caribbean examinations council caribbean ......pure mathematics unit 1 - paper 02 algebra, geometry...

IFORM TP 2016279

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TEST CODE 02134020 IMAY/JUNE 2016

CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@

PURE MATHEMATICS

UNIT 1 - Paper 02

ALGEBRA, GEOMETRY AND CALCULUS

2 hours 30 minutes

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

1. This examination paper consists of THREE sections.

2. Each section consists of TWO questions.

3. Answer ALL questions from the THREE sections.

4. Write your answers in the spaces provided in this booklet.

5. Do NOT write in the margins.

6. Unless otherwise stated in the question, any numerical answer that is notexact MUST be written correct to three significant figures.

7. If you need to rewrite any answer and there is not enough space to do soon the original page, you must use the extra lined page(s) provided at theback of this booklet. Remember to draw a line through your originalanswer.

8. If you use the extra page(s) you MUST write the question numberclearly in the box provided at the top of the extra page(s) and, whererelevant, include the question part beside the answer.

Examination Materials Permitted

Mathematical formulae and tables (provided) - Revised 2012Mathematical instrumentsSilent, non-programmable, electronic calculator

Copyright © 2014 Caribbean Examinations CouncilAll rights reserved.

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SECTION A

Module 1

Answer BOTH questions.

1. (a) Let f(x) = 2x3 - X2 +px + q.

(i) Given that x + 3 is a factor off(x) and thatthere is a remainder oflO, when f(x) isdivided by x + 1 show thatp = -25 and q = -12.

[7 marks]

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(ii) Hence, solve the equation f(x) = o.

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[6 marks]

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[6 marks]

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(b) Use mathematical induction to prove that 6n- 1 is divisible by 5 for all natural

numbers n.

(

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(c) (i) Given that p and q are two propositions, complete the truth table below:

p q p-q pVq pllq (pVq)_(pllq)

T T

T F

F T

F F

[4 marks)

(ii) State, giving a reason for your response, whether the following statements arelogically equivalent:

(p v q) _ (p II q)

[2 marks)

Total 25 marks

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2. (a) Solve the following equation for x:

[6 marks]

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(b) A function [is defined by [(x) = ~~ ~ , x f::. l.

Determine whether [is bijective, that is, both one-to-one and onto.

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(c) Let the roots of the equation 2x3- 5x2 + 4x + 6 = 0 be u, p and y.

(i) State the values of u + P + y, up + uy + py and upy.

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[3 marks]

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(ii) Hence, or otherwise, determine an equation with integer coefficients which has

111roots -2 ' 2 and -2 .a ~ 'I

Note: (a~? + (uy)2 + (~yf = (a~ + ay + ~yf - 2a~y (a + ~ + 'I)

a2 + ~2+ '12= (a + ~ + 'If - 2 (u~ + ay + ~y)

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3. (a)

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SECTIONB

Module 2

Answer BOTH questions.

(i) Show that see/ () = _c:.c::c0s::....::...ec:.......:()_cosec () - sin ()

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[4 marks]

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(ii) h . 1 h . cosec ()Hence, or ot erwise, so ve t e equation (). ()cosec - sm

for O:S () :s 2n.

43

[5 marks]

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(b) (i) Express the function fee) = sin 19 + cos 19 in the form r sin (19 + a), where

r > 0 and 0 < 19 < .!£ .- - 2

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[5 marks]

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(ii) Hence, find the maximum value of f and the smallest non-negative value of eatwhich it occurs.

[5 marks]

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(c) Prove that

tan A +tanB + tan C-tanA tanBtan Ctan (A + B + C) = -----------1-tanA tanB-tanA tan C-tanBtan C

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4. (a) (i) Given that sin e = x, show that tan e = x , where 0 < e < ~ ..-VI - J? 2

[3 mark

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(ii) Hence, or otherwise, determine the Cartesian equation of the curve definedn

parametrically by y = tan 2t and x = sin t for 0 < t <T"

[5 marks]

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1 2(b) Let u = -3 and v = 1 be two position vectors in R3

.

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(i) Calculate the lengths of u and v respectively.

(ii) Find cos e where e is the angle between u and v in R3.

[3 marks]

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[4 marks]

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(c) A point, P (x, y), moves such that its distance from the x-axis is half its distance from theongm.

Determine the Cartesian equation of the locus of P.

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[5 marks]

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(d) The line L has the equation 2x + y + 3 = 0 and the circle C has the equation X2 +Y = 9.Determine the points of intersection of the circle C and the line L.

[5 marks]

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Total 25 marks

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SECTIONC

Module 3

Answer BOTH questions.

1

(a) Use an appropriate substitution to find f (x + 1)3 dx.

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[4 marks]

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(b) The diagram below represents the finite region R which is enclosed by the curve y = x3- 1

and the lines x = 0 andy = o.

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--------~~~---x

Calculate the volume of the solid that results from rotating R about the y-axis.

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[5 marks]

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a a

(c) Giventhatj f(x)dx=f f(a-x)dx a>O,showthato 0

II e' 1o t!+ el -x dx = 2 .

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(d) An initial population of 10 000 bacteria grow exponentially at a rate of 2% per hour,where y =f (t) is the number of bacteria present t hours later.

(i) Solve an appropriate differential equation to show that the number of bacteriapresent at any time can be modelled by the equation y = 10 000 e 0.021 .

[7 marks]

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[3 marks]

Total 25 marks

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6. (a) Find the equation of the tangent to the curve f(x) = 2x3 + 5.r - x + 12 at the point wherex= 3.

[4 marks]

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(b)

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A function f is defined on R as

(i)

{

X2+2x+3f(x) =

ax+b

1"Calculate the imx~O-

x:sOx>O

f(x) and lim f(x)x~o+

[4 marks]

(ii) Hence, determine the values of a and b such that f(x) is continuous at x = o.

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[5 marks]

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(iii)lim f(O + t) - f(O)

If the value of b = 3, determine a such that f' (0) = t ---+ 0 t

[6 marks)

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(c) Use first principles to differentiate [(x) = -IX with respect to x.

END OF TEST

[6 marks]

Total 25 marks

IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.

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