Cambridge O Level

* 1 2 2 9 0 9 0 2 8 1 *

ADDITIONAL MATHEMATICS 4037/22

Paper 2 May/June 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 12 pages. Blank pages are indicated.

DC (LK) 198170
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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 Variables x and y are such that y = sin x + e -x . Use differentiation to find the approximate change in
r r
y as x increases from to + h, where h is small. [4]
4 4

2 DO NOT USE A CALCULATOR IN THIS QUESTION.

The point `1 - 5, pj lies on the curve

10 + 2 5
y= . Find the exact value of p, simplifying your
answer. x2 [5]

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3 Find the values of k for which the line y = x-3 intersects the curve y = k 2 x 2 + 5kx + 1 at two
distinct points. [6]

1
4 The three roots of p (x) = 0 , where p(x) = 2x 3 + ax 2 + bx + c
are x = , x = n and x =- n, where
2
a, b, c and n are integers. The y-intercept of the graph of y = p(x) is 4. Find p(x), simplifying your
coefficients. [5]

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

The points A and B are (4, 3) and (12, −7) respectively.

(a) Find the equation of the line L, the perpendicular bisector of the line AB. [4]

(b) The line parallel to AB which passes through the point (5, 12) intersects L at the point C. Find the
coordinates of C. [4]

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r
6 (a) Find the equation of the tangent to the curve 2y = tan 2x + 7 at the point where x = .
8
r
Give your answer in the form ax - y = + c , where a, b and c are integers. [5]
b

(b) This tangent intersects the x-axis at P and the y-axis at Q. Find the length of PQ. [2]

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7 Giving your answer in its simplest form, find the exact value of

y0 5x10+ 2 dx,
4
(a) [4]

`e 4x + 2j dx.
ln 2 2
(b) y0 [5]

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8 (a) Solve 3 cot 2 x - 14 cosec x - 2 = 0 for 0° 1 x 1 360°. [5]

sin 4 y - cos 4 y
(b) Show that = tan y - 2 cos y sin y. [4]
cot y

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9 5x
9 (a) Solve the equation = 243. [3]
27 x - 2

1
(b) loga b - = logb a, where a 2 0 and b 2 0.
2
Solve this equation for b, giving your answers in terms of a. [5]

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10 (a) The first 5 terms of a sequence are given below.

4 -2 1 - 0.5 0.25

(i) Find the 20th term of the sequence. [2]

(ii) Explain why the sum to infinity exists for this sequence and find the value of this sum. [2]

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(b) The tenth term of an arithmetic progression is 15 times the second term. The sum of the first
6 terms of the progression is 87.

(i) Find the common difference of the progression. [4]

(ii) For this progression, the nth term is 6990. Find the value of n. [3]

Question 11 is printed on the next page.

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11
A

C1 C2

The circles with centres C1 and C2 have equal radii of length r cm. The line C1C2 is a radius of both
circles. The two circles intersect at A and B.

(a) Given that the perimeter of the shaded region is 4r cm, find the value of r. [4]

(b) Find the exact area of the shaded region. [4]

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