20 25 30 35 40

(3)

pH



02468101214

5 10 15 20 25 30 35 40

(4)

pH



02468101214

5 10 15 20 25 30 35 40

(5)

pH



02468101214

5 10 15 20 25 30 35 40


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• Instruction for questions No. 31 to 40 : For each of the questions 31 to 40, four statements (a), (b), (c) and (d) are given; out of which, one/

more than one is/are correct. Select the correct statement/statements. Mark, if only (a) and (b) are correct. if only (b) and (c) are correct. if only (c) and (d) are correct. if only (d) and (a) are correct. if any other statement or combination of statements is/are correct.

31. In general, moving across a period from left to right the following trend is/trends are observed. (a) Melting point decreases. (b) Size of atom decreases. (c) Metallic character decreases. (d) Electronegativity decreases.

32. Consider particles of the size in the range from 0.1nm to 1μm. Which of the following statement/s is/are true?

(a) Carbon nano-tubes are used in electronic industry. (b) Fullerines can also the made up of lead. (c) Carbon nano-tubes are used in paint industry. (d) Carbon nano-tubes are used in water purification.

33. Consider the compound, CH3COCH2CH2COOCH3 Which of the following statement/s is/are true? (a) It gives CH3OH when heated with aqueous NaOH (b) It gives an orange coloured precipitate with Brady’s reagent. (c) It gives an alkane with a Grignard reagent. (d) It gives a brick - red coloured precipitate when heated with Fehling’s solution.

34. Consider the conversion CH3CH2CH = CH2 CH3CH = CHCH3

Which of the following represent the correct sequences of steps to be taken to effect the above conversion.

(a) (i) conc. H2SO4 (ii) H2O (iii) conc. H2SO4 / heat (b) (i) conc H2SO4 (ii) 170 0C (iii) H2O (c) (i) HBr (ii) alcoholic KOH (d) (i) HBr / peroxides (ii) alcoholic KOH

35. Consider the following statements regarding reaction rates. Which of the following statement/s is/are true?

(a) The relative rates of removal of different reactants involved in a chemical reaction depend on the stoichiometric coefficients of the reactants.

(b) The rate of an electrochemical reaction depends on the current. (c) The rate of a zeroth order reaction in the gaseous phase can be changed by changing the partial

pressure of the reactant cocerned. (d) Activation energy is determined by the effect of all other factors which affect the rate of

the reaction.


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1.0

0.0

D

C

BA

Pressure/atm

Temperature/K

36. The following statements are based on the diagram given below. Which statements shows the variation of vapour pressure of water with temperature.

(a) Water vapour can be represented by A. (b) No liquid water exists above the temperature correspond to B. (c) Point C represents the boiling point of water. (d) The curve CD represents the liquid - vapour equilibrium.

37. Which of the following statement regarding electrodes is/are true.

(a) At an electrode at equilibrium, a reduction reaction and an oxidation reaction occur at equal rates.

(b) When a potential more positive than the equilibrium potential is applied to an electrode at equilibrium, a net oxidation reaction occurs at that electrode.

(c) When a potential more negative than the equilibrium potential is applied to an electrode at equilibrium a net reduction reaction occurs at that electrode.

(d) When two redox electrodes which are connected by a salt bridge are externally connected using a conducting wire, a reduction reaction occurs at the electrode with a more negative potential.

38. Which of the following statement regarding the root mean square speed of an ideal gas is/are true? (a) It is directly proportional to the thermodynamic temperature. (b) It is inversely proportional to the square root of the relative molecular mass. (c) It is equal to the speed of a molecule having the average molecular kinetic energy. (d) It is independent of the pressure of the system.

39. What is/are the salt/salts that dissolves by the addition of excess solution of dilute HCl? (a) Barium sulphate (b) Iron(III) hydroxide (c) Nickel(II) hydroxide (d) Calcium oxalate

40. What is/are the correct processes/process that both ∆G and ∆H are negative? (a) Dissolution of solid NaCl (b) Burning of LP gas (c) Neutralization of an acid with a base (d) Decomposition of calcium carbonate at high temperature


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• Instruction for questions No. 41 to 50 : In questions No. 41 to 50, two statements are given in respect of each question. From the Table given below, select the response out of the responses (1), (2), (3), (4) and (5) that best

fits the two statements given for each of the questions and mark appropriately on your answer sheet.

Response First statement Second statement(1)(2)(3)(4)(5)

TrueTrueTrueFalseFalse

True, and it explains the first.True, but does not explain the first.FalseTrueFalse

Second statement Second statementThe amount of ammonia gas that dissolves in water at room temperature is much higher than that of dioxygen.

Unlike dioxygen ammonia gas reacts with water to form NH4

+ and some hydrogen bonding occurs between ammonia and water.

Unlike group 1 elements, all group 2 elements react with N2 gas to give nitrides.

Nitrides are formed only elements which can form divalent cations.

The boiling point of liquid HF is higher than that of liquid HCl.

Liquid hydrogen fluoride is polar.

Unlike oxygen, sulphur can form a stable fluoride, SF6.

Sulphur is less electronegative than oxygen.

Some properties of the first element of any group of s and p blocks differ from the rest of the group.

The first element in any group of the s and p blocks is smaller in size and higher in electronegativity than the rest of the elements in the group.

The reaction between ethanoyl chloride and water occurs more easily than the reaction between ethyl chloride and water.

The bond between carbon and chlorine in ethyl chloride is covalent.

CH3CONH2 is a stronger base than CH3NH2 The lone pair on N in CH3CONH2 is less available for donation to a proton compared to that of methylamine.

The concentration of Mg2+ in a 2ppm Mg2+ solution is greater than the concentration of K+ in a 2ppm K+ solution.

The molar charge of Mg2+ ion is higher than that of K+ ion.

Electrodeposition is an example for an energy storing chemical process.

There is a tendency for the temperature of an electroplating bath to increase during electroplating.

The amount composition of dioxygen in the Earth atmosphere is about four times lower than that of dinitrogen.

The mean speed of dioxygen molecules at room temperature exceed the escape velocity for the Earth.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

* * *


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02 - ChemistryPaper II

* Answer all questions in part A.* Answer four questions selecting two questions from part B and two question from part C.

Part A - Structured Essay

1. (a) Methyl isocyanate (CH3NCO) is a toxic gas that was responsible for the deaths of 3000 people when it was accidently released into the atmosphere in December 1984 in Bhopal, India.

The following questions are based on the molecule CH3NCO. The basic skeleton of CH3NCO is given below.

HC

-

-N-H -

H-C-O

(2)(1)

(i) Draw the most acceptable Lewis structure. (ii) Deduce the shapes around C(1) ,C(2) and N atoms using VSEPR Theory.

(iii) Draw resonance structures and identify the most important and least important structure used in describing the stability of resonance hybrid of CH3NCO. Give briefly the reasons for your choice.

(iv) Indicate the hybridizations of C(1) ,C(2) and N.

(2)(1)


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(v) Sketch the shape and indicate σ bonds, π bonds and lone pairs of the structure (i) above.

(vi) Identify the hybrid and atomic orbitals involved in forming the σ C - N and σ C - H bonds in the structure (i) above.

(b) Arrange the following in the increasing order of the property indicated in the parenthesis. State briefly the reasons for your choice.

(i) MgCO3, BaCO3, SrCO3 (decomposition temperature)

(ii) NO2-, NO2

+ (electronegativity of nitrogen atom)

2. An element A, belongs to the s block, reacts with cold water slowly liberating a colourless gas B with the formation of solution C. The element A when allowed to react with N2 gas gives a white solid D. When CO2 gas is bubbled through the solution C, a white precipitate E is formed which redissolved to give a clear solution. When E is heated to 900 °C a white solid F is formed, which when heated with carbon at 2000 °C gives a compound G. The compound G reacts with water to produce a hydrocarbon gas, H of commercial importance. When a flame test is carried with the white solid E, a brick red colouration is obtained.

(The letters A, B, C, D, E, F, G and H represent compounds or elements and not symbols of atoms.) (a) Identify the substances A to H and write their chemical formulae in the boxes given below.

A B C D

E F G H

porous tubegas mixture


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(b) Give balanced equation for the reaction of A with N2, and F with Carbon.

(c) Give one industrial use of each of E, F, G and H

3. (a) Write an expression for the compressibility factor, Z for a gas in the form of an equation. Identify all the terms used in the equation.

(b) Using from the equation you have given in part (a) above derive a relationship for the density of a gas.

(c) The mole fraction of argon in a gas mixture, which contains only argon and hydrogan, is 0.650. Calculate the density of the gas mixture at a pressure of 1.50 bar and a temperature of 3.00 × 102 K. State the assumption you used in this calculation.

Relative atomic masses: Ar = 40, H = 1


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(d) (i) Using the ideal gas equation and the molecular kinetic equation for ideal gases, derive an equation for the root mean square speed (√u 2) of a gaseous particle in terms of its relative molar mass.

(ii) Calculate the root mean square speed of He atoms in ms−1 at 25°C. (Escape velocity = 1.1 × 103 ms−1, He = 4)

(iii) Give one important reason for the very low abundance of He in the earth atmosphere.

4. (a) (i) Name the characteristic class of reaction shown by both aldehydes and ketones.

(ii) Draw the structure of the product obtained when acetaldehyde (ethanal) reacts with hydrogen cyanide.

(iii) State whether it is possible for the compound represented by the structure drawn by you in part (ii) above to exist in optically active forms. Give reasons for your answer.

(iv) Propose a mechanism for the reaction stated in part (ii) above.


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(b) (i) Describe how you would distinguish acetaldehyde from acetone (propanone) using a chemical test.

(ii) Explain the chemical basis of the test you have chosen in part b (i).

(iii) Write the structure and the colour of the product formed by the reaction of acetone with 2,4-dinitrophenylhydrazine.

(c) (i) Name the compound formed when carbon dioxide dissolves in water. (ii) Indicate the reaction taking place in part c (i) above using a balanced chemical equation.

(iii) Using your knowledge of organic reaction mechanisms, propose a mechanism for the reaction given in part c (ii).

* *


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Part B - Essay

5. (a) (i) Sketch a silver - silver chloride electrode and label all important components on it. (ii) Write the equilibrium electrode reaction. (iii) State two factors that determine the electrode potential of this electrode. (iv) Give one exam.0ple of the practical application of the silver- silver chloride electrode. (v) Briefly explain why it is more suitable than the standard hydrogen electrode as a practical

reference electrode.

(b) The compound A decomposes thermaly to compounds B and C in aqueous media according to the equation.

2A(aq) 2B(aq) + C(aq) Data collected during a kinetic study of the above reaction are given below. At a given constant temperature, the initial concentrations of A(aq) in two given solution are

1.0 × 10-2 mol dm-3 and 2.0 × 10-2 mol dm-3. The initial rate of disappearance of A(aq) from each solution were 1.2 × 10-5 mol dm-3 s-1 and 2.4 × 10-5 mol dm- 3 s-1 respectively. Also the concentration of A(aq) remaining in a solution of A(aq) was found to decrease from 0.5 × 10-2 mol dm-3 to 0.25 × 10-2 mol dm-3 during a period of 9.5 min.

(i) Write a mathematical expression for the rate of the above reaction. (ii) Using the above data calculate, (1) the order of the reaction with respect to A(aq) (2) the rate constant. (iii) Determine, (1) the half life of the reaction. (2) the time it takes, for the concentration of A(aq) remaining in the solution to decrease

from 2.4 × 10-3 mol dm-3 to 1.20 × 10-3 mol dm-3. (iv) (1) Calculate the average rate of the reaction. (2) State why the average rate is not be used for accurate determination of rate law for

all reactions. (v) Suppose that, you are provided with the data of remaining concentrations of A(aq)

measured at several selected times for the above reaction. Using not more than three sentences, briefly describe how you would treat the data to determine initial rate and the instantaneous rate at selected instancents of time.

(c) (i) State the Raoult law in the form of a mathematical expression. (ii) A small volume of a concentrated sodium chloride solution previously heated to

100 °C was added to a boiling water bath at the same temperature, while the heating was continued. An immediate cease of boiling was observed but boiling restarted after a short time. Explain this observation using the Raoult law.


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6. (a) The energy released in thermite reaction is used for welding of massive units such as propellers for large ships. The standerd state reaction coresponding to thermite reaction is given below.

2 X(s) + Fe2O3(s) X2O3(s) + 2 Fe(s) Where X is a p-block element which reacts with aqueous hydrochloric acid as well as aqueous

sodium hydroxide with evolution of dihydrogen gas. The standard enthalpy of formation of X2O3(s) and Fe2O3(s) at 25.0 oC are - 1675 kJ mol-1 and - 825 kJ mol-1, respectively.

(i) Identify the element X. (ii) Calculate ΔHo for the thermite reaction at 25.0 oC. (iii) The standard entropy change of the thermite reaction, ΔSo at 25.0 oC is 15.2 J K-1.

Calculate the standard Gibs energy change ΔGo for the thermite reaction at 25.0 oC. (iv) Calculate the amount (in moles) of X necessary to melt 5.20 g of chromium metal at its

melting point if reaction the energy released. (The enthalpy of fusion of Cr(s) is 14.6 kJ mol-1 , Relative atomic mass: Cr = 52.0)

(b) Consider the following gas phase equilibrium that occurs in a closed container at 500 K.

A(g) + B(g) 2C(g)

The symbols A, B and C represent the chemical formulae of the gasses. The initial amounts of each of the gases A(g) and B(g) were 2.0 mol and the C(g) was not present at the beginning.

(i) If the equilibrium constant, Kp of the above reaction is 4.0 at 500 K, calculate the percentage of initial A(g) converted at equilibrium.

(ii) If the percentage of A(g) converted is 40% at another temperature, calculate the equilibrium constant, Kp for the above reaction at that temperature.

7. (a) A hydrated sulphate of a 3d transition metal M gives a violet solution, A, when dissolved in water. Dropwise addition of concentrated aqueous ammonia to the violet solution A, gives a pale green precipitate, C, which slowly dissolves in excess concentrated ammonia to give a purple solution, D. The pale green precipitate (C) dissolves in aqueous NaOH to give a green solution E. Addition of H2O2 to the green solution and heating produces a yellow solution F. Treatment of solution F, with dilute H2SO4 gives an orange solution G. The solution F reacts with SO2 under basic conditions to give either the pale green precipitate C or green solution E depending on the basic strength of solution.

(i) Identify the transition metal M consistent with the above observations and give its electronic configuration and common oxidation state (s).

(ii) Identify the species present in A, B, C, D, E, F and G.

(iii) Give the balanced ionic equations for the formation of F and G and the reaction of F with SO2 under basic condition.

(iv) Give one use of the species present in solution G and the transition metal M identified in (i).

(v) Give the IUPAC name of E.


- 45 -

(b) (i) What do you mean by electromagnetic radiation? (ii) Sun burn occurs on exposure to sunlight of wavelength in the vicinity of 325 nm. Calculate the following coresponding to radiation with wavelength of 325 nm. (C = 3 × 108 ms−1, h = 6.63 × 10−34 Js) (1) Frequency of the radiation (2) Energy of a photon of the radiation (3) Energy of one mole of photon (4) Number of photons correspond to 1.0 × 10-3 J of energy. (iii) In what region of the electromagnetic spectrum would this radiation be found?

Part C - Essay

8. (a) The atmosphere contains 20% oxygen and 0.03% carbon dioxide. With the assumption that the atmosphere is a closed system and the relative molecular masses of all constituent gases are the same, calculate

(i) the dissolved oxygen content (in mg dm−3) of rain water, (ii) the approximate pH of rain water. For oxygen, KH = 1.3 × 10−8 mol dm−3 Pa−1. For carbon dioxide, KH = 3.8 × 10−7 mol dm-3 Pa-1. For carbonic acid, pKa1

= 6.37, pKa2 = 10.33

Relative atomic masses : C = 12, O = 16

(b)

+A

Ca(HCO3)2(l) more

E

F DE

H Bleachingpowder(s)

B CO2(g)

K

C2H2(g) Phosphatefertilizer(s)

C

IJ

L

1. F2. G

M

N

Consider the manufacture of phosphate fertilizer, bleaching powder and C2H2 as shown in the flow chart.

(i) Give the name of the starting material A in the triangle which is an abundant natural resource.

(ii) Write in the boxes and circles, the chemical formulae with physical states of the appropriate substances involved.

(iii) Give the conditions (M) and (N) used in the process.


- 46 -

9. (a) A salt Na2A was dissolved in water until the concentration of Na2A becomes 0.05 mol dm−3.

(i) A 25.00 cm3 portion of the above solution was titrated with 0.10 mol dm-3 sloution of HCl. Calculate the volume of HCl required and sketch the tritation curve. Lable the curve repactively pKa1

and pKa2 of H2A at the relevant temperature are 1.0 and 7.1

(ii) Derive the equation(s) and calculate the amount of HCl required to be added to 1.0 dm3 of the above salt solution to convert it to a buffer solution of pH 7.4.

(b) The following is a part of the chain of a commercial polymer. It is a product of condensation polymerisation. Draw the structural formulae of the monomers from which the above polymer is made.

CH2 N CH

-

O

=

H

-

N CH2C

O=[ ]

(c) (i) It is well known that a chlorine compound plays an important role in the process of depletion of ozone layer in the atmosphere. Explain the above statement using relevant chemical reactions involved in the process.

(ii) Today we are concerned about acid rain. State why we are not concerned about basic rain.

(iii) State five main stages in the electroplating industries that contribute to the environment pollution by metallic ions.

10. (a) Compound A (C4H10O), on heating with concentrated H2SO4, forms the compound B (C4H8). B reacts with water in the presence of a mineral acid to form the compound C. The compound C, is an isomer of A. C, on heating with concentrated H2SO4, forms B easily. C does not react with acidic K2Cr2O7 under mild condition.

(i) Identify A" B and C giving the reactions involved.

(ii) Give a chemical test with observations to distinguish between A and C.

(iii) Give the structures of D and E in the following reaction.

A PCC/CH2Cl2 D ; A

K2Cr2O7 / H+

E

* PCC = pyridinium cholorochromate

(iv) Give the structure of an isomer of A and C that can exhibit optical isomerism.

(b) Using your knowledge of the reactions of alkenes, alcohols and alkyl halides, indicate a method by which the following conversion can be carried out.

CH3 CH = CH2 CH3 CH O CH2 CH2 CH3

CH3

* * *


- 47 -

07 - Mathematics

Structure of the question paper

Paper I - Time : 03 hours. This paper consists of two parts. Part A - Ten questions. All questions should be answered. 25 marks

for each question - altogether 250 marks. Part B - Seven questions. Five questions should be answered. Each

question carries 150 marks - altogether 750 marks. Total marks for paper I 1000 fi 10 = 100

Paper II - Time : 03 hours. This paper consists of two parts. Part A - Ten questions. All questions should be answered. 25 marks


question carries 150 marks - altogether 750 marks. Total marks for paper II 1000 fi 10 = 100

Calculation of the final mark : Paper I = 100 Paper II = 100

Final marks = 200 fi 2 = 100

Important :

* In Mathematics I and II question papers 30 prototype questions for part A and 21 prototype questions for part B have been included. However, the question paper that will be given at the examination will be constructed according to the question paper structure given above.

* At the examination all questions in part A of both Mathematics I and II will have to be answered using the space provided for each question.

Mathematics I: Part A - 18 11 2010

Structure of the Question Papers and Prototype Questions for G.C.E.(Advanced Level) Examination – 2011 onwards

- 48 -

07 - Mathematics

Prototype Questions for Paper I

Part A

1. Let }10,9,8,7,6,5,4,3,2,1{=U be a universal set.Let }10,7,4,1{=A , }5,4,3,2,1{=B and }6,4,2{=C be three subsets of U.

Determine the following, in the usual notation:

CCBCA ofset Power )iii(,)ii(,)i( ′−∪′ .

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2. A survey revealed that in a certain town, 300 people read the newspaper ‘Lakderana’, 200people read the newspaper ‘Lakpahana’ and 50 people read both the newspapers.Find how many people of those who were surveyed read(i) at least one,(ii) exactly oneof these two newspapers.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 49 -

3. Let }:),{( nmnmR <= be a relation defined on Z.Determine whether this relation is(i) reflexive, (ii) symmetric, (iii) transitive.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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4. Consider the functions ,3)( 2xxf −= x ∈ IR and ,11)(x

xg+

= x ∈ IR and x ≠ −1.

Find g º f (x) and its range.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 50 -

5. Let 12)( 23 +−+≡ xkxxxP , where k ∈ IR . When )(xP is divided by )( kx − , the remainderis k. Find all possible values of k.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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6. Let the roots of the quadratic equation 02 =++ cbxx , where 0≠c , be α and β. Find the

quadratic equation whose roots are β

βα

α1 and 1

++ .

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- 51 -

7. Find the set of values of k such that the expression )43()(4 −−+ kkkxx is positive for all x.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. Find the partial fractions of )2)(2(

32

2

+−

−+

xxxx .

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- 52 -

9. Show that 38log7log6log5log4log3log 765432 =××××× .

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10. A student has 6 different books on Pure Mathematics and 4 different books on AppliedMathematics. Find the number of different ways in which the books can be arranged on ashelf such that(i) all the books on Pure Mathematics are together and all the books on Applied

Mathematics are together,(ii) all the books on Pure Mathematics are together.

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- 53 -

11. Find the number of different permutations of the letters of the word ERROR taking all at atime.Find the number of different selections of three letters which can be made from the lettersof the word ERROR.

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12. Write down the general term of the binomial expansion of 1432

+x

x .

Show that there are exactly two consecutive terms with equal coefficients.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 54 -

13. Using the Principle of Mathematical Induction, prove that

3

)12)(12()12(531 2222 +−=−++++

nnnnL , for any positive integer n.

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14. Using the standard formulae for 1∑=

n

rr and

1

2∑=

n

rr , prove that

)2)(1(31)1(

1++=+∑

=

nnnrrn

r.

Hence, find ∑=

−n

rrr

1)1( .

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- 55 -

15. Using the properties of determinants, show that

)cos)(sinsin1)(cos1(1sincos1sincos111

22θθθθ

θθ

θθ −−−= .

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16. Express the system of equations02 =+ ykx142 =+ yx

in matrix form, where k is a real constant.Hence, find the values of k for which the system has a unique solution.

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- 56 -

17. Let

−=

−=

1123

and2312

BA .

Find, in the usual notation, TTTT ABABBA and ,, , and verify that ( ) TTT ABAB = .

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18. Let

=

1032

A .

Show that OIAA =+− 232 , where I is the identity matrix of order 2 and O is the zeromatrix of order 2.Hence, find 1−A .

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- 57 -

19. The coordinates of the points A, B and C are )4,1( and )1,1(),3,2(− respectively. Find theequation of the perpendicular from the point B to the line AC and the coordinates of thefoot of that perpendicular.

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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

20. Find the equation of the line which passes through the point (1, 1) and the point ofintersection of the lines 023 and 072 =−+=+− yxyx .

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- 58 -

21. Find the coordinates of the points at which the bisectors of the angles between the lines022 and 072 =+−=+− yxyx intersect the x-axis.

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22. Obtain the equation of the circle having the straight line segment joining the points)1,6( and )3,4( −−−− as a diameter.

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- 59 -

23. Show that the line 5=x is a tangent to the circle 0118422 =+−−+ yxyx .Find the coordinates of the point of contact.

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24. Show that the point )0,6(−A lies outside the circle 0442422 =−+−+ yxyx and find thelength of a tangent from the point A to the circle.

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- 60 -

25. Find .cos1cotlim

0

+→ x

xxx

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26. The rate of increase of the surface area )(rA of a spherical balloon is 1.6 m 2 /min, where r

is the radius of the balloon in metres. Find the rate of increase of the radius when π2

=r .

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- 61 -

27. If ,3cos2 tey t−= show that 013dd4

dd2

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ty

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28. Show that the function xxexf −=)( has a maximum when 1=x .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 62 -

29. Find the equation of the tangent to the curve 842 +−= xey x at the point where the curvemeets the y-axis.

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30. Show that the function 1463)( 345 +−+= xxxxf has a stationary point at x = 0 anddetermine its nature.

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01 12 2010


- 63 -

Part B

1. (a) Let CBA and , be three subsets of a given set S.Stating clearly the laws of algebra in set theory used, prove, in the usual notation, that(i) ( ) ABAA =∩′− ,(ii) ( ) )()( CBACABA −∩=∩−∩ .

(b) A group of 96 students are expected to sit for History at the forthcoming G. C. E.(Advanced Level) Examination. Of these 96 students, 43 attend extra classes, 38 aregirls, and 18 are repeat students. Only 11 repeat students attend extra classes. 10 repeatstudents are girls. 7 girls attend extra classes. Among the students who attend extraclasses 6 girls have to repeat the examination.Find how many boys(i) attend extra classes,(ii) repeat the examination,(iii) will sit the examination for the first time without attending extra classes.

2. (a) Let },3,2,1{ L=Ζ+

The relation R is defined on +Ζ as ‘ yxR if and only if yx 2≥ ’.Determine whether the relation R is(i) reflexive,(ii) symmetric,(iii) transitive.

(b) The relation R is defined on Ζ as ‘ aRb if and only if (a + b ) is even’.Show that R is an equivalence relation onΖ .Write down the equivalence classes [1] and [2].Find ]2[]1[ ∩ .

(c) A functions f is defined as ,23

25:−

→x

xf x ∈ ]9,1[ .

(i) Show that f is a decreasing function and find its range.(ii) Show that f is one-one.(iii) Show that 1−f exists and find it explicitly.

3. (a) When the monic (leading coefficient is 1) cubic polynomial p(x) is divided by )1( −xand )3( −x the remainders are 7 and 13 respectively. Find the remainder, when p(x) isdivided by )3)(1( −− xx .Further, if 6)2( =p , find the quotient when )(xp is divided by )3)(1( −− xx .Hence, obtain )(xp .

01 12 2010


- 64 -

(b) Let f : IR → IR be the function defined by 12)( += xxf .Determine whether )(xf is(i) one – one, (ii) onto.

Further, let g : IR →− }0{ IR be the function defined byx

xg 1)( = .

Find explicitly, each of the composite functions gxf and fxg, indicating the domain andthe range.

4. (a) Let kxkxxf ++++≡ )1()1()( 2 .Find all real values of k such that the roots of the equation 0)( =xf are real.When 2−=k , find the axis of symmetry and the least value of )(xf .Hence, draw the graph of )(xfy = when 2−=k .

(b) Let 72)( 2 −+≡ axxxQ and bxQxxP +−≡ )()12()( , where a and b are real constants.If )(xQ is an even function and the remainder when )(xP is divided by x is 12, find thevalues of a and b.Express )(xP as a product of linear factors of x.

5. (a) Let βα and be the roots of the quadratic equation 02 =++ cbxax , where 0≠c . Find

the quadratic equation whose roots are α

αcb + and

ββ

cb + .

(b) Write down the binomial expansion of nx)2( + , where n is a positive integer.(i) If there are two consecutive terms with equal coefficients in the above expansion,

show that (n + 1) is a multiple of 3.Find the value of this coefficient when n = 8.

(ii) By choosing two appropriate values for x, show that, when n is even22213 1

33

11 −

−− +++=− nnnnnnn CCC L .

6. (a) Using the properties of determinants, show that

2

333

))()()(( cbaaccbbacbabcaacb

cbacba

++−−−−=

−+−+−+

.

(b) Express the system of equations58 =+− zyx22 =+− zyx

742 =+− yxin matrix form.Using Part (a) above and Cramer’s rule, solve the system of equations.

01 12 2010


- 65 -

7. (a) Find the number of different permutations that can be made using(i) all the letters,(ii) only four lettersof the word ‘EXAMINATION’.

(b) 15 persons are recruited to a certain institution.Find the number of ways that these 15 recruits can be assigned, five each, into threedifferent sections of the institution.

8. (a) Let

−=

3112

A and IAAA 75)( 2 +−=f , where I is the identity matrix of order 2.

Show that OA =)(f , where O is the zero matrix of order 2.Hence, find A-1.Verify your answer by an alternative method.

(b) Let

−

−

−

=

312321111

A and

−

−

−

=

135259123

B .

Compute BA .

Hence, solve the system of equations 2=+− zyx

232 =+− zyx232 =−+ zyx .

9. Let

−=

1111

P . Find PPT and TPP , where TP is the transpose of P.

Deduce 1−P , the inverse of P.Using 1−P find 1)( −TP . Any result used should be clearly stated.

Find a diagonal matrix B of order 2 such that

=

1111TPBP .

Let

=

3113

A . Show that 122 −+= PBPIA .

Hence, show that APP 1− is a diagonal matrix.

01 12 2010


- 66 -

10. (a) Using the Principle of Mathematical Induction, show that2

33332)1(321

+

=++++nnnL for any positive integer n .

Hence, find(i) 3333 )2(321 n++++ L ,

(ii) 3333 )12(531 −++++ nL .

(b) Find the partial fractions of )2)(1(

23++

+

xxxx .

Hence, find ∑=

n

rru

1, where ru is the rth term of the series

L+++5.4.3

114.3.28

3.2.15 .

Is the above series convergent? Justify your answer.

11. (a) Prove that, ab

bc

ca log

loglog = , where a, b and c are positive and not equal to 1.

Show that, 1logloglog =acb cba .Hence, obtain the value of 125log7log32log 4945 .

(b) Obtain the largest coefficient in the binomial expansion of 12)34( x+ .

12. A group of 10 students consisting of 5 boys and 5 girls have to sit around a table. Find inhow many different ways the sitting arrangements can be made if(i) all the boys sit together,(ii) no two girls sit together,(iii) a particular boy and a particular girl sit together,(iv) a particular boy and a particular girl do not sit together and no two girls sit together.(v) a particular girl refuses to sit while the other four girls agree to sit with the five boys

on the condition that no two girls sit together.

13. (i) Prove that, for any positive integer n, 2)1(321 +

=++++nnnL .

(ii) Show that 233 242)12()12( rrr +≡−−+ for any integer r.

Hence, prove that )12)(1(61

2 ++=∑=

nnnrn

r.

Using (i) and (ii) above, find the sum of the first n terms of the seriesL+++ 10.77.44.1 .

Is the above series convergent? Justify your answer.

01 12 2010


- 67 -

14. Let n be a positive integer. Write down the binomial expansion of nx)1( + .Let A be the sum of the terms of the odd powers of x and B be the sum of the terms of theeven powers of x in the binomial expansion of nx)1( + .Prove that(i) nxAB )1( 222 −=− ,

(ii) nn xxAB 22 )1()1(4 −−+= .

(a) Obtain the binomial expansions of nn )12( + and nn )12( − , where n is an integergreater than 1.Deduce that, nnnn nnn 2)12()12( +−>+ .

(b) Obtain the binomial expansion of 55 )32()32( xx ++− in ascending powers of x.

Hence, find the value of 55 )03.2()97.1( + , correct to four decimal places.

15. (a) A house buyer takes on a fixed-interest mortgage of 40000 Rupees on 1st January1990. It is to be repaid in 20 equal annual instalments, the first of which is due at theend of 1990. The interest rate is fixed at 10% per annum and the interest iscompounded annually. Find, to the nearest Rupee, the value of each instalment.

(b) A microbiologist models the population of a colony of microbes by the equation

baPtP

−=dd , where P is the number of microbes at time t days and a and b are

positive constants.

(i) Show that )(1 bea

P at += is a solution of the above equation.

(ii) Draw the graph of P given in (i) against t.(iii) The microbiologist makes two observations of the colony population and finds

that P = 1, when t = 1 and P = 3, when t = 2.Show that 0232 =−− bee aa .

16. Obtain the equations of the bisectors of the angles between the two straight lines given by0111 =++ cybxa and 0222 =++ cybxa .

Two sides of a square lie on the bisectors of the angles between the lines 034 =++ yxand 034 =−+ yx . One of the vertices of the square is (0, 2). Show that there are twosuch squares and find the equations of the sides of the two squares.

01 12 2010


- 68 -

17. (a) Obtain the conditions under which(i) two circles intersect each other orthogonally,(ii) a circle bisects the circumference of another circle.

(b) A circle 0=S passes through the points of intersection of the circle012222

1 =−+++≡ yxyxS and the line 02 =+≡ yxl .

(i) If the circle 0=S and the circle 0910222 =−++≡ yyxS intersect each other

orthogonally, find the circle 0=S .(ii) If the circle 0=S bisects the circumference of the circle

0722223 =−+++≡ yxyxS , find the circle 0=S .

18. Find the coordinates of the point A at which the two circles 042221 =+−+≡ yxyxS

and 02010222 =+−+≡ xyxS touch each other externally.

If P is a point such that the length of a tangent from P to 01 =S is half of the length of thetangent from P to 02 =S , show that the locus of the point P is a circle through the point A.Suppose that a circle 03 =S intersects the circle 02 =S at the points A and B(4, 2).(i) Show that the locus of the centre of the circle 03 =S is a straight line.(ii) Find the coordinates of the centre C of the circle 03 =S when CA is perpendicular

to CB.

19. (a) Evaluate )cos(tan1

tanlim2

0 xx

x −→.

(b) From first principles, find the derivative of x3cos .

(c) Given that 0sin 2 =−+ xyexy , show that yxyx

xyxyyxy

sincos)2(sin

dd

3 −+

−+= .

20. (a) Differentiate each of the following with respect to x:

(i) 2

22 1)13(x

xx +− , (ii) )lncos(22 xxe x− , (iii) xxsin

(b) Let θθ 33 sin,cos ayax == , where a is a constant.

Find xydd and 2

2

dd xy in terms of θ.

Hence or otherwise show that

−= yxyx

xy

xyxy

dd

dd

dd3 2

2.

01 12 2010


- 69 -

21 (a) From first principles, find the derivative of x .

(b) Using the substitution θtan=x , show that the equation

0)1(

1dd

12

dd

2222

2=

++

++ y

xxy

xx

xy can be reduced to the equation 0

dd2

2=+

θyy .

(c) A curve is given by the parametric equations ttx+

−=11 2

and 212tty

+= , where t is a

parameter. Find the Cartesian equations of the tangent and the normal drawn to thecurve at the point corresponding to t =1.

Mathematics II: Part A


- 70 -

07 - Mathematics

Prototype Questions for Paper IIPart A

1. Find all the values of x for which 042 ≥

−xx .

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2. Find the set of all values of x satisfying 8152>

+

xx .

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- 71 -

3. Find the range of x which satisfies xx 212 −≤− .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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4. Shade the region R in the xy-plane, containing the points which satisfy the inequalities 0≥+ yx , 4≤− xy and 1622 ≤+ yx .

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- 72 -

5. Solve the following linear programming problem:Maximize: yxz 23 +=Subject to 02 ≥− xy ,

,5≤x 1≥y .

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6. The table below gives the information in respect of the mix, prices and the standardquantities in the preparation of a large bag of cement:

PriceMix Year 1 Year 2 Standard quantity

A 1.50 3.00 8B 3.40 4.25 3C 10.40 8.84 1

Find the weighted average of the relatives.

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- 73 -

7. The information on the prices and the quantities sold in 2008 and 2009 by a departmentstore are given in the following table:

2008 2009Item Price Quantity sold Price Quantity soldVideo recorder 425 40 450 18

TV set 400 25 500 45

Find the price and the quantity relatives for 2009 (2008 = 100) for both items and interpretyour results.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. Draw an appropriate network for the following activity list indicating clearly, the dummyactivity required:

Activity Immediate predecessor activitiesA, B

CDE

-B

A, BC, D

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- 74 -

9. The following gives a part of a network:

Find the earliest event time and the latest event time for each event of the network.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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10. If 4π

=+ BA show that, 2)tan1)(tan1( =++ BA .

Hence, show that 128

tan −=π .

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3

1

2

4

A

2 C1

D

6

E

3B

4



- 75 -

11. In the usual notation, if )sin(2

tan CBA+= for a triangle ABC, show that ABC is a right

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12. Obtain the general solution of the equation xxx 3sin5sinsin =+ .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 76 -

13. Find ∫ +−x

xxd)3)(12(

7 .

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14. By a suitable substitution, evaluate ∫+

3

02

3d1x

x

x .

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- 77 -

15. Using integration by parts, evaluate ∫2

1

2 dln xxx .

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16. The mean and the variance of five observations 6, 4, 8, a and b are 6 and 2 respectively.Find the values of a and b.

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- 78 -

17. Find the mode of the frequency distribution given below in respect of the times taken by 55students to solve three mathematics problems:

Time (in minutes) 05 – 14 15 – 24 25 – 34 35 – 44 45 – 54Frequency 5 7 19 17 7

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18. Two random samples of 30 and 20 bags of milk powder are taken from a productionprocess. A summary of the results obtained is given in the table below:

Sample Size Mean weight (kg) Standard deviation (kg)1 30 12.10 0.902 20 11.70 1.10

Find, in kilograms, the mean and the standard deviation of the 50 bags.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 79 -

19. Events A and B are such that 4.0)( =AP and 25.0)( =BP . If A and B are independent,find the probability that(i) both A and B occur,(ii) only one event occurs.

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20. Two events A and B are such that 41)( =AP ,

32)|( =ABP and

21)|( =BAP .

Find )( BAP ∪ .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 80 -

21. Two events A and B are such that 74)( =AP ,

31)( =′∩ BAP and

145)|( =BAP .

Find )( BAP ∩ and )|( ABP .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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22. A firm is working independently on two separate projects. There is a probability 0.3 thateach of the projects will be finished on time. Find the probability that at least one of theprojects will finish on time.

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- 81 -

23. The probability mass function of a discrete random variable X is given by2][ cxxXP == , for x = 0, 1, 2, 3 and 4.

Find (i) the value of c,(ii) ]35.0[ ≤< XP .

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24. A bag contains 2 white balls and 1 black ball. A second bag contains 1 white ball and 2black balls. A ball is randomly chosen from each bag and is then placed in the other bag.Find the expected number of white balls in the first bag.

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- 82 -

25. A continuous random variable X has probability density function

≤<−

≤≤

=

otherwise ,032 ),32(20 ,

)( xxkxk

xf

Find (i) the value of k, (ii) ]5.21[ ≤≤ XP .

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26. A continuous random variable X has probability density function

≤≤−

=

otherwise ,0

60 ,)6(108

1

)(

2 xxx

xf

Find the variance of X .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 83 -

27. All the letters in a particular office are typed by Nimal who is a trainee typist. Theprobability that any letter typed by Nimal will contain one or more errors is 0.3. Find theprobability that a random sample of 4 such letters will include one or two letters free fromerrors.

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28. Customers arrive randomly at a super market at an average rate 3.4 per minute. Assumingthat the number of arrivals follows a Poisson distribution, find the probability that one ormore customers arrive in any 30 seconds.

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- 84 -

29. In the usual notation, ),(~ 2aaNX , where a is positive. Find ]0[ <XP and

> aXP23 .

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30. The one-step transition probability matrix of a Markov chain is

5.05.07.03.0

. If the initial

probability vector is ( )8.02.0 , find the probability vector after two steps.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 85 -

Part B

1. (a) Draw the graphs of 42 −+−= xxy and 32 −= xy in the same diagram.

Hence, find the values of x for which 3242 −>−+− xxx .

(b) Let 234

2

2

−−

+−=

xxxxE , where x is real.

Find the values of x for which(i) E is defined,(ii) E > 0.

2. In a certain factory two types A and B of electrical items are produced. Both types makeuse of two essential components, a motor and a transformer. Each item of type A requires 3motors and one transformer. Each item of type B requires 2 motors and 3 transformers. Dueto the restriction in budget allocation the total supply of components per month is limited to100 motors and 200 transformers. Due to the lack of human resources the production oftype A items per month is restricted to 50 items. By selling an item of type A the factoryearns a profit of 3000 Rupees and that for a type B item is 2000 Rupees.(i) Formulate the above situation as a linear programming problem in order to maximize

the monthly profit of the factory.(ii) Using the graphical method, find the number of items that should be produced per

month from each of the two types in order to gain the maximum profit.In this case, find the amount of this profit.

(iii) If the profit gained from any item of type A is reduced to 2000 Rupees, then find theoptimal solution.

3. A company produces three brands A, B and C of soft drinks, using two bottling plantslocated in town X and town Y. The capacity of each plant in terms of bottles per day isgiven in the following table:

Brand A Brand B Brand CPlant at X 3000 1000 2000Plant at Y 1000 1000 6000

The costs of operating the two plants at town X and town Y per day are 500000 Rupees and400000 Rupees respectively. The minimum demands in bottles per week for the threebrands A, B and C are 6000, 4000 and 12000 respectively. The management of thecompany needs to determine the number of days per week each plant should be operated soas to minimize the total cost of production while meeting all the demands. Assuming thatyou are one of the members of the management committee, formulate the above situation asa linear programming problem.Using the graphical method(i) indicate the feasible region,(ii) solve the problem for the optimal solution.


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4. (a) A survey of household expenditure in a certain country showed the following changesover the same week in each of three years for an average family:

Price (per unit)Item Quantity Purchased Year 1 Year 2 Year 3Sugar 2 kg 20 22 57Bread 5 loaves 21 23 32Tea 1 kg 18 21 24Milk 10 litres 13 13 22

Butter 0.5 kg 21 32 41

(i) Using year 1 as base, calculate the index numbers for year 2 and year 3.(ii) State giving reasons, whether the survey based on these items represents a

reasonable assessment of changes in the cost of food over the three year period.

(b) The quantities of potatoes harvested during the period from 2001 to 2008, in 1000tonnes are given in the table below:

Year 2001 2002 2003 2004 2005 2006 2007 2008Index

2003=100 97 102 100 112 128 121 145 149

(i) Convert the above information to show a set of chain base relatives.(ii) Given that the amount of potatoes harvested in 2004 was 587000 tonnes, calculate

the amount of potatoes harvested (to the nearest 1000 tonnes) each year from 2006to 2008.

5. A project consists of eight activities. The activity completion times and the immediateprecedence relationships are as follows:

Activity Completion Time(Weeks)

Immediate predecessoractivities

ABCDEFGH

57634265

---A

B,CCD

E,F

(i) Draw the appropriate network diagram.(ii) Calculate earliest start and finish times, latest start and finish times, and float times.(iii) Identify which activities are critical.(iv) If activity E is delayed by 3 weeks, how is the project completion time affected?(v) If activity F is delayed by 3 weeks, how is the project completion time affected?


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6. The following diagram shows a CPA–Network modelling the stages in the manufacturingprocess of a certain product:

Each arc is labelled with a letter denoting the activity it represents and a weight thatdenotes the number of days needed to complete that activity.(i) State what is meant by an event in a CPA–Network.

Define the earliest event time and the latest event time for an event.(ii) Explain briefly why a dummy activity must sometimes be included in a CPA-Network.

Illustrate your explanation by referring to suitable activities in the network shownabove.

(iii) Calculate the earliest event time and the latest event time for each event of thenetwork. State the shortest completion time of the manufacturing process and find thecritical path.

7. (a) Let

−

+=12

cot12

tan ππ xxy .

Prove that, xyy 2sin)1(21 −=+ .

Hence, show that for all values of x ,

−

+12

cot12

tan ππ xx cannot take values

between 31 and 3.

(b) Prove the identity θθθ 2sin431cossin 266 −≡+ .

(c) Solve the equation 0cos2cossin 848 =+− xxx .

StartFinish

1

2

3

4

5

6

7

8 9B

2

G8

H2

F3

L6

M2

A3

J8

D4

K

4

C4

E3 I1


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8. (a) Find all the values of θ in the range [0, π ] satisfying θθ 2cos3cos = .By expressing θθ 2cos3cos − in terms of θcos and factorizing or otherwise, show that

52cos π and

54cos π are the roots of the quadratic equation 0124 2 =−+ xx .

Hence, find the value of 52cos π .

(b) Express xx 2cos2sin3 + in the form )2sin( α+xR , where R and α are real.

Draw the graph of xxy 2cos2sin3 += for 1211

127 ππ

≤≤− x .

Draw also, the graph of xxy 2cos2sin3 +−= for 36ππ

≤≤− x .

9. (a) In the usual notation, state the Sine rule for a triangle ABC.In a triangle ABC, the median through A is of length m and makes angles θ and φ withAB and AC respectively.Prove that,(i) )sin(sin)sin(sin2 CBam −=− φθ ,

(ii) 2

sin)(2

sin2 Acbm −=

−φθ .

(b) Using the Cosine rule, prove that, in the usual notation, ))(())((

2tan2

cbaacbcbacbaA

++−+

−++−= .

10. (a) Using partial fractions, find ∫ ++x

xxd)13)(13(

82 .

(b) Using the method of integration by parts, find ∫ − xxe x d2cos .

(c) Calculate the area of the region bounded by xyxx sin,2

,0 ===π and xy cos= .

11. (a) By using the substitution, 2xu −= evaluate ∫ −1

0

3 d2xex x .

(b) Find ∫−

xx

x d42

.

Hence, evaluate ∫−

−8

52

2d

4

)4ln( xx

xx .

(c) By using the substitution θ2cos=x , evaluate ∫ −+2

1

0

d11 xxx .


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12. (a) State the Trapezoidal Rule in the usual notation.The values, correct to two decimal places, of a function )(xf for the indicated values of x,are given in the following table:

x – 1.0 – 0.5 0.0 0.5 1.0 1.5 2.0f(x) 7.00 5.00 3.50 4.00 5.50 6.00 6.50

Using the information given in the above table and the Trapezoidal Rule, obtain anapproximate value for the area below the curve )(xfy = and above the x-axis, betweenx = – 1.0 and x = 2.0.

(b) State Simpson’s Rule in the usual notation.

Using h = 0.2 and Simpson’s Rule, obtain an approximate value for ∫ +

1

03 d

10x

xx ,

correct to four decimal places.

13. Some information on the number of asthma sufferers whose first attacks came at variousages is given in the following table:

Age at first attack 0 – 10 10 – 20 20 - 30 30 – 40 40 – 50No. of cases 14 ? 27 ? 15

Data corresponding to the age groups (10 – 20) and (30 – 40) is missing from the table.However, it is found that the median and the mode of the distribution are 25 and 24respectively.(i) Calculate the missing frequencies.(ii) Obtain the mean and the standard deviation of the distribution.(iii) Find the coefficient of variability and the coefficient of skewness and identify the

shape of the distribution.

14. Define the mean and the variance of a set of n observations.The mean and the variance of a set of n observations },,{ 21 nxxx L were calculated. Itwas found later that the value x1 is incorrect and that it should be replaced by 1x′ .Show that the adjustment to be done to the variance to rectify this error is

+−′−+′−′

nTxxxxxx

n2)(1 11

1111 , where T is the sum of the original data

set },,{ 21 nxxx L .

The following gives the information regarding the employees in two factories A and Bbelonging to the same industry:

Factory A Factory BNumber of employees 50 100Average wage per employee per month(in Rupees) 12000 8500

Variance of the wages per month(in Rupees2) 9 16


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(i) Which factory A or B pays out a larger amount as monthly wages? Justify youranswer.

(ii) Find in which factory there is greater variability in individual wages.(iii) In factory B the wage of an employee is recorded incorrectly as 12000 Rupees instead

of 10000 Rupees.Find the corrected variance of factory B.Does this correction affect your answer in (ii)? Justify your answer.

15. Define the harmonic mean of n observations.(a) A train runs first 40 km at an average speed 48 km.p.h., second 80 km at an average

speed 64 km.p.h., then due to the maintenance of the track, travels for 15 minutes at anaverage speed 20 km.p.h. and finally covers the remaining distance at an average speed25 km.p.h. If the average speed of the train for the entire journey is 45 km.p.h., find thetotal distance that the train runs.

(b) The distribution in the following table gives the personal wealth of a certain crosssection of the population of a country for a particular year:

Personal wealth Number of persons(000,000)

Total personalwealth (000m)

0 - 20002000 - 50005000 - 1000010000 - 1500015000 - 2000020000 - 2500025000 - 5000050000 and over

1926744116851

2.47.855.549.225.716.815.06.3

Draw a Lorenz curve to illustrate the data and use it to estimate(i) the percentage of total personal wealth that the least wealthy 30% of persons have

at their disposal,(ii) the percentage of most wealthy who command one half of all personal wealth.

16. (a) Let A and B be two events such that 4.0)|(,6.0)|( =′= ABPBAP and5.0)|( =′BAP .

(i) Find )(),( BPAP and )|( ABP .(ii) Are A and B independent? Justify your answer.

(b) A production line is known to produce 8% of defective items, one quarter of which arerejected. If three items are selected at random from the production line, find theprobability that(i) the first item is rejected,(ii) the second item is rejected,(iii) none of the items are rejected,(iv) at least one of them is defective.


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17. Let N(t) be the number of customers who have visited a shop during t hours since the shopopens at 9.00 a.m. Assume that N(t) has a Poisson distribution with mean λt .(i) Find the probability that 20 customers have visited the shop by the time it closes at

5.00 p.m.(ii) Let T be the time that the first customer of the day visits the shop. Find )( tTP > and

hence, show that T has an exponential distribution.(iii) Till 10.00 a.m., no customers have visited the shop. Find the probability of the first

customer of the day arriving between 10.00 a.m. and 11.00 a.m.

18. The records of a company show that the weekly distance travelled by any salesman isnormally distributed with mean 800 km and standard deviation 90 km. The sales managerconsiders that salesmen who travel less than 600 km in one week perform poorly.(i) Find the probability that a randomly selected salesman has travelled more than 710 km

but less than 935 km.(ii) If the company employs 200 salesmen, how many would be expected to perform

poorly in a particular week?(iii) The sales manager wishes to estimate the number of kilometres travelled in a week,

above which only 1% of the salesmen are expected to exceed. Evaluate his estimate.

19. Three light bulbs each of whose lifetime T is a random variable with probability density

function 0,10001)( 1000 ≥=

−tetf

t

are fitted in a room.

(a) Find ][ tTP ≤ and hence, compute ][ tTP > .

(b) Assuming that all three bulbs are used at the same time, find the probability that(i) all three bulbs are,(ii) only two bulbs are,(iii) only one bulb isstill working after 1300 hours.(You may assume that exp(-1.3) = 0.273, correct to three decimal places.)

Deduce the probability that the room will be in the dark after 1300 hours.

20. A housewife is in the habit of buying one of the two brands A and B of tooth paste eachmonth. If she buys brand A in a month, then she buys brand B in the following month withprobability 0.1. If she buys brand B in a month, then she buys brand A in the followingmonth with probability 0.05. Let Xn be the brand of the tooth paste she buys in the nth

month.(i) Show that {Xn : n = 0, 1, 2, . . .} represents a Markov chain and find its one-step

transition probability matrix.(ii) If the housewife buys brand A in January, find the probability that she buys the same

brand in March.(iii) If the housewife has probability 0.4 of buying brand A in January, find the probability

that she buys the same brand in May. (iv) Find the conditional probability that she has bought brand A in January given that she

buys brand A in May.(v) In the long run, how often does she buy each of the two brands?


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21. A lorry driver delivers goods to Kandy, Galle and Haputale. He goes to one city each dayand never goes to the same city on successive days. If he goes to Kandy on a given day,then the next day he is equally likely to go to Galle or Haputale. If he goes to Galle on agiven day, then the next day he is twice as likely to go to Kandy as he is to go to Haputale.If he goes to Haputale on a given day, then the next day he is three times as likely to go toGalle as he is to go to Kandy.(i) Write down the one step transition probability matrix for this situation.(ii) Given that on a particular Monday he goes to Kandy, find the probability that he goes

to Galle on Wednesday.(iii) Show that over a long period of time the visits to the Kandy, Galle and Haputale are

in the ratio 18 : 21 : 16.


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08 - Agricultural ScienceStructure of the Question Paper


Paper II - Time : 03 hours This paper consists of two parts as Structured Essay type and Essay type. Part A - Four structured essay type questions. All questions should be answered. 100 marks for each question altogether 400 marks Part B - Six essay type questions. Four questions should be answered. Each question carries 150 marks altogether 600 marks Total marks for paper II = 1000 ÷ 10 = 100

Calculation of the final mark = Paper I = 100 Paper II = 100 Final mark = 200 ÷ 2 = 100


provided at the examination.)

• Use the following table to answer questions 1 - 3.

Type of rainfall Time of receiving the rainfallA. South-East monsoon P. September to DecemberB. North-East monsoon Q. January to MarchC. Inter-monsoon - I R. May to SeptemberD. Inter-monsoon - II S. June, JulyE. Cyclone T. January, February

1. Among the above the correct relationship between the type of rainfall and time of receiving the rainfall is (1) AP. (2) AR. (3) BQ. (4) CS. (5) DP. 2. Shifting of inter-tropical convergence zone to north from Sri Lanka is taken place due to (1) A. (2) B. (3) C. (4) D. (5) E. 3. Dry zone of Sri Lanka mainly receives rains from (1) A. (2) B. (3) C. (4) D. (5) E.

4. It was decided that following factors should be considered installing a rain-gauge in the weather station located in the school garden

A - The upper edge of the rain-gauge should be 30 cm above the ground level B - The floor should be uniform and level C - The distance between the rain-gauge and the trees or buildings of the surrounding should

be more than four times of the height of those obstacles Of above, the most important factor/s to be considered in installing a rain-gauge would be (1) A only (2) A and B only (3) A and C only (4) B and C only (5) All A, B and C

5. A student who tested the texture of the soil in the field conclude that it was a sandy soil. The most abundant mineral of the parent material which the soil was formed is

(1) Dolomite (2) Quartz (3) Mica (4) Apatite (5) Feldspar


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6. Following is a flow chart prepared in a study of the soil microorganisms

Dead bodies A Ammonia B Nitrate C Nitrogen

gas Above A, B and C processes are (1) nitrification, nitrification and denitrification respectively (2) denitrification, nitrification and ammonification respectively (3) ammonification, denitrification and nitrification respectively (4) denitrification, ammonification and nitrification respectively (5) ammonification, nitrification and denitrification respectively

7. Following are some statements regarding a particular soil A - pH is above 7 B - Large amount of the cation exchange complex (more than 15]) is Na+

C - This soil is found in arid and semi-arid areas The above soil is (1) acidic (2) neutral (3) saline (4) alkaline (5) basal

8. A group of students observed the following soil characteristics in the school garden, before and after the primary land preparation

A - Texture B - Structure C - True density

D - Bulk density E - Random roughness F - Porosity of above, the soil characteristics that the changes could be expected after primary land preparation are

(1) A,B,C, and D (2) A,C,D, and E (3) B,C,D, and E (4) B,D,E and F (5) C,D,E and F

9. Following are some statements regarding the nutrient availability in the soil A - Though essential nurtients are available in an acidic soil, plants could show nutrient

deficiencies B - Plants growing on a high acidic soil are unable to absorb iron C - Nutrient retention ability is high in a soil having organic matter Of above, the correct statements would be (1) A only (2) B only (3) A and B only (4) A and C only (5) B and C only

10. Various problems arise due to the over application of chemical fertilizer in paddy cultivation. Following are some affects of over application of urea A - eutrophication of water bodies B - becomes acidic soils C - high susceptibility of plants to pests Of above, the correct statements would be (1) A only (2) B only (3) A and B only (4) A and C only (5) B and C only

11. Amount of triple supper phosphate (45%P2O5) needed to prepare 1000 kg, of fertilizer mixture of 15:10:0 is approximately

(1) 450 kg (2) 333 kg (3) 222 kg (4) 45 kg (5) 22 kg

12. An agriculture instructor in the dry zone, instructed the farmers in his area to cary out zero tillage in their agricultural lands. The correct procedure of zero tillage is

(1) seedbed preparation after ploughing (2) drilling the soil and planting the seeds (3) levelling the soil and seedbed preparation (4) digging holes and planting seeds (5) preparation of ridges and furrows


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13. A farmer who intented to cultivate his upland which has not been cultivated for many years, used a disc plough to prepare the land. The main objective of using a disc plough is to

(1) crush the soil aggregates (2) break the hardpan in the soil (3) control the weeds (4) prepare the ridges and furrows (5) mix organic matter

14. The soil factors necessary to know in determination of irrigation requrement for a field crop cultivation would be

(1) moisture percentage, field capacity and bulk density (2) moisture pecentage, field capacity and the depth of the root zone (3) moisture percentage, structure and bulk density (4) texture, structure and bulk density (5) texture, structure and porosity

15. A farmer established a sprinkler irrigation system to supply water to his vegetable cultivation. He used a water pump in the system. The main objective of using a water pump is to

(1) lift water to surface of the earth (2) increase the potential energy of the water (3) increase the kinetic energy of the water (4) increase the volume of the water (5) increase the weight of the water

16. It was observed that the crop plants grown on a land were wilted; their leaves turned to pale green and dropped. When those plants were uprooted, it was observed that the lateral roots were dead and the depth of the root zone was short. The main reason for this situation is

(1) nitrogen deficiency (2) potassium deficiency (3) soil acidity (4) weak drainage (5) iron toxicity 17. Following are some statement regarding plant breeding A - New varieties produced through breeding are always fit to natural environment B - Potential yield of the crops can only be improved through breeding C - Mutation can create genetic variances Of above, the correct statements would be (1) A only (2) B only (3) A and B only (4) A and C only (5) B and C only

18. New plants have been produced by incorporating the gene having insecticidal property taken from Bacillus thuringiensis in to plants via a vector. This bio technological method is called

(1) Mutation breeding (2) Hybridization (3) Somaclonal variation (4) Gene recombination (4) Recombinant DNA technology

19. The parameter used to measure the nutrient availability of the media in hydroponics is (1) pH. (2) CEB. (3) EC. (4) BOD. (5) FC.

20. Following are some statements regarding soiless cultivation in solid media A - Metam Sodium can be used to disinfect the coir dust B - Drainage can be improved by mixing coir dust with paddy husk charcoal C - When nursery plants rearing, the nutrient media is directly applied to the media in the

nursery pan Of above the corrcet statements would be, (1) A only (2) B only (3) A and B only (4) A and C only (5) B and C only


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21. The diagram shows a simple propagatory structure prepared for rooting of a cutting. The techniques used in here to promote rooting are

(1) reducing of leaf area and increasing the humidity (2) increasing the humidity and temperature (3) reducing the leaf area and increasing the temperature (4) wetting the soil and reducing the humidity (5) reducing the temperature and humidity

22. The stomata are closed when the plant is subjected to a water stress. This is happen due to the reducing of the turgidity in guard cels. A hormone and a mineral responsible for this are respectively

(1) Auxin and Ca++ (2) Auxin and K+ (3) ABA and Ca++ (4) ABA and K+ (5) Gibberellin and Ca++

• Use following information to answer questions 23 - 24. Students who have observed a snakegourd cultivation, have recorded following information. Observations A - Polythene covered snakegourd B - Veins remained in the damaged leaves looked like a net 23. Observations A is used to prevent (1) Aulacophora damage (2) Fruit fly damage (3) Disease vector damage (4) Epilachna damage (5) Biting insect damage

24. The damage relevant to B observation is done by (1) Piercing and sucking insect (2) Rasping and sucking insect (3) Biting or Chewing insect (4) Leaf rolling insect (5) Disease vector insect

25. During the night, lighting of fire torches prepared by Kakuna oil and during the daytime placing a stage of yellow and red flowers while spreading the milk rice in the field were done by a farmer. By this activity, farmer expects to

(1) provide light to the paddy field during the night (2) control the insect pest damages in the field (3) chuck of rats coming to paddy field during the night (4) decorate the field (5) chuck the pests coming to the paddy field

• For questions 26 and 27, select the correct responses from (A), (B), (C) and (D) and then select the correct number.

1. If only responses A, B, D are correct, mark on 1 2. If only responses A, C, D are correct, mark on 2 3. If only responses B, C, D are correct, mark on 3 4. If only responses A, B, C are correct, mark on 4 5. If only responses C and D are correct, mark on 5 Mark in the answer sheet according to the instructions given

26. The characteristics of the order Homoptera (A) Fore wings are thicken uniformly (B) Rear wings are membranous (C) Shows complete metamorphosis (D) Has piercing and sucking mouth parts

27. Correct statements regarding pyrethrum are (A) The persistence is high (B) Extract from the chrysanthemum plant (C) Organic insecticide with plant origin (D) Effective insecticide but not much toxic to mammals


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• Use the information given in following table to answer questions 28 -30

A B CBatadallaAtawaraIlluk

PartheniumAlligator plant Salvinia

GirapalaBalunaguta Kadupahara

28. Weeds belong to group A are (1) Grasses (2) Broad leaves (3) Sedges (4) Grasses and sedges (5) Grasses and broad leaves 29. Of above, the group/groups of invasive allien weeds is/are (1) A only (2) B only (3) C only (4) A and B only (5) B and C only

30. Of above, group/groups of annual weeds is/are (1) A only (2) B only (3) C only (4) A and B only (5) A and C only

31. If a farmer expects to harvest his BG 300 rice crop during the period from 15th February to 15th March in the Maha season of the dry zone, the appropriate time for land preparation and sowing would be between

(1) 10th to 30th November and at the beginning of December respectively (2) 25th October to 15th November and at the middle of November respectively (3) 10th to 30th October and at the beginning of November respectively (4) between 20th to 30th September and at the middle of October respectively (5) between 10th to 30th December and at the end of January respectively

32. Following are some statement regarding the formation of empty seeds in rice A - Due to stem borer attack B - Due to drying stigma C - Due to the deposition of pollen of another variety on the stigma D - Application of nitrogen and potassium fertilizer during the flowering Of above, the correct statements are (1) A and B only (2) C and D only (3) A, B and C only (4) A, B and D only (5) A, C and D only

33. Nutrient content in 100g of some foods are given in the following table

Type of food Protein (gram) Fat (gram) Carbohydrate (gram)ABC

-196.8

81140.5

--

78.2

Examples for above A,B, C foods are (1) cheese, cowmilk and dhal respectively (2) butter, beef and raw-rice respectively (3) pork, chick pea and raw-rice respectively (4) curd, dhal and soyabean respectively (5) coconut, tuna fish, and bread respectively

34. Food group not subjected to enzymatic browning would be (1) apple, mango, and pears (2) butterfruit, papaya and ripe babana (3) ash plantain, brinjal and okra (4) grapes, guava and orange (5) potato, pineapple and beans


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35. When green vegetables are prepared as a food, it is common to chop into small pieces, mix with scraped coconut and make a malluma. The nutritional importance of this preparation is

(1) Adding coconut detoxify the toxic substances found in greens (2) Nutrients in greens dissolve in coconut milk and facilitate the absoption in to the body (3) Allows the absoption of fat soluble vitamins in greens into body (4) Chopping into small pieces facilitate the water soluble vitamins to come out (5) Reduce the cholesterol in coconut and transform into favourable state to the body

36. Following are some factors affecting the consumer demand on a particular good. A - Consumer requirement B - Cost of the inputs C - Purchasing power Of above, the correct would be (1) A only (2) C only (3) A and B only (4) A and C only (5) B and C only

37. By fixing a maximum price for a certain good, the government basically expects (1) to increase the demand for the good (2) to protect the producer (3) to protect the consumer (4) to buy the good by the govenrment (5) to provide a good profit to the producer for their goods

38. A farmer bought a land to caltivate chilli and the bought chemical fertilizer to apply the crop. The type of above cost are

(1) Fixed cost and total cost respectively (2) Variable cost and fixed cost respectively (3) Fixed cost and variable cost respectively (4) Variable cost and total cost respectively (5) Total cost and variable cost respectively

39. Students observed a herd of cattle commonly found in dry zone having a long tail, well developed dewlap and hump. They also observed that while some were grazing other were lying on the floor. These observations can be identified as

(1) Behavioral changes of animals for the temperature (2) Physiologocal changes of animals for the temperature (3) Morphological changes of animal for the temperature (4) Behavioral and pysiological changes of animals for the temperature (5) Behavioral and morphological changes of animal for the temperature

40. A farmer has a well managed cattle farm. He wants to improve the productivity of his indigenous milking cattle in a long run. The most appropriate method is

(1) Selection of animals (2) Inbreeding of animals (3) Provision of good nutrition to animals (4) Crossing with improved breeds (5) Improving with animal health


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41. Parts of the complex stomach of cattle were dipped in formalin and kept in seperate vessels in the school laboratory. But the lables of these vessels were displaced Photographs of the parts of the stomach found in those vessels are shown below.

A B C D Based on the characteristics‚ of above photographs, the labels relevent to each vessel would be (1) Reticulum, omasum, ruman and abomasum respectively (2) Abomasum, reticulum, ruman, and omasum respectively (3) Omasum, reticulum, ruman, and abomasum respectively (4) Abomasum, omasum, ruman and reticulum respectively (5) Ruman, reticulum, omasum, and abomasum respectively

42. In a poultry breeding farm having pure bred birds, poultry shed X has white colour, single combed broiler chicken while the poultry shed Y has white colour layer chicken Poultry breeds in sheds X and Y would be

(1) Sussex and leghorn respectively (2) Sussex and Bhrahma respectively (3) R.I.R and leghorn respectively (4) R.I.R and Newhamsphire respectively (5) Sussex and Cochin respectively

43. A farmer purchased two groups of day old chicks, one group for egg production and the other for broiler production. If he wants to buy poultry feeds for first three weeks, the feeds he should buy would be

(1) chicks ration and growers ration (2) growers ration and layers ration (3) broiler finishing ration and growers ration (4) chicks ration and broiler starter ration (5) growers ration and finishing ration

44. In an awareness programme for commercial poultry farmer, livestock development officer stated certain characteristics of the hens

A - Activeness B - Strong legs with warn nails C - Small hard (Rough) abdomen D - Short beak and vigor E - Dry small cloaca Of above, the characteristics of a good layer hen would be (1) A,B, and C only (2) A,B, and D only (3) A,C, and D only (4) B,C, and D only (5) B,D, and E only

45. Following are the certain points noted by a veterinary surgeon after visiting a livestock farm A - Udder of the certain cows were swollen and the milk of those cows had clots B - Certain cows were limping and saliva was coming out from their mouth C - Droppings with blood were observed in the poultry shed and remaing feed were found in the

containers According to above A,B,C symptoms it is suspected that farm animals were having (1) Mastitis, Haemoregic septecemia and ranicut respectively. (2) Milk fever, foot and mouth diseaces and coccidiosis respectively. (3) Mastities, foot infection and ranicut respectively. (4) Mastities, foot and mouth disease and coccidiosis respectively. (5) Milk fever, anthrax and bird flue respectively.


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• For question from 46 - 50 two statements for each question were given. Considering the pair of statements given in each question, select the most suitable answer for each question from the five answers given below and mark according to the instructions given on the answer script.

Response First statement Second statement(1)(2)(3)(4)(5)

TrueTrueTrueFalseFalse

True and explains the first statement correctly True and does not explain the first statement correctlyFalseTrueFalse

46'

47'

48'

49'

50'

First statement Second statementMaize is a monocotyledon seed. Maize seed has a single cotyledon and it is called

scutellumSeed dormancy is defined as the ability of a seed to germinate when factors needed for germination are provided

Seed viability can be secured by reducing the CO2 concentration in the stores

Self sterility avoids the self pollination in plants Self sterility means weakening of the growth of pollen tube after pollination

Rooting of cuttings are promoted in the simple propagaters

High humidity and temperature promote the rooting in cuttings

No variations occur in plants obtained from tissue culture

Genetic variations could be obtained from somaclonal variations

* * *


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08 - Agricultural SciencePaper II



1. Digestive tract of the cattle is given in the diagram below. Use this diagram to answer following ques-tions. (Select and write English letters relevant to the answers of questions (i) - (iii) of section A)

YX

W

P

QR

S

T U

V

(A) (i) State two places where food is subjected to mechanical digestion. a. b.

(ii) What is the place, where food is subjected to microbial digestion.

(iii) What is/are the main place/places, that food is subjected to enzymatic digestion.

(iv) Name the part of the digestion tract of pig resemble to the task of the part "S" of this digestive tract.

(v) Name the main method of nutrient absorption in the place W.

(B) Following are two digestive processes take place in the digestive tract of cattle.

a. Starch Place - 1

Enzyme - 1 Maltose Place - 2

Enzyme - 2

Glucose

b. Crude fiber Place - 3

Volatile fatty acids + gases

(i) Name places 1 and 2 of process a. Place - 1 Place - 2

(ii) Name two enzymes contributing to process a. Enzyme - 1 Enzyme - 2


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(iii) Name the place - 3 of digestive process b.

(iv) What is the main group of microbes responsible for process b ?

(v) Name the place of digestive tract of pig that the process b takes place.

(C) The types of food given to farm animals and examples for each type are given in the following table.

Type of food ExampleP Concentrated food W Hay, silageQ Wet roughphage X Hay, straw R Dry roughphage Y Coconut poonac, rice bran, molassesS Processed roughphage Z Desmidium, Brachiaria, Setaria

(i) Select the correct example for each of food types P, Q, R, S from W, X, Y, Z in the above table.

(a) P (b) Q (c) R (d) S

(ii) State two main differences between concentrated food and roughphage. a. b.

(D) Various agro-chemicals used in agriculture and their uses are given in the following table.

X YX1 Copper sando 4 Y1 Selective weedicide used in paddy cultivation.X2 Carbofuran Y2 Total weedicide used to control all weeds in a field X3 MCPA Y3 A safe insecticide extracted from plantsX4 ROUND UP Y4 Neuro toxic insecticideX5 Pyrethroid Y5 Fungicidused to control leaf spot disease

Match X and Y based on above information. (i) X1 (ii) X2 (iii) X3 (iv) X4 (v) X5


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2' (A) A student conducted a soil analysis and found following quantites of different ions in an oven dried 100g of soil.

Mg2+ = 4.5 milli equivalents Ca2+ = 6.0 milli equivalents K + = 3.6 milli equivalents H+ = 1.5 milli equivalents SO4

2- = 0.8 milli equivalents

(i) Using above data, calculate the cation exchange capacity of the soil sample.

(ii) State two importances of the cation exchange capacity in soil. a. b.

(iii) Calculate the base saturation of above soil sample.

(iv) What is the importance of calculating the base saturation of soil?

(v) Write two compounds that can be applied to soil to reduce the acidity of the soil. a. b.

(B) A famer digging a well in his land observed that the soil is hard when it is dry, muddy and sticky when it is wet and there is a quarts layer when the soil is dig to a one meter depth.

(i) Name the soil group in this land?

(ii) What could be the district that farmer is digging the well?

(iii) Name two crops suitable to cultivate in this farm. a. b.

(iv) Write two main problems that could arise when cultivating this land. a. b.


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(C) Following facts were found when a barren land is inspected for the suitability for crop cultivation. 1. The land has a mild slope 2. Top soil is eroded in most of the places 3. The soil is severely degraded 4. Poorly grown weeds are abandon

(i) What is the meaning of "the soil is severely degraded"?

(ii) Write four methods that could be followed for conservation of this soil. a. b. c. d.

(iii) Write three types of soil erosion that could occur in this type of lands. a. b. c. (iv) Name a suitable land preparation method for cultivating this land.

(v) If vegetables are cultivated on this land, name a suitable irrigation method.

(vi) Name two suitable weed control methods for this land. a. b.

3' (A) (i) Supply function for rice is given as QS = 4P - 50. Fill the following supply list using this supply function.

Price of the 1kg of rice Ammount supplied(kg)15 '''''''''''''''''''''

25 '''''''''''''''''''''

35 '''''''''''''''''''''

45 '''''''''''''''''''''

(ii) Plot the supply curve of this supply list.


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(iii) If Government provides fertilizer subsidy to paddy cultivation, state the impact of it to this supply curve by mean of a graph.

(iv) State two main reasons affecting the expansion and contraction of supply along the supply curve in section (ii).

a. b.

(B) (i) State the main difference that could be observed between following cost groups.

1. Fixed cost and variable cost

2. Avarage cost and marginal cost

(ii) The total cost of a broiler chicken farm is given below. C = 15,000 + 250 y Where, C is the total cost (Rs) and y is the number of production units (kg). 1. How much is the fixed cost?

2. How much is the marginal cost?

(iii) Define opportunity cost.

(C) (i) Malnutrition is a major nutritional problem in Sri Lanka. State the two main types of malnutrition.

a. b.

(ii) State the main disease condition occur due to iodine deficiency.


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(iii) What is the procedure followed by the government to avoid above disease condition?

(iv) Recent surveys disclosed that the diabetic conditions are prevalence among school children in Sri Lanka. State two main factors causing to this situation.

a. b.

(D) (i) A farmer, who intends to cultivate pineapple as an intercrop under his 10 year old coconut cultivation, met the agricultural instructor of the region for necessary advices.

a. The instructor did not approve the farmer's intention. What could be the main reason for it?

b. Name other two crops which could be grown as intercrops under coconut. 1'

2'

(ii) State two main advantages of Gliricidia (Gliricidia maculata) which is commonly used in street cropping to the soil.

a. b.

(iii) In addition to the advantages to soil, Gliricidia is recently used for another main alternate use. What is it?

(iv) Name two commonly used cropping systems in Sri Lanka. a. b.

4' (A) Vegetables and cereals are commonly propagated by seeds. Capsicum, tomato, finger millet, snake gourd, ridged gourd, okra, rice and maize are some examples.

(i) Classify above mentioned crop into monocots and dicots. a. Monocots b. Dicots

(ii) Of above, what are the crops having hypogeal germination?

(iii) Tomato seeds show dormancy due to the growth inhibitors. How to remove those?


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(iv) State two other factors causing seed dormancy and write relevant examples from above mentioned crops.

a. b.

(v) Write four advantages of seed propagation of plants. a. b.

c. d.

(B) (i) Complete the following table.

Crop Disease Causal agentPotatoRicePapaya

''''''''''''''''''''''''''''''''''''''''''''''

Leaf blightRoot knot disease

Phytophthera infestans'''''''''''''''''''''''''''''''''''''''''''''''

'''''''''''''''''''''''''''''''''''''''''''''''

a.b.c.

(ii) Fill in the relevant blanks with the nutrient mainly responsible for the following deficiency symptoms.

a. Stunted plants with purple colour leaves b. Complete yellowing of young leaves c. Burning of margins of mature leaves and burnt spots on the leaves d. Inter-veinal chlorosis e. Shortening of internodes at the apex and formation of leaf rosette

(C) A student who examined the snake gourd cultivation in the school garden, observed the following insect damage.

(i) Name the main insect making above damage.

(ii) What is the order of this insect?

(iii) State two characteristic features of above insect order. a. b.

(iv) State a method to control above insect.

(v) a. In addition to above insect damage name another main insect damage that could be seen in the snake gourd crop.

b. Name a non-chemical method to control the insect damage you named.

* *


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Part B - Essay

5. (i) (a) What are the main problems that could happen in agriculture due to climatic change? (b) What are the actions that could be followed to avoid such problems in order to succeed the

agricultural activities? (ii) (a) A farmer carried out a soil test before cultivating his land and results were shown to the

agricultural instructor. The instructor advised him to apply lime (CaCO3) to his soil. What are the reasons for this advice?

(b) What are the advantages that the farmer could obtain by following these advices?

(iii) Large milking cows having a hump and an undulated dewlap were found in a dry zone livestock farm.

(a) What is the type of breed of these milking cows? (b) Describe the responses and adaptations of these milking cows to unfavorable climatic

conditions?

6. (i) An agricultural teacher divided her class into three groups and instructed to carry out following tests.

Group one - Determination of moisture percentage in the soil Group two - Determination of field capacity of the soil Group three - Determination of permanent wilting point of the soil (a) Name the equipment needed by group - one to carry out their test. (b) State the readings to be taken by group - two to find out the field capacity of the soil. (c) Explain how the group - three calculate the permanent wilting point.

(ii) Following are some information taken from a report prepared by students who visited a home garden.

X - Plants such as Monarakudumbiya, Brinjal, Atthadi, Kuppamania, Atawara, Chille, Tomato, Ridge-gourd were found in the field.

Y - Labourers were involved in activities such as applying straw mulch to turmeric cultivation, hand weeding of Kuppamania, Sparaying a liquid on atawara.

(a) Classify the weeds found in this home garden based on the information mentioned in X. (b) What are the adaptations of these weeds for their survival? (c) Based on the information in Y, classify the weed control methods. Write the advantages and

disadvantages of each method.

7. (i) A farmer expects to commence a broiler chicken farm with an idea to send the production to the market during the New Year period in April.

(a) At what time he should bring the day old chicks to the farm? (b) Describe the main factors to be considered in rearing broiler chicken.

(ii) Describe the challenges of developing the agriculture production in present Sri Lanka.

8. (i) (a) Describe the most suitable rearing method for a farmer who expects to start a medium scale dairy farm in two hactare land.

(a) Sketch a floor plan for a cattle shed for 20 milking cows rearing according to above described method.

(ii) Describe the ways of improving the efficiency in agricultural marketing process.


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9. (i) (a) Define "agricultural product marketing". (b) Agricultural product marketing process is complicated, comparing to the marketing

processes of other products and services. Explain the reasons for this with examples. (ii) State the factors to be considered in construction of dairy cattle sheds.

10. A dry zone farmer intends to cultivate chillie in his two acre land for dry chillie production. (i) Explain the procedure that should be followed from land preparation to crop establishment in

order to obtain a healthy chillie crop. (ii) Explain how to use the concept of integrated plant nutrient system (IPNS) to maintain favourable

soil nutrient level for a productive crop. (iii) Describe the procedure that should be followed by the farmer from the time of harvesting to

minimize the post harvesting losses.

* * *


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09 - BiologyStructure of the Question Paper


Paper II - Time : 03 hours This paper consists of two parts as Structured Essay type and Essay type. Part A - Four structured essay type questions. All questions should be answered. 100 marks for each question - altogether 400 marks Part B - Six essay type questions. Four questions should be answered. Each question carries 150 marks - altogether 600 marks Total marks for paper II 1000 ÷ 10 = 100



provided at the examination.)

1. A hypothesis is (1) an opinion that cannot be proved experimentally. (2) an opinion to explain an observation. (3) an opinion confirmed by all experimental evidences. (4) an opinion not agreeing with accepted principles. (5) an unconfirmed conclusion of an experiment.

2. Both glycogen and starch (1) are made of glucose. (2) are straight chain polymers. (3) are products of photosynthesis. (4) stain blue with iodine. (5) are food substances stored by animals.

3. Competitive inhibitors of enzymes are (1) molecules that bind to substrate and inhibits its reaction. (2) molecules that bind to active sites of an enzyme and inhibit its reaction. (3) molecules that bind to an enzyme at sites other than active site and inhibits its reaction. (4) molecules that have similar chemical activities as substrate. (5) molecules that are structurally similar to an enzyme.

4. Which of the following characteristics of water is most helpful for minimizing temperature variation within organisms?

(1) Water remains as a liquid within a wide range of temperatures. (2) Water molecule makes Hydrogen bonds with many other molecules. (3) Water molecules are held together by cohesive forces. (4) Water has a high specific heat capacity. (5) Water has a high latent heat of evaporation.


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5.

-O - P - O - P - O - P - O - CH2

O O O

O- O- O-

H

OH

OH

H

OHH

NHC

C NCHNN C

C

NH2

Which of the following best describes the above molecular structure? (1) a nucleotide. (2) ATP molecule. (3) NAD molecule. (4) a nucleotide of the DNA molecule. (5) a coenzyme molecule.

6. Which of the following structure - function relationships is incorrect? (1) Nucleolus - Synthesis of rRNA (2) Golgi complex - Formation of secretory vesicles (3) Plasmodesmata - Exchange of material between cells (4) Glyoxysomes - Conversion of lipids to carbohydrates (5) Rough endoplasmic reticulum - Detoxification

7. Which of the following is an event that does not occur during prophase I of meiosis? (1) Duplication of each chromosome to produce two chromatids. (2) Shortening of chromosomes (3) Parallel arangement of homologous chromosomes making a synaptonemal complex. (4) Exchange of parts of chromatids between homologous chromosomes. (5) Disappearence of nuclear membrane

8. Which one of the following features is unique to the organism group indicated against it? Feature Organism group (1) Paired fins - Osteichthyes (2) Cilia - Ciliophora (3) Pedicellaria - Echinodermata (4) Exoskeleton - Arthropoda (5) Wings - Aves

9. Select the correct statement. (1) Birds are poikilotherms. (2) Post-anal tail is a chordate feature. (3) Nematodes are coelomates. (4) Polyps and medusa are present in anthozoans. (5) All platyhelminths are parasites.

10. The following are characters seen in Pterophyta. A - Distinct vascular tissues B - Independent gametophyte C - Independent sporophyte D - Haploid spores E - Motile reproductive cells Which of the above are not seen in Bryophyta? (1) A and B (2) A and C (3) A and E (4) B and D (5) B and E


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11. Which one of the following is an example for mutualism? (1) Orchid plants growing on a mango tree (2) Sea anemone living on hermit crab (3) Larval stages of Wuchereria bancrofti in mosquito (4) Mushrooms growing on a tree trunk (5) Rhizobium living in roots of legume plants 12. Select the correct statement with regard to human respiration. (1) duration of a respiratory cycle is about 4 - 5 seconds. (2) vital capacity is approximately 6 dm3. (3) diaphragm relaxes during inspiration. (4) bronchioles are lined by ciliated columnar epithelium. (5) rate of breathing is always controlled involuntarily.

13. With reference to phleom transport which of the following organs is most likely to function as a sink in potato plants?

(1) Mature leaf (2) Senescing leaf (3) Growing leaf (4) Underground stem (5) Germinating tuber

14. Five events that occur in phloem transport are given below. A - Water diffuses into sieve tubes B - Leaf cells produce sugar C - Companion cells load sugar into sieve tubes D - Sugar is transported from cell to cell in the leaf E - Sugar moves down the stem Select the order that correctly explains phloem transport (1) A, B, C, D, E (2) B, A, D, C, E (3) B, D, C, A, E (4) B, D, A, C, E (5) D, B, A, C, E

15. Along which one of the following paths, a red blood cell from upper arm passes through subclavian vein to lung for oxygenation?

(1) Brachio-cephalic vein Superior vena cava Heart Pulmonary vein (2) Common carotid vein Superior vena cava Heart Pulmonary vein (3) Brachio-cephalic vein Superior vena cava Heart Pulmonary artery (4) Jugular vein Superior vena cava Heart Pulmonary artery (5) Superior vena cava Heart Pulmonary vein

16. Select the correct statement regarding human heart. (1) Myocardium of left atria is thicker than that of the right atria. (2) Cuspid valves are closed by its muscle fibres. (3) Right atrium receives deoxygenated blood only by superior and inferior vena cava. (4) Corda tendineae join atrio-ventricular valves to papillary muscles. (5) Coronary circulation is associated with systemic circulation.

17. Which one of the following microorganisms is used in yoghurt manufacture. (1) Lactobacillus bulgaricus (2) Streptomyces griseus (3) Saccharomyces cerevisiae (4) Acetobacter aceti (5) Bacillus subtilis

18. Chemoautotrophic bacteria obtain energy and carbon respectively (1) from light and Carbondioxside. (2) from light and organic compounds. (3) by oxidation of inorganic compounds and from Carbondioxside. (4) by oxidation inorganic compounds and from organic compounds. (5) by oxidation of organic compounds and from Carbondioxside.


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19. Which one of the following hormones acts on both smooth muscles and cardiac muscles? (1) Oxytocin (2) Noradrenalin (3) ADH (4) Adrenalin (5) Cortisol

20. Select the incorrect statement regarding action potential of a neuron. (1) Its production requires a threshold stimulus. (2) In myelinated neurons, action potentials are produced only at the nodes of Ranvier. (3) Rising phase of the action potential is due to influx of Na+ into neuron. (4) During an action potential, polarity of the neurilemma is reversed. (5) Its duration is about 10 ms.

21. Stimulation of parasympathetic nervous system of man (1) dilates pupil of eye. (2) stimulates secretion of saliva. (3) inhibits secretion of intestinal juice. (4) increases blood pressure. (5) increases the rate of ventilation of lungs. 22. Select the incorrect statement regarding nitrogenous excretion. (1) Nitrogenous excretion is due to metabolism of proteins and nucleic acids. (2) Initial nitrogen containing waste material of all animals is ammonia. (3) Main site of urea synthesis is the kidney. (4) Four nitrogen atoms can be excreted in a single molecule by producing uric acid. (5) Creatinine is an end product of nitrogenous excretion.

23. Select the mismatched pair regarding movement. (1) Flagellar movement - Chlamydomonas (2) Ciliary movement - Paramecium (3) Bipedal movement - Pigeon (4) Nictinastic movement - Leaves of Mimosa (5) Phototropic movement - Opening of flowers

24. Spherical bacteria arranged in cuboidal packets are known as (1) Streptococcus. (2) Staphylococcus. (3) Diplococcus. (4) Sarcina. (5) tetrads.

25. Which of the following is an example of thigmotropism? (1) A sunflower turning towards light (2) Growth of pneumataphores in Sonneratia (3) Folding movement of Mimosa leaves (4) Coiling of tendrils of passion fruit (5) A seedling stem turning towards light

26. Which of the following is least suitable as an explant for a plant tissue culture experiment? (1) apical meristem (2) lateral meristem (3) leaf discs (4) bark (5) embryo

27. The main site of sperm storage in man is (1) seminiferous tubules. (2) epididymis. (3) vas deferens. (4) ejaculatory duct. (5) seminal vesicles.

28. Select the incorrect statement regarding human spermatogenesis. (1) It starts at puberty. (2) Spermatogonia undergo meiosis to produce primary spermatocytes. (3) Spermatids develop into sperms without undergoing cell division. (4) Testosterone regulates spermatogenesis. (5) The entire process takes about two months.


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29. Select the incorrect statement regarding human lactation. (1) Milk production occurs in the alveoli of mammary glands. (2) Oestrogen stimulates lactation. (3) Baby’s suckling is essential for maintenance of lactation. (4) Prolactin initiates milk production. (5) Oxytocin promotes milk ejection.

30. Select the correct statement regarding human uterus. (1) The bulk of its wall is formed from the endometrium. (2) Fertilization usually occurs in it. (3) Prostaglandins secreted by the placenta inhibit its contractions. (4) Its wall does not contribute to the formation of placenta. (5) Oxytocin induced its contractions are regulated by a positive feedback mechanism.

31. In the genetic code there are three codons which do not code for amino acids. These codons (1) are not transcribed into RNA from DNA. (2) found only in tRNA genes. (3) found only in inactive genes. (4) give stop signals in protein synthesis. (5) give start signals in protein synthesis.

32. Which of the following best describes what is known as gene cloning? (1) Making recombinant DNA molecules in test tubes by joining DNA molecules obtained from

different sources. (2) Making genetically modified organisms by introducing foreign DNA into live cells. (3) Producing a protein by introducing a foreign gene into a bacterium. (4) Introducing a foreign gene to the vector and transforming the host cell and multiplication of the foreign gene within the host cell. (5) Making a genetically modified bacterial strain by introducing a foreign gene.

33. Which of the following features that has evolved in terrestrial plants can not be considered as an adaptation to terrestrial life?

(1) Loss of requirement of external water for fertilization. (2) Presence of vascular tissues to transport water and solutes within plants. (3) Presence of secondary thickening in stems and roots. (4) Presence of leaves as organs specialized for photosynthesis. (5) Evolution of microspores as pollen which are dispersed by wind.

34. In a variety of corn, a dominant allele of a gene produces a purple colored kernel while its double recessive form produces colorless kernel. Another gene inhibits pigment synthesis. If two corn plants,both of them heterozygous for both genes were crossed, plants with colored kernels and color-less kernels are produced in 3 : 13 ratio among the progeny. Which of the following is most appropriate to describe this pattern of inheritance.

(1) Polyallelism (2) Complementary epistasis (3) Recessive epistasis (4) Dominant epistasis (5) Gene linkage

35. Genetic recombination (1) occurs at fertilization. (2) occurs at meiosis. (3) can occur at mitosis. (4) is a result of independent segregation of chromosomes. (5) occurs only in diploid organisms.


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36. A man was colour blind although his father was normal. What is the probability of his sister’s son becoming colour blind? Assume that his sister’s husband is normal.

(1) 1 (2) 0.75 (3) 0.5 (4) 0.25 (5) 0

37. Which one of the following statements best explains a flagship species? (1) A species that has been given priority in conservation efforts, because it permits a large number of

other species in the community to persist. (2) A species whose occupancy is large enough so that when it is protected many other species are

under that protection. (3) A popular species that serves as a symbol to rally people for conservation awareness and action.

(4) A species selected by a group of people for conservation awareness and action. (5) A species that is present only in restricted geographical areas.

38. What is the most critical global environmental problem which has received highest attention today? (1) Global warming (2) Depletion of the ozone layer (3) Desertification (4) Acid rain (5) Accumulation of solid wastes

39. Which one of the following controls trans boundary movement of hazardous waste? (1) Basel convention (2) CITES (3) Montreal protocol (4) Marpol convention (5) Kyoto protocol

40. The antibiotic erythromycin affects (1) protein synthesis in bacteria (2) the cell wall of bacteria (3) the cell membrane of bacteria (4) the replication of viruses (5) DNA replication of bacteria

• For each of the questions 41 to 50 one or more of the responses is/are correct. Decide which of the reponse/responses is/are correct and then select the correct number.

If only A, B and D are correct ........................................... 1 If only A, C and D are correct ........................................... 2 If only A and B are correct ........................................... 3 If only C and D are correct ........................................... 4 If only other response or combination of responses is correct ........................................... 5

Directions summarised1 2 3 4 5

A, B, Dcorrect

A, C, Dcorrect

A, Bcorrect

C, Dcorrect

Any other response orcombination of responses correct

41. Which one or more of the following relationships is/are correct? (A) Irregularly thickened cell walls - Collenchyma (B) Chondrin - Compact bone (C) Collagen fibres - Areolar tissue (D) Fibrinogen - Blood (E) Intercalated discs - Nervous tissue

42. Which of the following can be considered as evolutionary steps that occurred in evolution of gymnosperms from ferns.

(A) Confinement of male gametophyte to the pollen. (B) Retention of female gametophyte in megasporangium. (C) Dioecious sporophyte. (D) Development of pollen tube as a means of obtaining nourishment for the male gametophyte. (E) Enclosure of embryo within a seed as a dispersal unit.


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43. Question 43 is based on the skeletons of the animal groups given below. â) Annelida ^b) Arthropoda ^c) Amphibia ^d) Mollusca ê) Echinodermata Which of the following groups show/ shows the type of skeleton seen in the animal groups given above

in the sequence of exoskeleton, hydrostatic skeleton and endoskeleton? (A) d a e (B) b a c (C) b c d (D) e a c (E) e a d

44. Which of following is/are seen in Domain Archaea? (A) Cell walls contain peptidoglycan. (B) Only one type of RNA polymerase is present. (C) Generally found in extreme environments. (D) Do not show sensitivity to streptomycin. (E) Motile forms are common.

45. Which of the following hormone/hormones is/are synthesized in the hypothalamus? (A) TRH (B) GnRH (C) ACTH (D) ADH (E) LH

46. Select the correct statement/statements regarding human nephron. (A) Cortical nephrons are the predominant type of nephrons. (B) Convoluted tubule is the site of obligatory reabsorption of water. (C) Podocytes are found in the outer wall of the Bowman’s capsule. (D) H+ is secreted in the distal convoluted tubule. (E) K+ is reabsorbed in the descending limb of loop of Henle.

47. Which of following is/are example/examples of passive immunity? (A) Immunity developed in new born babies by antibodies passing through the placenta of the

mother. (B) Immunity developed in a person who has cantracted measles. (C) Immunity developed by vaccination of a healthy persons with polio vaccine. (D) Immunity developed after vaccination of a person who has been bitten by a rabid dog. (E) Immunity developed in certain individuals against malaria.

48. Which of the following character/characters is/are seen in the Phylum Phaeophyta? (A) In many forms the vegetative body is differentiated into a holdfast, stalk and thallus. (B) The chloroplasts contain chlorophyll a and c. (C) Occurs primarily in fresh water. (D) Contains the pigment phycobilin. (E) Life cycle has no flagellated cells.

49. Hookworm infections can be controlled by (A) refraining from defaecating outdoors. (B) refraining from using human faeces as fertilizer. (C) washing hands thoroughly using soap before consuming food. (D) wearing footwear. (E) keeping food material always covered.

50. Select the correct statement/statements regarding coconut mite. (A) damages young nuts as well as mature nuts. (B) spreads due to transportation of nuts. (C) is clearly visible to the naked eye. (D) can be successfully controlled by indigenous methods. (E) is abundant in the base of young coconut fronds.

* * *


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09 - BiologyPaper II



1. (A) (i) What feature of the ATP molecule makes it suitable to function as an energy carrier?

(ii) What is meant by oxidative phosphorylation?

(iii) Name the site of oxidative phosphorylation.

(iv) What is the source of energy for oxidative phosphorylation?

(B) (i) Name three major characters which are used to group living organisms into the three Domains. (a)

(b) (c) (ii) Indicate the Domain to which each of the groups indicated below belongs.

Group DomainCyanobacteriaMethanococcus

ChitridiomycotaCiliophora

(a) (b) (c) (d) (C) (i) Name an external structural characteristic feature that is seen in both birds and reptiles.

(ii) Give an external feature that could be used to distinguish a chondrichthyes from an osteichthyes.

(iii) Using visible external features allocate a leech and Nereis to their respective classes.

Animal Visible external feature ClassLeechNereis


- 118 -

2. (A) (i) Name the three main heterotrophic nutritional modes seen in living organisms.

(ii) Name the mode of nutrition of each organism given below. Nitrosomonas ................................................. Aspergillus ................................................. (B) The following experiment was carried out to determine the water potential of potato cells. Four equal sized strips each measuring 60 mm in length were cut from a potato tuber. One strip was

placed in a test tube with distilled water, while the others were placed in test tubes containing different concentrations of sucrose solutions. The tubes were labeled A, B, C and D. After 2 hours the potato strips were taken out and their lengths were measured again. The data are shown in the table below.

Tube Length of potato strip after 2 hours (mm) A 60 B 65 C 58 D 62

(i) Which tube contained a solution with a water potential equivalent to that of the potato cells?

(ii) Which tube contained distilled water?

(iii) Arrange the tubes A to D, in the descending order of water potentials of the liquids.

(iv) (a) What is a blood count?

(b) What is the equipment usually used in a lab to get a blood count?

(C) (i) Why is excretion essential for animals?

(ii) State one advantage in producing ammonia as a nitrogenous excretory product.

(iii) In humans, which type of nephrons possesses long loops of Henle?

(iv) What is meant by ultrafiltration in human renal physiology?

(v) What does the presence of proteins in a human urine sample generally indicate?


- 119 -

3.(A) (i) What is lactation ?

(ii) State one advantage of giving breast milk over powdered milk.

(iii) State one deficiency of human semen which can cause male infertility.

(iv) State two main functions of seminal vesicle fluid in man. (a) (b)

(v) What is the role of inhibin in the control of sperm production in man?

(vi) What is the role of positive feedback control of oxytocin release at parturition?

(B) (i) State the main function of each of the following type of RNA in the process of protein

synthesis. Type of RNA Function mRNA

rRNA

tRNA

(ii) Which one of the above types of RNA would change if a point mutation occurs in a gene coding for an enzyme?

(C) (i) State two important changes that have taken place in the evolution of the angyosperm ovule from a megasporangium, as found in Lycophyta.

(a) (b)

(ii) (a) What is known as double fertilization? (b) What is the main advantage of double fertilization?


- 120 -

4. (A) (i) What is an ecological niche?

Questions (ii) and (iii) are based on the hypothetical food web of a fresh water ecosystem given below.

Cormorants

Snakeheads Grass carpsBarbs

Common carps

Zooplankton Aquatic insects

Phytoplankton Aquatic macrophytes

(ii) Name the organism / organisms that occupy more than one trophic level.

(iii) If aquatic macrophytes are removed from the above ecosystem, what will happen to the population density of the common carp?

(B) (i) In the table below, some characters seen in fungi are given in the first column. If a given character

is present in a phylum indicate with a (√) and if absent with a (×).

Character Chitridiomycota Zygomycota AscomycotaHabitat is mostly aquaticUnicellular members may bepresentProduce zoospores

(C) (i) What are stem cells? (ii) Name a structure from which stem cells are mainly obtained for human stem cell therapy

today. * *


- 121 -

Part B - Essay

5. (a) Discribe the general composition and stucture of proteins. (b) Giving suitable examples, list the differant types of proteins found in living organisms

according to their functions. (c) Why a denatured enzyme does not show its normal activity?

6. (a) Briefly describe the structure of a sarcomere in human skeletal muscle fibre. (b) Briefly explain how a human skeletal muscle fibre contracts.

7. (a) Describe the structure of the Phloem tissue. (b) Explain the pressure flow hypothesis of phloem translocation. (c) What are the major differerences between phloem translocation and xylem translocation?

8. (a) What is meant by air pollution? (b) Briefly describe the effects of air pollutants on enviroment.

9. (a) What are prions? (b) Name a disease caused by prions in cattle. (c) Describe the general methods available for, (i) control and (ii) cure of diseases caused by micro-organisms

10. Write short notes on the following. (a) External factors affecting enzyme activity (b) Epistasis (c) Human skin receptors

* * *


- 122 -

10 - Combined Mathematics










Important :

* In Combined Mathematics I and II question papers 30 prototype questions for part A and 21 prototype questions for part B have been included. However, the question paper that will be given at the examination will be constructed according to the question paper structure given above.

* At the examination all questions in part A of both Combined Mathematics I and II will have to be answered using the space provided for each question.


- 123 -


Prototype Questions for Paper I

Part A

1. α and β are the roots of the quadratic equation ,02 =++ cbxax where c ≠ 0. Show that the

roots of the quadratic equation 0)2( 22 =+−− acxacbacx are βα and

αβ .

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2. Let .2)( 23 cxbxaxxf +−+≡

If (i) the remainder is )( 16 +x when )(xf is divided by ),( 2 xx + and (ii) )( 1−x is a factor of ),(xf then, find the values of ., cba and '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 124 -

3. Using the substitution xy 3= , solve the equation .3333 3212 xxx +=+ ++

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4. Find the set of all real values of x satisfying the inequality .212

−<−xx

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- 125 -

5. Let .A

−=

1113

By the Principle of Mathematical Induction, show that

−

−+= −

nnnnnn

22

2 1A

for every positive integer n.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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6. Using the properties of determinants show that

).)(( 222 cabcabcbacbabacacbcba

−−−++++−=

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- 126 -

7. Let 1a and 2a be the coefficients corresponding to x and 2x respectively in the binomial

expansion of ,)23( nx+ where n is a positive integral index. If 12 2aa = , find the value of n.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. Find the number of different arrangements that can be made using all the letters of the wordAPPLE.Also, find the number of different three-letter selections that can be made from the letters ofthe same word.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 127 -

9. Express the complex number ii

+

+

532 in the form ),1( i+λ where λ is real.

Hence, show that 4

532

+

+

ii is real.

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10. Find .cos1tanlim

0

−→ x

xxx

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- 128 -

11. If ,10 ,sin 1 <<= − xxy show that .0dd

dd)1( 2

22 =−−

xyx

xyx

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12. Find the turning points of the function .51232)( 23 +−+= xxxxfHence, determine the range of x in which the function )(xf increases.

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- 129 -

13. Using integration by parts, evaluate .d1

0

12∫ + xxe x

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14. Find ∫ −+

+ .d2

52 xxx

x

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- 130 -

15. Find the equations of the tangent and the normal to the curve given by the parametricequations 25tx = and 13 += ty at the point (5, −2).

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16. The point P divides a line BC internally in the ratio 1 : 2. If B = (−1, 1) and C = (1, 4), findthe equation of the straight line passing through the points P and (2, 3).

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- 131 -

17. A line 0=l perpendicular to the line ,092 =+− yx intersects the positive sides of the xand y axes at the points A and B respectively. If the area of the triangle OAB, where O isthe origin, is 16 sq. units, find the equation of the line .0=l

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18. Find the equation of the bisector of the acute angle between the lines 012 =++ yx and.02 =+ yx

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- 132 -

19. Find the equations of the two lines that pass through the point (1, 2) and form an angle 4π

with the line .022 =−− yx'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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20. If the tangents to the circle given by 04222 =−−+ yxyx at the points (2, 4) and (3, 3)intersect at the point P, find the distance between P and the centre of the circle.

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- 133 -

21. Find the equation of the circle passing through the origin and the points of intersection ofthe line 052 =++ yx and the circle .053222 =++++ yxyx

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22. Show that the triangle with vertices (0, 0), ),( 22 22 and

−

223

223 , is a right

angled triangle. Find the equation of the circle which passes through the vertices of theabove triangle.

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- 134 -

23. Find the coordinates of the points of intersection of the line 3+= xy and the circle withcentre (3, 4) and radius .52

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24. The circle 0=S with centre (−1, 6) touches the circle 0114622 =+−−+ yxyx externally.Find the equation of the circle .0=S

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- 135 -

25. The points A and B lie on the x and y axes respectively. Prove that, as A and B vary on the xand y axes respectively such that the length AB is a constant, the locus of the midpoint ofAB is a circle.

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26. Show that the locus of the centre of the circle which passes through the point (a, 0) andtouches the line 0=+ ax is a parabola.

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- 136 -

27. P is a variable point on the circle .0422 =−+ yx The point N is the foot of theperpendicular from the point P to the y-axis. If M is the mid point of NP, show that thelocus of M is an ellipse.

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28. Find the solutions of the equation θθθ 3sin5sinsin =+ in the range [0, π].'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 137 -

29. For a triangle ABC, ,675accbba +

=+

=+ where a, b and c have the usual meaning. Using

the Sine rule show that sin ,sin BA and Csin are consecutive terms of an arithmeticprogression.

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30. Show that .43

1tan21tan 11 π

=+ −−

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- 138 -

Part B

1. Let ,22)( 2 +++≡ kkxxxf x ∈ IR, where k is a constant.(a) (i) Find the range of values of k for which )(xf is positive for all values of x.

(ii) If α and β are the roots of the equation 0)( =xf ,

show that )2(4)( 22 −−=− kkβα .(iii) Determine the values of k for which the roots of the equation 0)( =xf differ by 4.

(b) If k ≠ − 2 , find a quadratic equation whose roots are 1βα

+ andαβ

+1 .

2. (a) Let ,1284)(

2

+

+=

xxxf x ∈ IR,

21

−≠x .

Show that f(x) cannot lie between – 8 and 4.Find the values of x for which(i) 8)( −=xf ,(ii) 4)( =xf .Hence, draw the graph of )(xf .

(b) Draw the graph of the curve 12 −= xy .

Deduce the graph of 12 −= xy .

Draw the graph of 72 −= xy in the same diagram.

Hence or otherwise, solve the inequality 17 22 −>− xx .

3. (a) The matrix

=

srqp

A satisfies IAA =T , where I is the identity matrix of order 2.

Show that(i) if 0=p , then 0=s and ,122 == rq(ii) if p ≠ 0, then ),(or ),( qrpsqrps =−=−== .

(b) Let

−−=

3243

M .

21 and λλ are the solutions of ,0)(det =− IM λ where I is the identity matrix of order 2.Find 21 and λλ .Find a non-zero column vector ix such that OxIM =− ii )( λ , i = 1, 2, where O is thezero column vector.

Let )(222 1xxP =× .

Verify that

=−

2

110

0λ

λMPP .


- 139 -

4. (a) Using the properties of determinants find all the factors of 63

42

2

111

xxxxxx

.

(b) Let

−

−

=

110211221

C .

Show that ,23 ICCC =−− where I is the unit matrix of order 3.Hence, find the inverse of C.

Write down the following system of equations in matrix form:

202122

−=+

=+−

=+−

zyzyxzyx

Hence, solve the system of equations.

5. (a) Using the Principle of Mathematical Induction, prove that 12633 −− nn is divisibleby 676 for all positive integral values of n.

(b) Find the partial fractions of )2)(1(332 2

++

−+

xxxx .

Hence or otherwise, find nv such that un, the nth term of the series

...41

5.424

41

4.311

41

3.22 32

+

+

+

can be written in the form )(42

1−−+= nnnn vvu .

Show that n

n nnS

+

+−=

41

)2(321

61 , where nS is the sum of the first n terms of the series.

Find nn

S∞→

lim .

6. (a) For any two real numbers a and b, prove that abba 222 ≥+ .Deduce that, for any three positive numbers a, b and c, abccba 3333 ≥++ .

Hence, prove that abccabcabcba 9))(( ≥++++ .

(b) The polynomial 42)( 234 −−++≡ xbxaxxxp has factors )2( and )1( +− xx .(i) Show that 2 and 3 == ba .(ii) Factorize )(xp completely.(iii) Find the values of x for which )(xp is positive.


- 140 -

7. (a) How many numbers can be formed by taking any number of different digits at a timefrom the six digits 0, 1, 2, 3, 4 and 5?

(b) How many different arrangements can be made from all the letters of the word‘ENGINEERING’?In how many of these will the three E’s stand together and in how many will they standtogether first?

(c) Let 22

33

2210

2 ...)1)(1( +++++++=++ nn

n xaxaxaxaaxx .Find possible values of n if 210 and , aaa are three consecutive terms in an arithmeticprogression.Write down the complete expansion of the expression in powers of x, for each value of n.

8. (a) Prove that, if a , b and c are any positive real numbers not equal to 1 then,cbc baa logloglog ×= .

Deduce that, b

aa

b log1log = .

If bcx alog= , cay blog= and abz clog= , prove that 2+++= zyxxyz .

(b) If 2≥n , find A, B and C such that !)!1()!2(!

122

nC

nB

nA

nnn

+−

+−

≡++ .

Hence, show that 15...!34

!23

!12 222

−=+++ e , where ∑∞

=

=0 !1

r re .

9. (a) Let ivu

z+

=1 , where u and v are both nonzero and real.

Prove that, in the Argand diagram, the complex number z lies on a circle when(i) v varies while u is a constant,(ii) u varies while v is a constant.

Prove also that the centres of the two circles obtained above lie on two fixed lines whichare perpendicular to each other.

(b) Prove that, if 042Re =

+

−

ziz then the locus of z is a circle with radius 5 .

Find its centre.

If 042Im =

+

−

ziz find the least value of 1−z .


- 141 -

10. (a) Evaluate xx

x sin)cos(sin1lim

0

−→

.

(b) Let xxy 11 sin)sin1( −−+= .

Prove that, xyx

xyx

dd

dd)1( 2

22 −− is independent of x.

Hence, find n

n

xy

dd at x = 0 for n = 1, 2, 3, 4.

(c) A rectangle is inscribed in a quadrant of a circle of radius a such that two sides of therectangle lie along the bounding radii of the quadrant and one of the vertices of therectangle lies on the arc of the circle. Find the maximum area of the rectangle.

11. (a) Using first principles find the derivative of 2)1(1x

y+

= .

(b) Let 21)(

xbaxxf

+

+= , where a and b are real numbers. If the point (−2, −1) is a turning point

of )(xf , determine the values of a and b.Draw the graph of )(xfy = by considering the first derivative.Deduce all possible values of k for which the equation kxf =)( has distinct real roots.

12. (a) Considering the sign of the first derivative of the function xxxy sincos −= over the

interval ),0( π , show that xxsin is a decreasing function of x.

Deduce that, xxx<≤ sin2

π for 0 < x ≤

2π .

(b) A curve is given by the parametric equations )( 22 atay −= and )2(2 atax += , wheret is real and a is a constant. Find the equation of the normal drawn to the curve at thepoint corresponding to t = t0.The point of intersection of the normals drawn to the curve at the points correspondingto t = t1 and t = t2 lies on the curve. Show that 221 =tt .

(c) If 3 3

22

1

)1(

x

xxyx

+

+= for 0>x , find

xydd .


- 142 -

13. (a) Using partial fractions, find ∫ +−

+− xxxxx d

)1()1(43

22

2.

(b) It is given that ayyx 222 =+ , where a is independent of x and y.

Using the substitution txy = , find ∫ yxd .

(c) Differentiate 2

tanlncos xx with respect to x.

Hence, evaluate xxx d2

tanlnsin2

3

∫

π

π

.

14. (a) Using the substitution 2

tan xt = , find ∫ + xxsin1d .

Hence or otherwise, find ∫ ++ xxxx d

sin1sin3cos2 .

(b) Using integration by parts, find ∫ − xx dsin 1 .

(c) Using a suitable substitution, show that ∫∫ −=aa

xxafxxf00

d)(d)( , where a is a positive

constant.

Hence, evaluate ∫ −1

0

2 d)1( xxx .

15. (a) Find ∫ +

+ xxx d)13(562 .

(b) Let ∫= d)lncos( xxaI and ∫= )dln(sin xxaJ , where a is a non-zero real number.Show that,(i) )lncos( xaxaJI =− , when x is positive

(ii) )lncos( xaxaJI =+ , when x is negative.Obtain another relationship between I and J in each interval of x.Hence, evaluate the integrals I and J.

(c) Using the substitution 0cos1 =+ xt show that 23lnd

cos1tan3

0

=+∫

π

xxx .


- 143 -

16. (a) The sides PQ, QR and RP of a triangle PQR are given by the equations 053 =+− yx ,03 =−− xy and 013 =−− yx respectively. The line through the point P perpendicular

to QR meets the line through Q parallel to PR at a point S. Find the equations of QSand PS.Hence, show that PQSR is a rhombus.

(b) Find the equation of the circle S = 0, whose centre is at the point ( – 3, – 1) and whichpasses through the point )1,2(−=A .Show that the tangent drawn at the point A to the circle S = 0 passes through theorigin O.Find the coordinates of Q, the point of contact of the second tangent drawn to thecircle S = 0 from the origin.Show that OA is perpendicular to OQ.

17. Show that the equation of any straight line can be written in the form pyx =+ αα sincos ,where 0≥p and πα 20 <≤ .Hence, find the necessary and sufficient condition for a straight line given by

0=++ nmylx to be a tangent to the circle given by .0222 =−+≡ ryxSThe tangents t1 and t2 drawn to the circle S = 0 are parallel to the y-axis. Two straight linesthrough the origin O are drawn in such a way that one line cuts the tangent t1 at a point T1

and the second line cuts the tangent t2 at a point T2. If 2

ˆ21

π=TOT prove that, the line T1T2

is a tangent to the circle S = 0.

18. The equations of two straight lines OA and OB are 02 =+ yx and 02 =− yx respectively.A line drawn from a point P parallel to OA, meets the line OB at a point M and a linedrawn from the point P parallel to OB, meets OA at a point L. The line drawn through thepoint L perpendicular to OA meets the line drawn through the point M perpendicular toOB at a point Q. Find the coordinates of the point Q in terms of the coordinates of thepoint P. If the point P varies on the parabola given by xy 82 = , show that the locus of thepoint Q is also a parabola and find its focus, directrix and vertex.

19. (a) In the usual notation, state the Sine rule for a triangle ABC.In the same notation, prove that, .0)sin()sin()sin( =−+−+− BAcACbCBa

(b) Let )cos(sin4)( 44 xxxf += .Prove that, xxf 4cos3)( += .

Hence or otherwise, draw a rough sketch of )cos(sin4)( 44 xxxf += for 2π

≤x .

Calculate the area bounded by the curve )(xfy = and the lines given by 2π

±=x and

2=y .

(c) Prove that

+

=

+

−−−−

53tan

41tan

31tan

21tan 1111 .


- 144 -

20. (a) In the usual notation, state the Cosine formula for any triangle ABC .If, in the same notation, )(2 222444 baccba +=++ for a triangle ABC, show that

43or

4ππ

=C .

(b) Suppose that

−

+=12

cot12

tan ππ xxy .

Prove that xyy 2sin)1(21 −=+ .

Hence, show that for any real value of x, the expression

−

+12

cot12

tan ππ xx does

not take values between 31 and 3.

21. (a) State and prove the Sine rule for a triangle ABC, in the usual notation.Hence, prove, in the same notation, that

−=

+2

cos2

sin)( CBaAcb and

−=

−2

sin2

cos)( CBaAcb .

Deduce the Cosine formula for a triangle ABC.

(b) Solve for x:(i) 4)sin(cos42sin =−+ xxx ,

(ii) )1(4cottan2tan 2111 xxx −=− −−− .


- 145 -


Prototype Questions for Paper II

Part A

1. A cyclist rides along a straight path with uniform velocity u and passes a motor car which isat rest. At the same instant, the motor car starts to move in the same direction with uniformacceleration a until it attains its greatest velocity v ( > u). Draw, in the same diagram, thevelocity-time graphs for the motions of the cyclist and the motor car.Deduce that within the period in which the motor car is behind the cyclist, the maximum

distance between them is au2

2.

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2. A motor-boat which can attain a maximum speed ,hkm 1−u is to intercept a ship whichmoves with a constant speed 1hkm)2( −< uv in the North-West direction. Initially theship is located south of the motor-boat. If the motor-boat move at its maximum speed, findthe direction in which it should move in order to intercept the ship, using a velocity triangle.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 146 -

3. The position vector r of a moving particle at time t, is given byjir )1095()205( 2ttt −+++= , where i and j have their usual meaning.

(i) Find the initial velocity of the particle.(ii) At time t = T , the particle is moving along a direction perpendicular to its initial

direction of motion. Find the value of T and the distance to the particle at t = T from itsinitial position.

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4. A wedge of mass m2 is free to move on a rough horizontal plane with coefficient of frictionµ . A particle of mass m , slides down along a line of greatest slope of the smooth face of thewedge, inclined at an angle α to the horizontal. Find the acceleration of the wedge.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 147 -

5. A particle is projected under gravity with speed u at an angle 4π to the horizontal.

Show that the maximum height attained by the particle is gu4

2.

Deduce that the particle can never pass over a wall of height h if ghu 42 < .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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6. A particle of mass 2m is drawn up along a line of greatest slope of a smooth plane inclinedat an angle α to the horizontal, by a light inextensible string which passes over a smoothpulley fixed at the top of the plane, and has a mass 3m hanging freely from the other end. Ifthe system is released from rest, find the accelerations of the particles and the tension in thestring.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 148 -

7. A light inextensible string passes over a smooth pulleyfixed at the edge of a smooth horizontal plane and arounda movable smooth light pulley C which carries a particleB of mass 2m. A particle A of mass m is attached to oneend of the string and the other end of the string is fixed tothe ceiling as shown in the figure. The particle A isinitially at rest on the horizontal plane. Find theacceleration of the pulley C and the tension in the stringwhen the system is released from rest.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. A train of mass M is ascending on a smooth track having an inclination of 1 in n. When thevelocity of the train is v its acceleration is f. Assuming that the resistance against the motion

of the train is negligible, prove that the effective power of the engine is )( nfgnMv

+ .

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C

B

A

2mg


- 149 -

9. A smooth small ball of mass m which is at rest, falls under gravity from a height 8h to ahorizontal smooth plane and rebounds to a height h2 . Find the coefficient of restitutionbetween the ball and the plane.Show that the loss of kinetic energy due to the impact is mgh6 and find the impulse betweenthe ball and the plane.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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10. One end of a light inextensible string of length 2 meters is attached to a fixed point in theceiling and the other end carries a particle of mass kg2 . The particle oscillates in a vertical

plane. If the velocity of the particle is 1sm3 − when the angular displacement of the string

from the downward vertical is 3π , find the tension in the string at this instant.

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- 150 -

11. A particle of mass m is attached to one end of an elastic string of natural length a andmodulus of elasticity λ. The other end of the string is attached to a fixed point O on asmooth horizontal plane. Initially the particle is kept at rest at a distance a2 from the

point O and then released. Show that it is in a simple harmonic motion for a time λ

π ma2

.

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12. A particle P of mass m which lies on a smooth horizontal plane is attached to one end of alight spring of natural length l and modulus of elasticity 2mg, whose other end is attachedto a fixed point O on the plane. Initially the spring is unstretched and the particle isdisplaced a distance less than l, along the direction of OP and released. Prove that the

displacement x of the particle at time t, satisfies the equation 02=+ x

lgx&& .

If the greatest velocity of the particle is 2gl , find the amplitude of the motion.

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- 151 -

13. Forces of magnitude 2P, 3P and 4P act respectively along the sides AB, BC and CA of anequilateral triangle ABC of sidea . Find the magnitude and the direction of the resultant.Find also, the distance from A to the point where the line of action of the resultant meetsAC.

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14. Three forces ji PP 3+ , ji PP −− 2 and ji PP 2− act at the points whose position vectors areji 52 + , j4 and ji +− respectively, where i and j have their usual meaning. Represent the

forces in a diagram in component form, indicating the coordinates of the respective points.Show that the system is equivalent to a couple of moment 10P, and indicate the sense ofthe couple.

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- 152 -

15. Two equal uniform heavy rods AB and BC are jointed smoothly at B. The system issuspended from the point A and the mid points of the rods are connected by aninextensible light string such that AB and BC are perpendicular to each other. If the systemis in equilibrium with the rod AB making an angle θ with the downward vertical, show

that 31tan =θ .

Show also, that the reaction at B is along the direction BC.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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16. The figure shows a framework which consists of four smoothlyjointed light rods carrying a weight W at the point C. AB, BCand AC are of equal length. The framework is smoothlyhinged at a fixed point D with B resting against a smoothsupport such that the rods AD and BC are horizontal while Blies vertically below D. The framework is kept in equilibriumin a vertical plane. Using Bow’s notation, find graphically, theforces acting on the rods AC, BC and AB.

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B

D A

C

W


- 153 -

17. A light inextensible string of length l4 passing over a smooth peg is connected to the endsof a uniform rod of weight w, and the rod is in equilibrium. If a weight w is attached toone end of the rod, show that the system can be in equilibrium again after the string slidesa distance l over the peg.

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18. A uniform right circular cylinder of height 40 cm and base radius 6 cm, is placed on arough horizontal plane with its axis perpendicular to the plane. The coefficient of frictionbetween the cylinder and the plane is 0.4. Determine whether the cylinder slides first ortopples first, if the horizontal plane is slowly tilted.

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- 154 -

19. The events A and B are such that 2011)( and

52)( ,

31)( =′== BA|PBPAP , where B′ is the

complementary event of B.Find (i) ),( BAP ∩

(ii) ),( BAP ∪(iii) ).( |BAP ′

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20. The events A and B are such that . )( and 1.02 )( ,2.0 )( xBAPxBPxAP =∩+=+=(i) If 0.7)( =∪ BAP , find the value of x.(ii) Verify that the events A and B are independent.(iii) Find )|( BAP ′ , where B′ is the complementary event of B.

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- 155 -

21. Let A and B be two events such that 51)( and

31)( ,

158)( === A|BPBPAP .

Find the probability that(i) both events occur,(ii) only one of the two events occurs,(iii) neither event occurs.

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22. In a street, there are two parking lots, each of which can accommodate only one vehicle.The probability that a parking lot is empty at 9.00 am on a Monday is 0.3. Find theprobability that, at 9.00 am on a Monday,(i) neither of the parking lots is empty,(ii) at least one of the parking lots is empty,(iii) both parking lots are empty.

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- 156 -

23. Neela and Mala enter a cake competition. The probability that Neela wins a prize is 61 and

that for Mala is 72 .

Assuming that the two events are independent, find the probability that(i) either Neela or Mala, but not both, wins a prize,(ii) at least one of them wins a prize.

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24. An engineer examines the operation of three groups of devices: 20% of all the devices ingroup A, 30% of all the devices in group B and 50% of all the devices in group C areexamined. The probabilities of failure of a device in each of the three groups are 0.002,0.003 and 0.005 respectively.(i) Find the probability that a failure in a device will occur in any of those groups.(ii) If a failure in a device is observed, find the probability that that device would have

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- 157 -

25. These 9 observations 5, 6, 13, 5, 10, 13, 3, x, y have a mean of 8 and a mode of 5.Find (i) the values of x and y,

(ii) the median of the set of observations.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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26. The mean and the standard deviation of 5 observations are found to be 10 and 2respectively. At the time of checking, it was found that one observation was erroneouslyrecorded as 6.Calculate the mean and the variance if the incorrect observation is(i) omitted,(ii) corrected as 12.

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- 158 -

27. The grouped frequency distribution of the term test marks of 30 students is given in thefollowing table:

Class interval 48 – 55 56 – 63 64 – 71 72 – 79Frequency 5 12 10 3

Find the mean, median and mode of the distribution.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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28. A company has 10 sales territories with approximately the same number of salesmenworking in each territory. The sales orders (in thousands) achieved during the last monthare as follows:

150, 130, 140, 150, 140, 300, 110, 120, 140, 120.Find the(i) inter-quartile range,(ii) mean deviationof the sales data.

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- 159 -

29. The mean and the standard deviation of the marks obtained in an examination by a groupof students are 42 and 15 respectively. The marks are now adjusted by using a linear scaleso that the mean and standard deviation become 50 and 20 respectively. Find the adjustedmark of a student who scores 54 marks at the examination.

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30. The length X (in cm) and weight Y (in gm) of 50 small insects are measured. The sum andthe sum of squares corresponding to those quantities are given below:

,21250

1=∑

=iix ,.8902

50

1

2 =∑=i

ix 26150

1=∑

=iiy and 61457

50

1

2 .=∑=i

iy .

Which is more varying, the length or the weight? Justify your answer.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 160 -

Part B

1. (a) An express train travels from station A to its next stop at station B. The distancebetween the two stations is d km. The uniform acceleration and retardation of the trainare f km s–2 and λ f km s–2 respectively, where λ is a positive constant. The greatestvelocity that the train can attain and maintain is v km s–1. The train starts from station Aat rest and stops at station B in a minimum time.(i) Draw the velocity-time graph for the motion of the train.

Hence, show that, if

+≥λ11

2

2

fvd then, the total time taken for the train to reach

station B is

++λ11

2 fv

vd .

(ii) If

+<λ11

2

2

fvd , make the necessary modification in your velocity-time graph

and hence, show that in this case, the total time for the train to reach station B is

+λ112

fd .

Find the average speed of the train.

(b) A person travelling due East with velocity u, feels the wind blowing from an acuteangle α North of East. When he starts travelling due North with velocity 2u, the windappears to blow from an acute angle β North of East. Draw, in the same figure, thevelocity triangles for both cases.Hence, find the direction of the wind.

2. (a) A ship is proceeding on a straight course with a uniform speed u km h–1. The distancefrom a port to the nearest point A on the course is a km. When the ship is b ( > a) kmaway from the port and has not reached the point A, a boat leaves the port to interceptthe ship. Prove that, the least uniform speed the boat must have in order to reach the

ship is bau km h–1.

Draw the path of the boat in a diagram.

Prove that, if the boat can go at a speed of v km h–1

>>bauvu , it can intercept the

ship in one of two specific positions and that the times taken by the boat for the two

positions differ by 22

22222vuuavb

−

− hours.

(b) A motor car of weight W has maximum power H. When the car is moving up a slopeinclined at an angle α to the horizontal, its maximum speed is v. When it is movingdown the same slope its maximum speed is 2v. Assuming that the resistances againstthe motion of the car in both cases are the same, find the resistance in terms of Wand α.


- 161 -

3. (a) A balloon released from rest at a point A at time t = 0, rises vertically with uniformacceleration f. At time t = T, a particle is projected from the point A vertically upwardsunder gravity with velocity u. Draw, in the same diagram, the velocity-time graphs forthe motions of the balloon and the particle.If the particle just touches the balloon, use the above graphs to prove that

))(( gfffTu ++= .

(b) Two particles A and B of masses m and 2m respectively are attached to the ends of aninextensible light string of length l which passes through a fixed smooth ring O. Theparticle B is vertically below O and the particle A is held in the same level as B suchthat the string is taut and A is away from B. The particle A is projected in a directionperpendicular to OA such that A describes a horizontal circular path with centre B andconstant angular velocity. Find the ratio AO : OB and the angular velocity of the particle A.

4. (a) A particle is projected under gravity from a point O on the ground with speed u at anangle α to the horizontal, in a plane perpendicular to a wall at a distance a from O . Ifthe particle is at a height y when it is at a horizontal distance x from O show that

).cossin(2sec 22 ααα yxugx −=If the particle just passes over the wall and falls on the ground at a point of distance d

from the wall, show that the height of the wall is αtanda

ad+

.

Find the maximum height attained by the particle in terms of a, d and α .

(b) A light inextensible string passes over a smooth pulley fixed to a ceiling and has aparticle of mass M attached to an end and a light smooth pulley attached to the otherend. Another light inextensible string passes over the second pulley and carries a massm1 at one end and a mass m2 at the other end. If the system is released from rest, show

that the particle of mass M will remain at rest provided that 21

114mmM

+= .

5. State (i) the law of conservation of momentum,(ii) the Newton’s law of restitutionas applied to the collision of particles.

Three equal smooth elastic particles P , Q and R are at rest at the points A , B, and Crespectively on a smooth horizontal table so that B is the mid-point of AC. The particle Pis projected along the table with velocity u and collides directly with the particle at Q. Ifthe particle P is involved in two collisions and that the particle R is involved in only onecollision, show that 223−>e , where e is the coefficient of restitution.


- 162 -

6. One end of a light elastic string is attached to a fixed point of a ceiling and the other end toa particle which hangs in equilibrium and causes an extension l in the string. Theequilibrium of the particle is disturbed, at time t = 0, by giving it a velocity gl2vertically downwards.Prove that(i) the maximum extension of the string is 3l,

(ii) the string becomes slack after a time gl

67π ,

(iii) the particle will not hit the ceiling, provided that the natural length of the string

exceeds 23l ,

(iv) the time needed for the particle to reach its maximum height is gl

+673 π when

the natural length of the string is greater than 23l .

7. A smooth hollow right circular cylinder of radius a is fixed with its axis horizontal. Let Obe a point on the axis of the cylinder and A, a point on the inner surface of the cylinderwith OA being horizontal and perpendicular to the axis of the cylinder. A smooth particle Pof mass m is projected from A vertically downwards with speed ag10 . When the particleP reaches the lowest point of the surface it collides directly with a smooth particle Q of

mass 2m which is at rest. The coefficient of restitution is 21 .

(i) Find the velocity of the particle Q just after the collision.(ii) Find the reaction between the particle Q and the inner surface of the cylinder when

OQ makes an acute angle θ with the upward vertical.

Deduce that the particle Q leaves the surface when OQ makes an angle

−

31cos 1

with the upward vertical.(iii) Show that the maximum height that the particle Q reaches above the horizontal level

through O is 2713a .

8. A smooth wedge of mass M stands on a smooth fixed horizontal plane. Each of the twosmooth sloping faces of the wedge is inclined at an angle θ to the horizontal. Two particlesP and Q with masses m1 and m2, are sliding down in a vertical plane along the lines ofgreatest slope, one on each face.(i) If the wedge is fixed to the plane, find the accelerations of the two particles P and Q.(ii) If the wedge is free to move on the horizontal plane find the accelerations of the two

particles and the wedge.Obtain the reactions of the wedge on the two particles.


- 163 -

9. (a) A particle is projected under gravity with speed u at an angle α inclined to thehorizontal. Obtain the expressions for the vertical and the horizontal components of thevelocity of the particle and the position of the particle after time t.If h is the greatest height and R is the horizontal range of the projectile show that

+=

hRhgu16

22

2 .

(b) A lorry of mass kgM with its engine working at constant power H kW, has a maximum

speed 1sm −u on a horizontal road. When the engine is working at the same power asbefore, the maximum speed of the lorry up a road inclined at an angle α to thehorizontal is 1sm −v . The resistance against the motion of the lorry is a constant. Showthat ( ) αsin1000 MguvvuH =− .

10. (a) Define the dot (scalar) product of two vectors.Let a and b be two non-zero vectors such that the dot product of and baba −+ iszero. Show that the vectors a and b are equal in magnitude.Using vector methods prove that, if the diagonals of a parallelogram are perpendicularto each other, then it is a rhombus.

(b) A uniform circular disc, whose plane is vertical, is free to turn about a smoothhorizontal axis through its centre C and perpendicular to the plane of the disc. Threeparticles of weights p, q and r are fixed to the rim of the disc at the points P, Q and Rrespectively, where PQR is an equilateral triangle. In the position of equilibrium thepoints P and Q are on the same side of the vertical line through C. Find the acuteangle that CP makes with the vertical.Deduce that the disc will be in equilibrium at any position if rqp == .

11. (a) The figure represents a frameworkconsisting of five smoothly jointedlight rods. The framework is smoothlyhinged at a fixed point A and carries aload W at C. The framework is kept inequilibrium in a vertical plane with AChorizontal and AB vertical by a force Papplied at B in a direction parallel toAC. Find the magnitude of P and thehorizontal and vertical components ofthe reaction at A.

Draw a stress diagram for the framework, using Bow’s Notation.Hence, determine the stresses in the rods in terms of W, distinguishing betweentensions and thrusts.

30o

D

60o

30o

A

B

C

W

P


- 164 -

(b) Four equal uniform rods PQ, QR, RS and SP each of weight w, are smoothly jointedto form a rhombus PQRS. It is suspended from the point P and kept in equilibrium ina vertical plane with the angle QRS equal to 2tan2 1− by a light inextensible stringconnected to the midpoints of the rods PS and RS.(i) Find the reactions at the joints R and S.(ii) Show that the tension of the string is 4w.

12. Find the position of the centre of gravity of a uniform circular arc, subtending an angle 2αat the centre of the arc.Hence, determine the centre of gravity of a uniform semicircular(i) arc,(ii) lamina.

A body as shown in the figure is formed by connectinga handle to a uniform semicircular lamina of radius 2aand mass m. The handle consists of a uniform rod oflength 4a and mass m together with a uniformsemicircular arc of radius a and a mass m. Find thecentre of gravity of the body.

The body can rest in equilibrium with thecircumference of the semicircular lamina in contactwith a smooth horizontal plane so that the plane of thebody is vertical. If the straight part of the handle makesan angle θ with the vertical prove that

)19(23tan−

=ππ

θ .

13. (a) Two uniform rods AB and BC of the same length but with different weights W1 andW2 (> W1) respectively, are freely jointed together at B. The system stands in a verticalplane with A and C on a rough horizontal table. Show that equilibrium is possible ifand only if µθ )3(tan)( 2121 WWWW +≤+ , where µ is the coefficient of frictionbetween the table and the rods, and θ is the angle that each rod makes with thevertical.

(b) A system consists of three forces F, λF and λ2F, where λ is a positive constant, actingrespectively along the sides BC, CA and AB of an acute-angled triangle ABC. Find themagnitude and the direction of the resultant. If the resultant meets AC at the point D,find the length of AD.If the resultant passes through the orthocentre of ABC, show that

)cos(coscos1 2

BABA +=+

λλ .


- 165 -

14. (a) Find the position of the centre of gravity of a solid uniform right circular cone.

The base radius and height of a solid uniform right circular cone are a and hrespectively. The ends of a light inextensible string of length l which passes over twofixed smooth pegs on the same horizontal level, are connected, one to the vertex andthe other to a point on the circumference of the base of the cone.If the cone is in equilibrium with the axis of the cone horizontal, prove that

222 )(4)2)(( dhadhlhlh −=−+− , where d is the distance between the two pegs.

(b) The figure below shows a framework consisting of seven smoothly jointed lightuniform rods of equal length:

The framework is smoothly hinged at A and two weights of 100 N each are placed at Band C. It is kept in equilibrium in a vertical plane with the rods AE, ED and BChorizontal, by a force P applied at D in the direction which makes an angle 60o to thehorizontal. Find the magnitude of P.

Using Bow’s notation determine the stresses in the rods B C , C D and ED,distinguishing between tensions and thrusts.

15. Find the position of the centre of gravity of a uniform hemispherical shell.Hence, determine the centre of gravity of a uniform solid hemisphere.

A body is formed by removing a hemispherical region of radius a from a uniform solidhemisphere of radius 2a, with the centres of both hemispheres coinciding. Determine thecentre of gravity of the body.

The body is placed in such a way that the curved surface is in contact with a smoothvertical wall and a rough horizontal plane. The coefficient of friction between the curvedsurface of the body and the horizontal plane is µ. If the body is in equilibrium regardlessof the inclination of the plane surface of the body, show that 45112 ≥µ .

A

B C

DE

P60o

100 N 100 N


- 166 -

16. The foot of a uniform ladder of weight 2w and length 4a rests on a rough horizontalground, and the top of the ladder rests against a smooth vertical wall. The ladder is inequilibrium in a vertical plane, making an angle θ with the downward vertical. Thecoefficient of friction between the ladder and the ground is µ .(a) Show that a man of weight 6w can climb safely to the top of the ladder provided that

θµ tan87

≥ .

(b) Suppose that 83 and

6<= µ

πθ

(i) Show that the minimum couple required to be applied to the ladder, for the manto climb safely to the top of the ladder is 8aw.

(ii) Find the maximum distance that he can climb along the ladder, carrying a weightof w.

17. (a) Four light rods are freely jointed at their ends and kept in a form of a parallelogramABCD by means of a fifth light rod BD. The angle BAD is α, AD = a and AB = b.The framework is kept in equilibrium on a smooth horizontal plane by introducingequal and opposite forces of magnitude P at A and C along CA and AC respectively.Find the forces in the rods AB and AD.

Prove that the thrust in BD is α

α

cos2cos2

22

22

abbaabbaP

++

−+ .

(b) A uniform circular disc of weight w and radius a can turn freely about a fixedhorizontal axis through its centre O perpendicular to its plane. Four particles ofweights w, 2w, 3w and 4w are attached to the disc at points A, B, C and D respectivelyon its rim such that ABCD is a square. Find the moment of the couple required tomaintain the equilibrium of the disc when AC makes an angle θ with the vertical.Hence, find the position of equilibrium of the disc if no couple is applied.

18. (a) A square hole is punched out of a uniform circular lamina. One diagonal of the squarehole is a radius of the lamina. Show that the centre of gravity of the circular lamina

with the hole, is at a distance 24 −π

a from the centre of the lamina, where a is the

radius of the lamina.

(b) A light rod AB of length 4a is pivoted at A and a weight w is attached at B. Twosmooth rings are fixed at the points C and D where the point C is at a height 2avertically above A and the point D is at a depth a vertically below A. The rod is kept inequilibrium with AB horizontal by connecting one end of a light inextensible string tothe point B with the string then passing through C and D and the other end connectedto the middle point of the rod. Find the tension in the string and the reaction at A.


- 167 -

19. (a) A box contains 6 identical balls out of which 5 are black and the other is white. Aniland Bimal are engaged in a game where a ball is selected randomly from the box at atime with each player taking alternate turns. The person who selects the white ball firstwill be the winner of the game. Anil starts the game. Assuming that the selected ball isnot replaced after each draw, construct the tree diagram that indicates how Bimal winsthe game. Hence, find the probability that Bimal wins the game.The game is now redefined so that the selected ball is replaced after each attempt.Find the probability that Anil wins the game(i) in his first attempt,(ii) in his third attempt,(iii) eventually.

Deduce the probability that Bimal wins the game eventually.Verify your answer by an independent method.

(b) The marks obtained by two students Amal and Kamal for the three subjects Sinhala,English and History at the yearend examination, and the weight to be applied to eachsubject are given in the following table:

Subject Sinhala English HistoryMarks of Amal 80 72 46

Marks of Kamal 64 82 40Weight 2 x 3

If the weighted mean marks of Amal and Kamal are the same, find the value of x andcalculate the weighted mean mark.

20. (a) Drivers are classified by an insurance company as low, average or high risk drivers.The company estimates that at present they have 25% low, 60% average and 15%high risk drivers in their records. The probabilities of such drivers encountering agiven number of accidents during a year are indicated in the following table:

RiskLow Average High

0 x 0.93 0.741 0.01 y 0.102 0.00 0.01 z3 0.00 0.00 0.01

Number of accidents peryear

≥ 4 0.00 0.00 0.00

(i) Find the appropriate values of x, y and z.(ii) Find the probability that a randomly selected driver had no accident in the year.(iii) If A had no accidents in the year, find the probability that he is a high risk driver.(iv) If B had no accidents for 4 such years, find the probability that he is a low risk

driver.


- 168 -

(b) The sum and the sum of squares of the times (in minutes) taken by 20 students to dotheir homework in Mathematics are 320 and 5840 respectively.(i) Calculate the mean and the standard deviation of the distribution of times taken

by the 20 students to do their homework in Mathematics.(ii) The time taken by another student to do his homework in Mathematics is added

and it is found that the mean is unchanged. Show that the standard deviation isdecreased.

(iii) The sum and the sum of squares of the times (in minutes) taken by another 10students to do their homework in Mathematics are 130 and 2380 respectively.Find the mean and the standard deviation of the times taken by all 30 students.

21. (a) An electronic system has 3 components R, S and T which work independently. Theprobabilities that the components R, S and T work during a year are 0.95, 0.90 and0.93 respectively.(i) If the system is designed so that at least 2 components must work for the system

to function, find the probability that the system will work during a year.(ii) If the system is redesigned so that it functions as long as at least one component is

working, find the probability that the system will work during a year.

(b) In an examination the pass mark of Mathematics is 30. The distribution of the marksof the students who passed the examination is as follows:

Number of studentsRange of Marks Boys Girls30 – 40 5 1540 – 50 10 2050 – 60 15 3060 – 70 30 2070 – 80 5 580 – 90 5 0

(i) Find the mean and the standard deviation of the marks of the(α ) 70 boys who passed,(β ) 90 girls who passedthe examination.

(ii) The overall mean and the standard deviation of the marks of boys including 30boys who did not pass are 48 and 21.5 respectively. The corresponding valuesfor girls including 10 girls who did not pass are 50 and 15 respectively.

Find the mean and the standard deviation of the marks of the(α ) 30 boys who did not pass,(β ) 10 girls who did not passthe examination.

(iii) Deduce the mean and the standard deviation of the marks of the students who(α ) passed,(β ) did not passthe examination.


- 169 -

11 - Higher Mathematics










Higher Mathematics I: Part A - 30 12 2010


- 170 -

11 - Higher MathematicsPaper I

Important :

* Answer all questions of part A.* Answer five questions only of part B.

Part A

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2. Let R be the relation defined on IR × IR by cbdadcRba +=+⇔),(),( . Show that R is anequivalence relation on IR × IR .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 171 -

3. Let :f IR →

−21 IR

−21 be defined by

123)(−

+=xxxf . Show that 1−f exists. Without

finding 1−f , show that f −1 = f.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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4. If a, b and c are positive, prove that 32

2

2

2

2

2

≥++≥++ab

bc

ca

ac

cb

ba .

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- 172 -

5. The transformation Axx =′ , where

=

2312

A ,

=yx

x and

′

′=′yx

x , maps the point

(x, y) of the xy-plane to the point ( )yx ′′, .Find(i) the equation of the straight line to which the straight line 0=+ yx is mapped,(ii) the values of m for which the straight line mxy = is mapped onto itselfby the above transformation.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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6. Prove that iii

−=−

+6

3

)1()31( .

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- 173 -

7. Show that the function ,2 if ,2

2 if ,28

)(

3

=

≠−

−

=

x

xxx

xf

is not continuous at 2=x .Do the necessary modification to )(xf so that the function is continuous at 2=x .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. Prove that the function

<

≥=

1if ,1if ,

)( 2 xxxx

xf

is not differentiable at x = 1.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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- 174 -

9. )(xφ is an integrable function of x such thatbxax =−+ )()( φφ for all real values of x, where a (> 0) and b are constants.

Show that 2

d)(0

abxxa

=φ∫ .

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10. In the same diagram draw the two curves 1Γ and 2Γ given respectively, in polar

coordinates, by 23ar = and θcos4ar = , where

20 and0 π

θ ≤≤>a .

Shade, in the diagram, the region given by θcos823 ara ≤≤ .

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* *

Higher Mathematics I: Part B - 30 12 2010


- 175 -

Part B

11. (a) In a certain school, there are 120 students in grade 12 classes, of which 35 are girls. 65students including all the prefects participate in sports. 13 of the girls do notparticipate in sports and there are 7 girl prefects. There are 38 students whoparticipate in sports but who are not prefects. Represent this information in a Venndiagram.Find the number of boys who(i) are not prefects but participate in sports,(ii) do not participate in sports,(iii) are prefects.

(b) Let A, B and C be subsets of a universal set S.Prove that { } { } { }ACBBCACBACBA −∩∪−∩=∩∩−∩∪ )()()()( ,stating clearly all the laws of algebra of sets used.

12. (a) Let 1

2)( 2

2

+

++=

xbaxxxf , where a, b ∈ IR and a ≠ 0. Show that there are two distinct

real values k1 and k2 for k such that f(x) – k is of the form 1)(

2

2

+

+

xBAx , where A and B

are independent of x.Show that

(i) )1)(1(])1[()(

2

2

+−

+−=−

xkaxk

rkxfr

r , for r = 1, 2,

(ii) (1 – k1)(1 – k2) −= a2.Deduce that for all real values of x, )(xf lies between k1 and k2.

(b) Factorize 333 ))(())(())(( cbababacacacbcb −+−+−+−+−+− .

13. Using De Moivre’s theorem for any positive integral index, obtain expressions for θ5cosand θ5sin in terms of θcos and θsin .Hence, obtain θ5tan in terms of θtan .Considering the equation ,05tan =θ prove that the roots of the equation

05102 =+− xx are

5

tan2 π and

52tan2 π .

Hence, show that 11252sec

5sec 44 =

+

ππ .

Higher Mathematics I: Part B - 30 12 2010


- 176 -

14. (a) Find the height of the right circular cylinder of maximum volume that can be inscribed

in a sphere of radius a. Show that the maximum volume of the cylinder is 3

934 aπ .

(b) Find the constants a and b such that xbxay sincos += is a solution of the differential

equation xyxy

xy sin2dd

dd2

2=−− .

(c) Find the general solution of the differential equation xeyxy

=+ 2dd .

15. (a) Evaluate ∫π

+0

dsin1

xx

x .

(b) Let 2

2

cos211)(

xxxxf

+−

−=

θ, where θ is independent of x. Show that the Maclaurin

expansion of )(xf upto the term in x2 is θθ 2cos2cos21 2xx ++ .

16. (a) Let )2,( 2 apapP and )2,( 2 ararR be two distinct points on the parabola 042 =−≡ axyS .Find the equation of the line PR in its simplest form.If PR is a focal chord of the parabola S = 0, obtain the condition that must be satisfiedby p and r.QT is also a focal chord of the parabola S = 0. Show that both pairs of opposite sides ofthe quadrilateral PQRT intersect on the directrix of the parabola S = 0.

(b) The distance between the foci of an ellipse is 8 units and that between the directrices is18 units. Find the equation of the ellipse referred to its principal axes.

17. (a) If a, b and c are constants and 222 cba >+ , show that there exist two distinct valuesα and β between 0 and π2 which satisfy the equation cba =+ θθ sincos .

Prove that, 22

22

2cos

bac+

=

− βα .

(b) Let θθθ 66 sincos)( +=f , where θ is real. Express )(θf in the form θkBA cos+ ,where A, B and k are constants.Hence solve the equation 054sin2)sin(cos4 66 =−−+ θθθ

(c) Solve the equation 4

2tantan 11 π=+ −− xx .

* * *

Higher Mathematics II: Part A - 30 12 2010


- 177 -

11 - Higher MathematicsPaper II

Important :

* Answer all questions of part A.* Answer five questions only part B.

Part A

1. Let the sides OA and OB of the triangle OAB represent the vectors a and b respectively.

If a=a and b=b , using the scalar product show that,

(i) abc ba += is parallel to the internal bisector of the angle AOB,

(ii) c and ba – ab are perpendicular.

Hence, write down a vector parallel to the external bisector of the angle AOB.

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2. With respect to the system of coordinates Oxyz, the coordinates of the points A, B and C are)0,0,1( , )0,2,0( and )3,0,0( respectively. Forces of magnitude P and P2 at A and B

act in the directions of BA and AC respectively. If the system is equivalent to a single forceR and a couple of moment G at the originO , find R and G.

Compute R.G and determine whether the system can be reduced to a single force.

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- 178 -

3. A uniform square sheet of side a, is completely immersed in a liquid, with its plane makingan angle θ with the vertical and two of the sides of the sheet being horizontal. The centre ofthe sheet is at a depth of H below the free surface. Find the total liquid thrust that acts on thesheet.

Determine the location of the centre of pressure.

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4. Let a and b be constant vectors and r(t), be the position vector, at time t, of a point Prelative to a fixed origin O.

(i) What can be said of the direction of a × b in relation to that of a or b?

(ii) Show that rarar & . ) (2) (dd 2 −=−t

.

Given that ) ( arbr −×=& , deduce that ar − is a constant.

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- 179 -

5. Two thin circular discs A and B have the same mass and same thickness. The densities of Aand B are )( and 121 ρρρ > respectively. Let 21 and RR be the respective radii of A and B.Let 21 and II be the moments of inertia of A and B respectively, about their axes through

the centres and perpendicular to the discs. Find 1

2ρρ and

1

2II in terms of

1

2RR .

Deduce that 21 II > .

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6. A car of mass M is cruising at a steady velocity u on a level straight road with its engine atfull power. If the engine is switched off at time t = 0, and the car is allowed to coast along

the same road, its velocity v at time t is given by constant. a is where,)0(,1

αα

≥+

= tt

uv

Find the power at t = 0, in terms of M, u and α.

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- 180 -

7. A sales manager receives 6 telephone calls on average between 9.00 a.m. and 10.00 a.m. ona weekday. Find the probability that he will receive exactly 2 calls between 9.00 a.m. and9.30 a.m.'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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8. The continuous random variable X has probability density function),2()( xkxxf −= 20 ≤≤ x .

(i) Show that 43

=k .

(ii) Find )1( ≤XP .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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9. Let [ ]4 2, ~ UX . Show that 31)Var( =X .

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10. Let 64) (10, ~ NX . Find the value of a such that 95.0)( =< aXP .'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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Higher Mathematics II: Part B


- 182 -

Part B

11. A system of forces is equivalent to a single force R and a couple of moment G at thepoint O. If the system is equivalent to a single force R′ and a couple of moment G′ atthe point O′ , derive the expressions forR′ and G′ .Show that the necessary condition for the system to reduce to a single force is 0. =GR .Three forces Li, Mj and Nk act at the points of position vectors ck, ai and bj respectively,where a, b and c are non-zero constants and i, j and k have their usual meanings. Findthe equivalent single force R and the moment G of the couple at the point O .( i ) Can the system be reduced to a single force ? Justify your answer.(ii) If the system is equivalent to a single forceR′ and a couple of moment R′λ at some

point, show that cLMbNLaMNNML ++=++ )( 222λ .

12. A circular lamina of radius a, is completely immersed in a liquid, with its plane makingan angle θ to the vertical. If the centre of the lamina is at a depth H below the free surfaceof the liquid, find the total fluid thrust acting on the lamina.( i ) A solid uniform hemisphere of radius a is completely immersed in a liquid such that

its plane surface makes an angle θ to the vertical and the curved surface is below theplane surface. When the centre O of the plane surface is at a depth H below the freesurface, determine the inclination to the horizontal of the resultant liquid thrust onthe curved surface.

(ii) As θ varies while H is fixed, determine θ when the line of action of the resultantliquid thrust on the curved surface passes through O.

13. A steady draught of air in a fixed vertical tube blows vertically downwards with velocityu. An insect of mass m jumps up vertically with velocity V from a point O on the tube. Inaddition to the gravitational force mg, the insect experiences an air resistance which is mktimes the velocity of the draught of the air relative to the insect, k being a positiveconstant.Write down the equations of motion for the insect on its way vertically upwards andvertically downwards.By considering appropriate equations of motion, find(i) the time taken for the velocity of the insect to be zero,(ii) the velocity of the insect when it reaches the point O.

14. Three identical particles P1, P2 and P3 , each of mass m, interact with each other according

to the law of gravitation 2rGmMF = , in the usual notation. At time t, let the particle P1,

P2 and P3 be at the points with position vectors r1, r2, and r3 with respect to the origin G,the centre of mass of the three particles.(i) Show that 0rrr =++ 321 .(ii) Write down the equation of motion for P1.(iii) If the system moves with constant and equal distances d between all three pairs of

particles, show that 1313 rrdGm

=&& .

(iv) Show that )cos(11 εω += tar , where a1 is a constant vector and ω, ε are constants ,is a possible solution of the equation of motion given in (iii) and determine ω.

r1

:

Higher Mathematics II: Part B


- 183 -

15. Find the moment of inertia of a uniform circular ring of mass m and radius a, about theaxis through the centre and perpendicular to the plane of the ring.Deduce the moment of inertia of the ring about an axis through a point on thecircumference of the ring and perpendicular to the plane of the ring.A uniform circular ring of mass m and radius a rolls without slipping on a roughhorizontal plane such that the plane of the ring is vertical. When its velocity is V, the ring

collides with a smooth peg A of vertical height 5a above the level of the horizontal plane.

Determine the angular velocity of the ring about A, just before the collision.Show that if the ring leaves the peg immediately, then gaV 8081 2 ≥ .If the ring goes past the peg, find the possible values of V.

16. State under what conditions a Geometric distribution, whose probability mass function is...,2,1,0;)1()( 1 =−== − xppxXP x , where 10 << p , can be used in a real life situation.

Find )(XE .

[You may use the result 21 )1( r

rxrx

x

−=∑

∞

=

.]

In a certain city, the probability that any telephone box is occupied is 51 . Let X be the

number of telephone boxes which will have to be tried before a person finds a telephonebox which is not occupied.(i) Write down the probability mass function of X.(ii) Find )(XE .(iii) Find the probability that a person wishing to make a call will find a telephone box

which is not occupied only at the 6th telephone box tried.

(iv) Given that the variance of X is 165 , determine the mean and the variance of Y, where

45 −= XY .

17. The probability that a randomly chosen flight from an airport is delayed by more than x

hours is 2)10(1001

−x , 100 ≤≤ x .

Assume that no flights leave early and none is delayed for more than 10 hours. Let thedelay, in hours, of a randomly chosen flight be denoted by X.(i) Find the cumulative distribution function and hence, find the median of X.

(ii) Obtain the density function of X and show that 310)( =XE .

(iii) Compute the probability that a randomly selected flight is delayed between 2 and 3hours.

A random sample of 3 such flights is taken. Find the probability that at least one of these3 flights is delayed between 2 and 3 hours.

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GCE (Advanced Level) Examination