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DC (SLM) 65996

UCLES 2012 [Turn over

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral Certificate of Education Ordinary Level

ADDITIONAL MATHEMATICS 4037/23

Paper 2 October/November 2012

2 hours

Candidates answer on the Question Paper.No Additional Materials are required.

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer allquestions.Give non-exact numerical answers correct to 3 significant figures, or 1 decimal

place in the case of angles in degrees, unless a different level of accuracy isspecified in the question.The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or partquestion.The total number of marks for this paper is 80.

For Examiners Use

1

2

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6

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12

Total

www.Xtrem

ePapers.com

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Mathematical Formulae

1. ALGEBRA

QuadraticEquation

For the equation ax2+ bx+ c= 0,

x b b ac

a=

24

2

Binomial Theorem

(a+ b)n= an+ (n

1 )an1b+ (

n

2 )an2b2+ + (

n

r)anrbr+ + bn,

where nis a positive integer and ( nr)= n!(n r)!r!

2. TRIGONOMETRY

Identities

sin2A+ cos2A= 1

sec2A= 1 + tan2A

cosec2A= 1 + cot2A

Formulae for ABC

a

sinA =b

sinB =c

sin C

a2= b2+ c2 2bccosA

=1

2bcsinA

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1 Solve the equation|5x+ 7| = 13. [3]

2 (i) Given thatA= 7 84 6 , find the inverse matrix, A1. [2]

(ii) Use your answer to part (i)to solve the simultaneous equations

7x+ 8y= 39, 4x+ 6y= 23. [2]

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3 Without using a calculator, simplify (3 3 1)2

2 3 3, giving your answer in the form a 3 + b

3, where

a and bare integers. [4]

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4 The pointsX, YandZ are such that

XY= 3

YZ. The position vectors ofXandZ, relative to an

originO, are 427and 207respectively. Find the unit vector in the directionOY. [5]

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5 Find the set of values of mfor which the line y= mx+ 2 does not meet the curvey= mx2+ 7x+ 11. [6]

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6 (a) Given that cosx=p, find an expression, in terms ofp, for tan2x. [3]

(b) Prove that (cot+ tan)2= sec2+ cosec2. [3]

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7 (a) Find (x+ 3) xdx. [3]

(b) Find

20

(2x+ 5)2dxand hence evaluate

0

10 20

(2x+ 5)2dx. [4]

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8 Solutions to this question by accurate drawing will not be accepted.

The pointsA (4, 5),B(2, 3), C(1, 9) andD are the vertices of a trapezium in whichBCisparallel toADand angleBCDis 90. Find the area of the trapezium. [8]

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9 The table shows experimental values of two variablesxandy.

x 1 2 3 4

y 9.41 1.29 0.69 1.77

It is known thatxandyare related by the equation y=a

x2+ bx, where a and bare constants.

(i) A straight line graph is to be drawn to represent this information. Given thatx2yis plottedon the vertical axis, state the variable to be plotted on the horizontal axis. [1]

(ii) On the grid opposite, draw this straight line graph. [3]

(iii) Use your graph to estimate the value of aand of b. [3]

(iv) Estimate the value ofywhenxis 3.7. [2]

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10

A

D

B

C

200m

80m

x

m

A track runs due east fromAtoB, a distance of 200 m. The point Cis 80 m due north ofB. A cyclist travels on the track fromAtoD, whereDisxm due west ofB.The cyclist then travels

in a straight line across rough ground fromDto C. The cyclist travels at 10 m s1on the trackand at 6 m s1across rough ground.

(i) Show that the time taken, Ts, for the cyclist to travel fromAto Cis given by

T=

200 x

10 +

(x2+ 6400)

6 [2]

(ii) Given thatxcan vary, find the value ofxfor which Thas a stationary value and thecorresponding value of T. [6]

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11 (a) Solve (2x2)12= 100,giving your answer to 1 decimal place. [3]

(b) Solve logy

2 = 3 logy

256. [3]

(c) Solve65z2

36z= 216

z1

363z. [4]

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12 Answer only oneof the following alternatives.

EITHER

(i) Express 4x2+ 32x+ 55 in the form (ax+ b)2+ c, where a, band care constants and aispositive. [3]

The functions f and g are defined by

f :x 4x2+ 32x+ 55 forx> 4,

g :x 1x

forx> 0.

(ii) Find f1(x). [3]

(iii) Solve the equation fg(x) = 135. [4]

OR

The functions h and k are defined by

h :x 2x 7forxc,

k :x 3x 4x 2

forx> 2.

(i) State the least possible value of c. [1]

(ii) Find h1(x). [2]

(iii) Solve the equation k(x) =x. [3]

(iv) Find an expression for the function k2, in the form k2:x a+ bxwhere aand bare

constants. [4]

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Start your answer to Question 12 here.

Indicate which question you are answering.EITHER

OR

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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the

publisher will be pleased to make amends at the earliest possible opportunity.

University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of

Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Continue your answer here if necessary.

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