Specimen Paper: Additional Mathematics

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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Ordinary Level

4037/02

ADDITIONAL MATHEMATICS

For Examination from 2013

Paper 2
SPECIMEN PAPER

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 all the questions.
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 is specified 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 part question. The total number of marks for this paper is 80.

This document consists of 15 printed pages and 1 blank page.

© UCLES 2012

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2
Mathematical Formulae

1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=

− b ± b 2 − 4ac
.
2a

Binomial Theorem
n
(a + b)n = an +   an–1 b +
1


 n  n–2 2
 a b +…+
 2


 n
n!
.
where n is a positive integer and   =
r
(n − r )! r!


2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1.
sec2 A = 1 + tan2 A.
cosec2 A = 1 + cot2 A.
Formulae for ∆ABC

a
b
c
.
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A.
∆=

© UCLES 2012

1
2

bc sin A.

4037/02/SP/13

 n  n–r r
  a b + … + bn,
r


3
1

2

13 6 
–1
Given that A = 
 7 4  , find the inverse matrix A and hence solve the simultaneous equations 


13x + 6y = 41,
7x + 4y = 24.
[4]

For
Examiner's
Use

Variables x and y are connected by the equation y = (2x – 9)3. Given that x is increasing at the rate of 4 units per second, find the rate of increase of y when x = 7. [4]

© UCLES 2012

4037/02/SP/13

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4
3

Find the set of values of m for which the line y = mx + 2 does not meet the curve y = x2 – 5x + 18. [5]

4

(a) A sports team of 3 attackers, 2 centres and 4 defenders is to be chosen from a squad of 5 attackers, 3 centres and 6 defenders. Calculate the number of different ways in which this can be done.

[3]

© UCLES 2012

4037/02/SP/13

For
Examiner's
Use

5
(b) How many different 4-digit numbers greater than 3000 can be formed using the six digits 1, 2, 3, 4, 5 and 6 if no digit can be used more than once?
[3]

5

(i) Differentiate x ln x with respect to x.

[2]

∫ ln x dx.

For
Examiner's
Use

[3]

(ii) Hence find

© UCLES 2012

4037/02/SP/13

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6
6

Solve the following equations.
(i)

For
Examiner's
Use

4x
24x
= x −3
2 5− x 8

[3]

(ii) lg (2y + 10) + lg y = 2

© UCLES 2012

[3]

4037/02/SP/13

7
7

For
Examiner's
Use

Q

48 m

1.4 ms –1

P
The diagram shows a river with parallel banks. The river is 48 m wide and is flowing with a speed of 1.4 ms–1. A boat travels in a straight line from a point P on one bank to a point Q which is on the other bank directly opposite P. It is given that the boat takes 10 seconds to cross the river. (i) Find the speed of the boat in still water.

[4]

(ii) Find the angle to the bank at which the boat should be steered.

[2]

© UCLES 2012

4037/02/SP/13

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8
8

The function f is defined, for 0 Y x Y 2π, by

For
Examiner's
Use

f(x) = 3 + 5 sin 2x.
State
(i) the amplitude of f,

[1]

(ii) the period of f,

[1]

(iii) the maximum and minimum values of f.

[2]

Sketch the graph of y = f(x).

© UCLES 2012

[3]

4037/02/SP/13

9
9

The line y = 2x – 9...
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