CSEC Maths January 2006

7/30/2019 CSEC Maths January 2006

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rEsr coDE 01234020FORM TP 2006017 TANUARy 2006CARIBBEAN EXAMINATIONS COUNCIL

SECOIYDARY EDUCATION CERTIFICATE. EXAMINATIONMATHEMATICS

Paper 02 - General Profrciency2 hoars 40 minutes

04 JANUARY 2006 (a.m.)

INSTRUCTIONS TO CANDIDATESl. Answer ALL questions in section I, and ANy rwo in section II.2. Write your answers in the booklet provided.3- All working must be shown clearly.4. A list of formulae is provided on page 2 of this booklet.

Examination Materials

t

Electronic calculator (non-programmable)Geometry setMathematical tables (provided)Graph paper (provided)rL

DO NOT TURN THIS PAGE T]NTIL YOU ARB TOLD TO DO SOCopyright @ 2004 Caribbean Examinations [email protected] rights reserved.

O 123 40/20 J ANUARY/F 2006


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Page3CTION I

Answer ALL the questions in this section.All working must be clearly shown.

1_ I52(ii) correct to 3 significant figures, rhe value of 1g.75 _ (zJDz. ( 3 marks)

(b) A loan of $12 000 was borrowed from a bank at l4vo per annum.Calculate

( 4 marks)

( 2 marks)( lmark)

(i) the interest on the loan at the end of the first year(ii) the total amount owing at the end of the first year.A repayment of $7 800 was made at the start of the second year.Calculate

(iii) the amount still outstanding at the start of the second year ( I mark )(iv) the interest on the outstanding amount at the end of the second year.( lmark)Total 12 marks

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2. (a)(b)

Given that m = -2 and n = 4, calculate the value of (2m + n)(2m - n) -Solve the simultaneous equations5x+6y=372x-3y- 4.Factorise comPletelY(i) 4xz - 25(ii) 6p - 9Ps + 4q - 6qs(iii) 3x2 + 4x - 4.

Page 4

( 2 marks)

( 4 marks)( 2 marks)( 2 marks)( 2 -aSqL-

Total 12 marks

( 3 marks)

(c)

(a)

o)3. Given the formula t = Lr{" + v)r, express r in terms of v' s, and t't,

on a certain day, 300 customers visited a bakery that sells bread and cakes'70 customers bought cakes onlY80 customers bought neither bread nor cakes2x customers bought bread onlYr customers bought both bread and cakes

(ii)(iii)

(i) urepresents the setof customers visiting thebakery on thatday,-Brepresents theset of customers who bought bread, *a C represents the set of customers whobought cake. Copy anO comptete the Venn diagram to illustrate the information'( 3 marks)Write an expression in x to represent the TOTAL number of customers whovisited the bakery on that day. ( 2 marks)calculate the number of customers who bought bread ONLY- ( 3 marks)

Total ll marksGO ON TO TI{E NEXT PAGE

a


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4. ("4 The equation of the line / is ! = 4x + 5.(i)

(ii)State the gradient of any line that is parallel to /.

Page 5

( lmark)

( 2 marks)( 4 marks)

Total 12 marks

Determine the equation of the line parallel to I that passes rhrough the point (2,4).( 3 marks)(b) An answer sheet is provided for this question.

The diagram above shows the positions of three cities, A, B and C on the north coastof Africa. The scale of the map is I : 20 000 000.(Jse your answer sheet when onr*l,"ring the following questions. Show all lines andangles used in your calculations-(i) Measure and state, in centimetres, the length of the line segmenl, BC.( 2 marks)

*\rtiil Hence, calculate in kilometres, the\ city c. actual shortest distance from City B totrBsing a protractor, determine the bearing of B from A.T:=--

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5. The curved surface area of a cylinder = 2nrh,surface area of a sphere is 4n f .Page 6

where r is the radius and ft is the height, and the

The diagram above, not drawn to scale, shows a solid glass paperweight which consists of ahemisphere mounted on a cylinder.The radius of the hemisphere is 3 cm, the radius of the cylinder is 3 cm and its height is 8 cm.(a) Calculate, using n = 3.14,

(D the curved surface area of the cylinder ( 2 marks)(ii) the surface area of the hemisphere ( 2 marks)(iii) the TOTAL surface area of the solid paperweight. ( 2 marks)Using a ruler, a pencil, and a pair of compasses, construct the parallelogram KLMN, inwhich KL = 8cm,KN = 6cm, andZLKN=60?. ( Smarks)

Total 11 marks

(b)

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6.PageT

An answer sheet is provided for this question.The diagram below shows quadrilateral PQRS, and its image, quadrilateral p'e'R,y,after ithas been rotated.

(a)

' (b)(c)

State the coordinates of the points R' and S.Describe the rotation completely.


Total 10 marks

on the answer sheet provided, drarv and label quadrilaterql p'e,RT,,which is-theimage of'P'Q'R'S' after it has been?eflebted'in the x-axis. - ( 3 marks)Describe completely, the single transformation that maps quadrilateral ppR,i ontoP'Q'R'S'. ( 2 marks)(d)

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7.

Page 8

The data below are the lengths, to the nearest centimetre, of the right foot of the 25 students ina class.

t415t616t7

State the lower boundary of the class interval 14 - 16.State the width of the class intervalz0 -22.

to 20 cm.State the modal.length of a student's right foot.

18 2018 2018 2t19 22t9 22

22 2422 2s22 2523 2623 27

(a) Copy and complete the following grouped frequency table for the data above.Length of Right Foot(cm) Frequency

t4-16l7-t920-2226-28

4

852

o)(c)(d)

( 2 marks)( lmark)( lmark)

( 2 marks)( lmark)

A student's right foot measured 16.8 cm. State the class interval in which this lengthwould lie. ( lmark)A student was chosen at random from the group, and the length of his right foot wasmeasured. Calculate the probability that the length was GREATER THAN or EQUAL

Calculate an estimate of the mean length of a student's right foot using the midpoints ofthe class intervals in (a) above. ( 3 marks)Total ll marks

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Page 9

8. The path of a ball thrown in the air is given by the equation h = 20t - 5P where ft is thevertical distance above the ground (in metres) and t is the time (in seconds) after the ball wasthrown.The table below shows some values of r and the corresponding values of h, correct to 1 decimalplace.

Copy and complete the table of values. ( 2 marks)Drawthegraph of h=2ot - SPfor 0


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9.

Page l0SECTION II

Answer TWO questions in this sectionALGEBRA AND RELA-TIONS, FUNCTIONS AND GRAPHS

(a) Solve the pair of simultaneous equations( 6 marks)

3 in the form a (x + h)2 + ft, where a, h and,ft are real numbers.( 3 rmrks)using your answer from (b) above, or otherwise, carculate(i) the minimum value of 4x2 - lb - 3 ( r mark )(ii) the value of -r for which the minimum @curs ( I mark )(iii) the values of x for which 4x2 _ lbc _ 3 = 0 expressing your answers to 3significant figures. ------p '--^ l' i'_".u"1

2x+y=7*-ry=6-Express 4x2 - l2x -b)

(c)

Total 15 marks


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10. (a)page ll

A shop stocks -r $onix and, y zentradios. It has shelf space for up to 20 radios.(i) Write an inequality to represent this information. ( lmark)The owner of the shop spends $150 to purchase each sonix radio and $300 for eachzentradio, she has $4 500 to spend on the purchase of these radios.(ii) Write an inequality to represent this information. ( lmark)The owner of the shop decides to stock at least 6 Sonix and at least 6 Zent radios.

(b)(iii) Write TWO inequalities to represent this information.(i) Using a scale of 2 cm to represent 5 Sonix radios5 Zent radios, draw the horizontal axis for 0 < x


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11.

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GEOMETRY AND TRIGONOMETRY

In the diagrym above, not drawn to scale, O is the centre of the circle, Z AOB = 130",Z DAE = 30o, and, AEC and, BED are chords of the circle.(a) Calculate the size of EACH of the following angles, givirig reasons for EACH step ofyour answers.

(i) Z ACB(iD z cBD(iiD z AEDShow that LBCE and MDE are similar.Given that CE = 6 cm, EA = 9.1 cm and DE = 5cm,(i) calculate the length of EB ( 3 marks)(ii) calculate correct to 1 decimal place the areaof MED. ( 3 marks)

Total 15 marks

, ( 2 marks)( 2 marks)( 2 marks)( 3 marks))

(c)

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13.

Page 14

VECTORS AND MATRICESThe points A (1,2), B (5,2), C (6, 4) and D (2,4) are the verrices of a quadril ateral ABCD.(a) /"\Express in the -* [, J

(i) the position vectors A,A,(ii) the vectors B unO --)DC

-) -+OC, and OD where O is the origin (0, 0)( 2 marks)( 2 marks)

- r-+r - -+calculate lABland hence determine the unit vector in the direction of AIi.I I (2marks)Using the answers in (a) (ii),(D state TWO geometrical relationships between the line segments AB and DC(iD explain why ABCD is aparallelogram.


(d) Using a vector method, determine the position vector of G, the midpoint of the line AC.Hence, state the coordinates of the point of intersection of the diagonals AC and, BDof parallelogranABCD. ( 5 marks)Total 15 marks

(b)

(c)

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14. (a)

(b)

rhe marrix t = (; ^.)(i) Calculate, in terms of x, the determinant of L.(ii) Calculate the values of x given that L is singular.rhe matrix , = (1 |(D Find lfl, the inverse of M.

- (ii) Hence, calculare the value of .r and of yfor which , g) = (13 )( 6 marks)(c) The image, (/, t'), of any point, (x, y), under a transformation N is given by the equation

(r= (a :) (;). (:,)Calculate(i) the image, (f ,t'), of (3, -l) under.ty' ( 3 marks)(ii) the coordinates of the point, (r, y), which is mapped by N onto (7, 4).( 3 marks)

END OF TEST

Page 15

( 3 marks)

Total 15 marks

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CSEC Maths January 2006