Cambridge O Level

* 8 1 9 4 9 5 6 9 6 6 *

ADDITIONAL MATHEMATICS 4037/13

Paper 1 October/November 2020

2 hours

You must answer on the question paper.

No additional materials are needed.

INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.

INFORMATION
● The total mark for this paper is 80.
● The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Blank pages are indicated.

DC (LK) 206977
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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax 2 + bx + c = 0 ,

- b ! b 2 - 4ac
x=
2a

Binomial Theorem
n n n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1 2 r

n
where n is a positive integer and e o =
n!
r (n - r) !r!

Arithmetic series un = a + (n - 1) d
1 1
Sn = n (a + l ) = n {2a + (n - 1) d}
2 2

Geometric series un = ar n - 1
a (1 - r n )
Sn = ( r ! 1)
1-r
a
S3 = ( r 1 1)
1-r

2. TRIGONOMETRY

Identities

sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A

Formulae for ∆ABC

a b c
= =
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2

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1 (a) On the axes below, sketch the graph of y = (x - 2) (x + 1) (3 - x) , stating the intercepts on the
coordinate axes.

O x

[3]

(b) Hence write down the values of x such that (x - 2) (x + 1) (3 - x) 2 0 . [2]

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e 2x - 3 dy
2 (a) Given that y = , find . [3]
2
x +1 dx

(b) Hence, given that y is increasing at the rate of 2 units per second, find the exact rate of change of x
when x = 2 . [3]

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3 (a) f (x) = 4 ln (2x - 1)

(i) Write down the largest possible domain for the function f. [1]

(ii) Find f - 1 (x) and its domain. [3]

(b) g (x) = x + 5 for x ! R

3
h (x) = 2x - 3 for x H
2
Solve gh (x) = 7 . [3]

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4 (a)

xm
150

100

50

0 50 65 85 125 ts

The diagram shows the x–t graph for a runner, where displacement, x, is measured in metres and
time, t, is measured in seconds.

(i) On the axes below, draw the v–t graph for the runner. [3]

v ms–1

4
3
2
1

0 50 65 85 125 ts
-1
-2
-3
-4

(ii) Find the total distance covered by the runner in 125 s. [1]

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(b) The displacement, x m, of a particle from a fixed point at time t s is given by x = 6 cos b3t + l.
r
3
2r
Find the acceleration of the particle when t = . [3]
3

(1 + x) b1 - l
xn 25
5 Given that the coefficient of x 2 in the expansion of is , find the value of the
2 4
positive integer n. [5]

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It is known that y = A # 10 bx , where A and b are constants. When lg y is plotted against x 2 , a

2
6
straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.

(a) Find the value of A and of b. [4]

Using your values of A and b, find

(b) the value of y when x = 2 , [2]

(c) the positive value of x when y = 4 . [2]

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7 The polynomial p (x) = ax 3 + bx 2 - 19x + 4 , where a and b are constants, has a factor x + 4 and is
such that 2p (1) = 5p (0) .

(a) Show that p (x) = (x + 4) (Ax 2 + Bx + C) , where A, B and C are integers to be found. [6]

(b) Hence factorise p (x) . [1]

(c) Find the remainder when pl (x) is divided by x. [1]

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8 In this question all lengths are in centimetres.

1.45 r

O
C 0.5 r r B

The diagram shows the figure ABC. The arc AB is part of a circle, centre O, radius r, and is of length
1.45r. The point O lies on the straight line CB such that CO = 0.5r .

(a) Find, in radians, the angle AOB. [1]

(b) Find the area of ABC, giving your answer in the form kr2, where k is a constant. [3]

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(c) Given that the perimeter of ABC is 12 cm, find the value of r. [4]

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9
A

Y
X

O b B C

The diagram shows the triangle OAC. The point B is the midpoint of OC. The point Y lies on AC such
that OY intersects AB at the point X where AX : XB = 3:1. It is given that OA = a and OB = b .

(a) Find OX in terms of a and b, giving your answer in its simplest form. [3]

(b) Find AC in terms of a and b. [1]

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(c) Given that OY = h OX , find AY in terms of a, b and h. [1]

(d) Given that AY = mAC , find the value of h and of m. [4]

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1 2 5x + 12
10 (a) Show that + can be written as . [1]
x + 1 3x + 10 2
3x + 13x + 10

(b)
y

P
5x + 12
y= 2
3x + 13x + 10

x=2
Q
O x

5x + 12
The diagram shows part of the curve y = 2 , the line x = 2 and a straight line of
3x + 13x + 10
gradient 1. The curve intersects the y-axis at the point P. The line of gradient 1 passes through P
and intersects the x-axis at the point Q. Find the area of the shaded region, giving your answer in
the form a + ln `b 3j , where a and b are constants.
2
[9]
3

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Additional working space for question 10

Question 11 is printed on the next page.

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11 (a) Given that 2 cos x = 3 tan x , show that 2 sin 2 x + 3 sin x - 2 = 0 . [3]

(b) Hence solve 2 cos b2a + l = 3 tan b2a + l for 0 1 a 1 r radians, giving your answers in
r r
4 4
terms of r. [4]

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© UCLES 2020 4037/13/O/N/20