NA
MINISTRY OF EDUCATION, SINGAPORE
in collaboration with
CAMBRIDGE ASSESSMENT INTERNATIONAL EDUCATION
General Certificate of Education Normal (Academic) Level
S
*0123456789*
ADDITIONAL MATHEMATICS
Paper 2
4051/02
For examination from 2021
SPECIMEN PAPER
1 hour 45 minutes
Candidates answer on the Question Paper.
No Additional Materials are required.
READ THESE INSTRUCTIONS FIRST
Write your centre number, index number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE ON ANY BARCODES.
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 approved scientific calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
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 70.
This document consists of 15 printed pages and 1 blank page.
© UCLES & MOE 2019
[Turn over
2
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
- b ! b 2 - 4ac
2a
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) =
tan A ! tan B
1 " tan A tan B
sin 2A = 2 sin A cos A
cos 2A = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A
tan 2A =
2 tan A
1 − tan 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
∆=
© UCLES & MOE 2019
1
bc sin A
2
4051/02/SP/21
3
1
Express
2x 4 − x3 − 10x 2 + 13x − 4
in the form ax2 + bx + c , where a, b and c are integers.
x2 + x − 4
© UCLES & MOE 2019
4051/02/SP/21
[3]
[Turn over
4
2
Leaking water is forming a circular patch on a flat horizontal surface. The radius of the patch is increasing
at a rate of 4 cm/min. Find the rate at which the area of the patch is increasing at the instant when the area of
the patch is 100 cm2.
[5]
© UCLES & MOE 2019
4051/02/SP/21
5
3
(a) Express x2 + 12x + 13 in the form (x + a)2 + b where a and b are constants.
[3]
(b) Hence state the coordinates of the vertex of the curve y = x2 + 12x + 13 .
[2]
© UCLES & MOE 2019
4051/02/SP/21
[Turn over
6
4
A curve has equation y = x3 – 3x2 + 5x – 7. Find the equation of the tangent to the curve at the point x = 2.
[5]
© UCLES & MOE 2019
4051/02/SP/21
7
5
The polynomial p(x) is given by p(x) = 8x3 + ax2 + bx – 15 , where a and b are constants. It is given that
2x – 3 is a factor of p(x) and when p(x) is divided by x – 1 the remainder is –21.
(a) Show that a = 12 and find the value of b.
[4]
(b) Using the values from part (a), find the remainder when p(x) is divided by 2x + 1.
[2]
© UCLES & MOE 2019
4051/02/SP/21
[Turn over
8
6
(a) Given that y =
dy
x4 + 3
, find
.
dx
2x + 1
(b) Given that y = x2(4x + 1)5, find
quadratic expression.
© UCLES & MOE 2019
[3]
dy
, giving your answer in the form (4x + 1)4 f(x) where f(x) is a
dx
[4]
4051/02/SP/21
9
7
y
y=x+2
(2, 4)
x
O
y = – x 2 + 2x + 4
The diagram shows the curve y = –x2 + 2x + 4 and the line y = x + 2. The line and the curve intersect at
the point (2, 4). Find the area of the shaded region.
[7]
© UCLES & MOE 2019
4051/02/SP/21
[Turn over
10
8
It is given that f(x) =
7x
2
, where x ≠ – , x ≠ 4 .
3
(3x + 2) (x − 4)
(a) Express f(x) in partial fractions.
© UCLES & MOE 2019
[4]
4051/02/SP/21
11
(b) Hence find f ′(x) .
(c) Given that a curve has equation
negative.
© UCLES & MOE 2019
[3]
y = f(x) , explain why the gradient of every point on the curve is
[2]
4051/02/SP/21
[Turn over
12
9
(a) Write 4 cos x – 2 sin x in the form R cos (x + α), where R > 0 and 0° < α < 90°.
[3]
(b) Hence solve the equation 4 cos x – 2 sin x = 1 for 0° < x < 360°.
[4]
© UCLES & MOE 2019
4051/02/SP/21
13
(c) Using your answer from part (a), or otherwise, find the greatest value of 15 – 4 cos x + 2 sin x and the
smallest positive value of x for which this occurs.
[4]
© UCLES & MOE 2019
4051/02/SP/21
[Turn over
14
10
A circle has equation x2 + y2 + 4x – 6y – 87 = 0 .
(a) Find the radius and the coordinates of the centre of the circle, C.
[3]
The line y = 2x – 3 cuts the circle at two points A and B.
(b) Find the coordinates of the midpoint, M, of AB.
© UCLES & MOE 2019
4051/02/SP/21
[6]
15
(c) Find the shortest distance of the centre, C, from the line AB.
© UCLES & MOE 2019
4051/02/SP/21
[3]
16
BLANK PAGE
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable 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.
Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local
Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
© UCLES & MOE 2019
4051/02/SP/21