r % TEST CODE 02234020 -lMAY/JUNE 2OI7FORM TP 2017300

CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@

PURE MATHEMATICS

UNIT 2 -Paper 02

ANALYSIS, MATRTCES AND COMPLEX NUMBERS

2 hours 30 minutes

Examination Materials Permitted

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

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

l. 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. lf 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.

ffiCopyright @ 2015 Caribbean Examinations Council

All rights reserved.

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SECTION A

Module I

Answer BOTH questions.

1. (a) Find the first derivative of the function f (x): cos ' (sin-' x;

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[3 marks]

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[5 marksl

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A function, w, is defined as w(x,-y): ln l# I

(i) Given that X : - + at the point (4, yo), calculate the value ofyo

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[5 marks]

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(ii) Showthat *-z dt :ooyox qi

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(c) (i) Findthecomplexnumbers u=x]-lysuchthatxandyarereal and*=-15+8i

[7 marksl

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(ii) Hence, orotherwise, solve the equation z' - (3 + 2i) z+ (5 + i): 0,for z

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[5 marksl

Total25 marks

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2. (a) (i) Use integration by parts to derive the reduction formula

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(ii) Hence, or otherwise, determine f e3'dx

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[4 marksl

[6 marksl

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r -10- -l(b) Calculate

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I 0

sin-' x

-dxJt -,'

[5 marks]

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(c) (i) Use partial fractions to show that

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2l-x+4 I x Ia

,r:+4x x l++ l++

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[5 marksl

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[5 marksl

Total25 marks

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-lrN-'4(ii) Hence, or otherwise, determine J f + 4x dx

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SECTION B

Module 2

Answer BOTH questions.

(i) Determine the Taylor series expansion about x : 2 of the functionf (x) = ln (5 + x) up to and including the term in x3.

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3. (a)

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[6 marksl

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(ii) Hence, obtain an approximation for f (7) - ln (7).

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[2 marksl

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(b) (i) Use mathematical induction to prove that

13 +23 + +n3: Irrrn+ l)2, forn € N

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[9 marks]

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(ii) Hence, or otherwise, show that I it = (2n + l)2 (n + l)22rr+ I

(iii) Use the results of Parts (b) (i) and (ii) to show that

[3 marksl

[5 markslTotal25 marks

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I (2i - l)t: (n + t)z (2n2 + 4n + t) .

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Eight boys and two girls are to be seated on a bench. How many seating arrangements

are possible if the girls can neither sit together nor sit at the ends?

[5 marks]

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4. (a)

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Show that the binomial expansion of (l + : x)8 up to and including the term inxois 8

7735l+x+-*+-l+-x416 64 2048

[4 marks]

(ii) Use the expansion to approximate the value of ( I .0 I 25)8

[4 marksl

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(i) Use the intermediate value theorem to show that f (x) : J; - cos x has a root inthe interval [0, l].

[3 marksl

(ii) Use two iterations of the interval bisection method to approximate the root of fin the interval [0, l].

[4 marksl

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of f (x):-Zf -3x+9.

[2 marksl

(ii) Apply the iterative formula with initial approximation .r, : l, to obtain a third

approximation, rr, of the root of the equation.

[3 marksl

Total25 marks

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-l(d) (i) Show that x--, : ,W- is an appropriate iterative formula for finding the rootn+t 1l 2

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SECTION C

Module 3

Answer BOTH questions.

The following Venn diagram shows a sample space, S, and the probabilities of two events,A and B, within the sample space S.

s

(i) GiventhatP (Aw B):0.7,calculate P (A only)

[3 marksl

(ii) Hence, determine whether events A and B are independent. Justify youranswer,

[2 marksl

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s. (a)

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=0.2

B

03

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Two balls are to be drawn at random without replacement from a bag containing 3 red

balls, 2 blue balls and I white ball.

(i) Represent the outcomes of the draws and their corresponding probabilities on atree diagram.

[4 marks]

(ii) Determine the probability that the second ball drawn is white.

[3 marks]

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-l(b)

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(i)

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show thatl B:201.

[5 marksl

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(ii) Hence, deduce the inverse, l-r, of the matrix l.

(iii) Hence, or otherwise, solve the system of linear equations given by

x- y+ I

x-2y+42:5x+3y+92:25

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[2 marksl

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[6 marksl

Total25 marks

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dy

dx6. (a) (i) Find the general solution of the differential equation (l + l) + 2xy: 4l;

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[7 marksl

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(ii) Hence, given thaty:2 when r:0, calculatey (l)

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(b)

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(i) Use the substitution u: y' to show that the differential equationy" + 4y' : 2cos 3x - 4sin 3x can be reduced to u' * 4u:Zcos3x - 4sin 3x

[3 marksl

[2 marksl

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(ii) Hence, or otherwise, find the general solution of the differential equation.

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[3 marksl

Total25 marks

END OF TEST

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

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