Chapter-Wise SolutionsQUESTION BANK

Mathematics

Strictly Based on the Latest Syllabus issued by CBSE Board for 2016 Examination

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11.05

CONTENTS

• Syllabus v - viii

• Solved Paper (KVS), 2015 9 - 16

1. Sets 1 - 12

2. Relations and Functions 13 - 20

3. Trigonometric Functions 21 - 37

4. Principle of Mathematical Induction 38 - 46

5. Complex Numbers and Quadratic Equation 47 - 57

6. Linear Inequalities 58 - 69

7. Permutations and Combinations 70 - 79

8. Binomial Theorem 80 - 91

9. Sequences and Series 92 - 107

10. Coordinate Geometry 108 - 126

11. Conic Sections 127 - 145

12. Introduction to Three Dimensional Geometry 146 - 150

13. Limits and Derivatives 151 - 163

14. Mathematical Reasoning 164 - 171

15. Statistics 172 - 184

16. Probability 185 - 199

Note : BSM - Board Supplementary Material, NCT - National Capital Territory, KVS - Kendriya VidyalayaSangathan

CBSE always believes in Global Trends of Educational Transformation. The CBSE curriculum gets its lead from National Curriculum Framework – 2005 and Right to Free and Compulsory Education Act – 2009. The aim of CBSE Curriculum is not just to let learners obtain basic knowledge but to make them life-long learners. CBSE always updates and reviews the syllabus to make it more relevant with educational transformation and in last few years the chapters and topics which CBSE has added are very interesting and increase practical knowledge.

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PREFACE

(iv)

Highlights of Curriculum Document for the examination to be held in March, 2016

Curriculum 2014-15 Curriculum 2015-16 final for the examination(Printed in 2014) to be held in March 2016

SyllabusCourse Structure

One Paper Total Hours—Periods of 35 Minutes eachThree Hours Max. Marks. 100

No. Units No. of Periods Marks

I. Sets and Functions 60 29

II. Algebra 70 37

III. Coordinate Geometry 40 13

IV. Calculus 30 06

V. Mathematical Reasoning 10 03

VI. Statistics and Probability 30 12

Total 240 100

*No chapter/unit wise weightage. Care to be taken to cover all the chapters.

Unit-I : Sets and Functions

1. Sets : 20 Periods

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of aset of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams.Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets.

2. Relations & Functions : 20 Periods

Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finitesets. Cartesian product of the sets of reals with itself (upto R x R x R). Definition of relation, pictorial

Unit-I: Sets and FunctionSub unit-2: Relations & Functions :

Unit-VI: Statistics and ProbabilitySub unit-1: Statistics

Question Paper Design (XI-XII)

Understanding Based - Number of LA-IIquestions- 1, Total Marks-16, % Weightage- 16%

Application Based - Number of LA-I questions-3,Total Marks-25, % Weightage- 25%

HOTS- Number of LA-II questions- 2, TotalMarks-21, %weightage-21%

Evaluation and Multi- disciplinary Based -Number of LA-I questions- 3, Total Marks-18, %Weightage- 18%

Topic(s) Added :Domain and range of exponential, logarithmicfunction (Page 95)

Topic Added: Range (Page 97)

Question Paper Design(XI-XII) (Page 98,103)

Understanding Based- Number of LA-II questions-2, Total Marks-22, % Weightage- 22%

Application Based - Number of LA-I questions-4,Total Marks-29, % Weightage- 29%

HOTS- Number of LA-II questions- 1, TotalMarks-15, %weightage-15%

Evaluation Based - Number of LA-I questions- 2,Total Marks-14, % Weightage- 14%

[ vi ]

diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorialrepresentation of a function, domain, co-domain and range of a function. Real valued functions,domain and range of these functions; constant, identity, polynomial, rational, modulus, signum,exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, productand quotient of functions.

3. Trigonometric Functions : 20 Periods

Positive and negative angles. Measuring angles in radians and in degrees and conversion from onemeasure to another. Definition of trigonometric functions with the help of unit circle. Truth of theidentity sin2 x + cos2 x=1, for all x. Signs of trigonometric functions. Domain and range of trignometricfunctions and their graphs. Expressing sin (x ± y) and cos (x ± y) in terms of sinx, siny, cosx & cosyand their simple application. Deducing the identities like the following :

tan (x ± y) =

tan tan

1 tan tanx y

x y , cot (x ± y) = cot cot 1

cot cotx yy x

sin ± sin = 2sin1

2( ± ) cos

1

2( )

cos + cos = 2cos1

2( + ) cos

1

2( - ),

cos – cos = – 2sin1

2( + ) sin

1

2( - ),

Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric

equations of the type siny = sina, cosy = cosa and tany = tana.

Unit-II : Algebra

1. Principle of Mathematical Induction : 10 Periods

Process of the proof by induction, motivating the application of the method by looking at naturalnumbers as the least inductive subset of real numbers. The principle of mathematical induction andsimple applications.

2. Complex Numbers and Quadratic Equations : 15 Periods

Need for complex numbers, especially 1 , to be motivated by inability to solve some of the quardraticequations. Algebraic properties of complex numbers. Argand plane and polar representation of complexnumbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with realcoefficients) in the complex number system. Square root of a complex number.

3. Linear Inequalities : 15 Periods

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representationon the number line. Graphical representation of linear inequalities in two variables. Graphical methodof finding a solution of system of linear inequalities in two variables.

4. Permutations and Combinations : 10 Periods

Fundamental principle of counting. Factorial n. (n!)Permutations and combinations, derivation offormulae for

rpn and rcn and their connections, simple applications.

5. Binomial Theorem : 10 Periods

History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle,General and middle term in binomial expansion, simple applications.

[ vii ]

6. Sequence and Series : 10 Periods

Sequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean

(G.M.), relation between A.M. and G.M. Formula for the following special sum

2 3

1 1 1, and

n n n

k k kk k k

Unit-III : Coordinate Geometry1. Straight Lines : 10 Periods

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line andangle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Equationof family of lines passing through the point of intersection of two lines. Distance of a point from aline.

2. Conic Sections : 20 PeriodsSections of a cone: circles, ellipse, parabola, hyperbola; a point, a straight line and a pair ofintersecting lines as a degenerated case of a conic section. Standard equations and simple propertiesof parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three–dimensional Geometry : 10 PeriodsCoordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance betweentwo points and section formula.

Unit-IV : Calculus1. Limits and Derivatives : 30 Periods

Derivative introduced as rate of change both as that of distance function and geometrically.Intutive idea of limit. Limits of polynomials and rational functions, trignometric, exponential andlogarithmic functions. Definition of derivative, relate it to slope of tangent of a curve, derivative ofsum, difference, product and quotient of functions. Derivative of polynomial and trignometricfunctions.

Unit-V : Mathematical Reasoning1. Mathematical Reasoning : 10 Periods

Mathematically acceptable statements. Connecting words/phrases - consolidating the understandingof “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”,“there exists” and their use through variety of examples related to real life and Mathematics. Validat-ing the statements involving the connecting words difference between contradiction, converse andcontrapositive.

Unit-VI : Statistics and Probability1. Statistics : 15 Periods

Measures of dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

2. Probability : 15 PeriodsRandom experiments; outcomes, sample spaces (set representation). Events; occurrence of events,‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic)probability, connections with theories of earlier classes. Probability of an event, probability of ‘not’,‘and’ and ‘or’ events.

[ viii ]

Mathematics (Code No. 041)

Question Paper DesignClass XI (2015-16)

Time : 3 Hours Max. Marks. 100

S. Typology of Questions Very Long Long Marks % No. Short Answer I Answer Weightage

Answer (4 marks) II(1 marks) (6 marks)

1. Remembering-(Knowledge based Simplerecall questions, to know specific facts,terms, concepts, principles, or theories, 2 3 1 20 20%Identify, define, or recite, information)

2. Understanding-(Comprehension to befamiliar with meaning and to understand 2 2 2 22 22%conceptually, interpret, compare, contrast,explain, paraphrase information)

3. Application (Use abstract information inconcrete situation, to apply knowledge to 1 4 2 29 29%new situations, Use given content tointerpret a situation, provide an example,or solve a problem)

4. High Order Thinking Skills (Analysis& Synthesis- Classify, compare, contrast,or differentiate between different pieces of 1 2 1 15 15%information, Organize and/or integrateunique pieces of information from a varietyof sources)

5. Evaluation -(Appraise, judge, and/orjustify the value or worth of a decision – 1+1 1 14 14%or outcome, or to predict outcomes based (Valueon values) based)

TOTAL 6x1=6 13x4=52 7x6=42 100 100%

QUESTION WISE BREAK UP

Type of Question Mark per Question Total No. of Questions Total Marks

VSA 1 6 06

LA-I 4 13 52

LA-II 6 7 42

Total 26 100

1. No chapter wise weightage. Care to be taken to cover all the chapters.2. The above template is only a sample. Suitable internal variations may be made for generating similar templates

keeping the overall weightage to different form of questions and typology of questions same.

SECTION - ‘A’ 1. Find the A∩ (B ∩ C) if A = {1, 3, 5, 8}; b = {3, 5, 7}

and C = {2, 4, 6, 8} 2. Find the coefficient of x5 in (x + 3)8. 3. Find the equation of the parabola with vertex at (0,

0) and focus at (0, 2) 4. Write the negation of the statement. Chennai is the capital of Tamil Nadu. 5. Write the contrapositive of the statement : If x is a prime number, then x is odd.

6. Write the converse of the statement : If a number is divisible by 10, it is divisible by 5.

SECTION - ‘B’ 7. Let A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}. Find

(i) A × (B ∩ C) (ii) (A × B) ∩ (A × C)

8. Let A = {1, 2, 3, ...,................ 14} and R is a relation defined by R = {(x, y) : 3x – y = 0}, where x, y ∈A}. (i) Write relation R in Roster form (ii) Find its domain, codomain and range.

9. Prove that : Cos 6q = 32 cos6 q – 48 cos4 q + 18cos2 q – 1.

10. Find the general and principal solution of

cos x =

32

OR

In ABC if a = 18, b = 24 and c = 30 find cos A and cos B.

11. Find the sum of the following series up to n terms :

5 + 55 + 555 + ....................

12. Convert complex number – 3 i+ in polar form.

13. Prove the following using mathematical induction : 102n–1 + 1 is divisible by 11 for all n ∈ N.

14. If the 3rd, 4th and 5th terms in the expansion of (x + a)n are 84,280 and 560 respectively, find x, a and n.

15. How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?

OR

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’ s not come together ?

16. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has

(i) no girl ?

(ii) at least one boy and one girl ?

(iii) at leat 3 girls ?

17. Find equation of the line passing through the point (– 1, 3) and perpendicular to the line 3x – 4y – 16 =0.

OR

If ‘p’ and ‘q’ are lengths of perpendiculars from the origin to the lines

xsec q + ycosec q = k and xcos q + ysin q

= kcos 2q, then prove that 4p2 + q2 = k2.

18. Find the co-ordinates of the foci, the vertices, the length of major axis and eccentricity of the ellipse 4x2 + 9y2 = 36.

19. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, – 10) and R (8, 14, 2c),

KENDRIYA VIDHYALAYA SANGTHANSESSION ENDING EXAMINATION 2014-15

SUBJECT : MATHEMATICSCLASS–XI

(SOLVED PAPER)

Duration : 3 Hrs. M.M. : 100

General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 26 questions divided into three sections A, B and C. Section A comprises of 6 questions

of 1 mark each, Section B comprises of 13 questions of 4 marks each and Section C comprises of 7 questions of 6 marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. (iv) There is no overall choice, However, internal choice has been provided in 4 questions of 4 marks each and 2 questions

of 6 mark each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted.

Oswaal CBSE Question Bank chapter-wise solutions For Class 11 Mathematics

Publisher : Oswaal Books ISBN : 9789351275794 Author : Panel Of Experts

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