DSE Maths Core · Past Paper Questions

2022 · Paper 1 Q1 Laws of integral indices

() Simplify

\frac{(a^{5}b^{-2})^{4}}{a^{-5}b^{6}}

and express your answer with positive indices. (3 marks)

2022 · Paper 1 Q2 Linear equations in two unknowns

() Let

x

and

y

be two numbers. The sum of

x

and

y

is

456

while the product of

7

and

x

is

y

. Find

x

. (3 marks)

2022 · Paper 1 Q3 Algebraic expressions

Simplify

\frac{3}{k-9} + \frac{2}{5k+6}

(3) Simplify

\frac{3}{k-9} + \frac{2}{5k+6}

(3 marks)

2022 · Paper 1 Q4 Polynomials

(a)

(i)

9c^{2}-6c+1

(b)

(i)

(4c+d)^{2}-9c^{2}+6c-1

(4 marks)

2022 · Paper 1 Q5 Using percentages

A fan is sold at a discount of

30\%

on its marked price. After selling the fan, the profit is

\

78

and the percentage profit is

26\%$. Find the marked price of the fan. (4 marks)

2022 · Paper 1 Q6 Linear inequalities in one unknown

Consider the compound inequality

-2(3x+2) > x+10

or

2x \leq -8

\ldots

(*)

.

(a) Solve

(*)

.

(b) Write down the greatest integer satisfying

(*)

. (4 marks)

2022 · Paper 1 Q7 Rectangular coordinate system

The coordinates of the points

S

and

T

are

(12, -5)

and

(-3, -7)

respectively.

S

is rotated anticlockwise about

O

through

90^{\circ}

to

S'

, where

O

is the origin.

T'

is the reflection image of

T

with respect to the

x

-axis.

(a) Write down the coordinates of

S'

and

T'

.

(b) Find the slope of

S'T'

.

(4 marks)

2022 · Paper 1 Q8 Congruent triangles

(a) Prove that

\Delta ABC \cong \Delta AED

.

(b) If

\angle ABC = 39^{\circ}

and

\angle DAE = 87^{\circ}

, find

\angle ACD

.

2022 · Paper 1 Q9 Measures of dispersion

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the times taken to complete a

3\text{ km }

race by a group of students.

(a) Write down the value of

x

.

(b) Find the mean of the distribution.

(c) Find the probability that the time taken to complete the

3\text{ km }

race by a randomly selected student from the group is less than

19.5

minutes. (5 marks)

2022 · Paper 1 Q10 Functions and graphs

(a) Find

f(x)

.

(b) Write down the

x

-intercept(s) of the graph of

y=8f(x)

. (1 mark)

(c) Let

k

be a real constant. Find the range of values of

k

such that the equation

f(x)=k

has two distinct real roots. (2 marks)

2022 · Paper 1 Q11 Measures of dispersion

The stem-and-leaf diagram below shows the distribution of the ages of the players of a football team.

Stem (tens) | Leaf (units)
1 | 7 8 9
2 | 0 a a 8 8 9
3 | b b 5 5 6 6 6 6 7 8
4 | 3

The inter-quartile range and the median of the distribution are

14

and

31

respectively.

(a) Find

a

and

b

.

(3 marks)

(b) A player now leaves the football team.

(i) Is there any change in the mode of the distribution due to the leaving of the player? Explain your answer.

(ii) If the range of the distribution is decreased, find the greatest possible standard deviation of the distribution.

(3 marks)

2022 · Paper 1 Q12 Equations of circles

The equation of the circle

C

is

x^{2}+y^{2}-154x-128y+224=0

. Denote the centre of

C

by

G

. The coordinates of the point

H

are

(65, 48)

.

(a) Find the distance between

G

and

H

.

(3 marks)

(b) Let

P

be a moving point on

C

. When the area of

\Delta GHP

is the greatest,

(i) describe the geometric relationship between

GH

and

GP

;

(ii) find the perimeter of

\Delta GHP

.

(4 marks)

2022 · Paper 1 Q13 Mensuration

There are two solid metal spheres. The ratio of the surface area of the smaller sphere to the surface area of the larger sphere is

4:9

. The radius of the larger sphere is

9\text{ cm}

.

(a) Express, in terms of

\pi

, the volume of the smaller sphere.

(3 marks)

(b) The two spheres are melted and recast into two solid right circular cones. Denote these two circular cones by

A

and

B

. It is given that the height and the base radius of

A

are

10\text{ cm}

and

6\text{ cm}

respectively. A student finds that the base radius of

B

is

12\text{ cm}

. The student claims that

A

and

B

are similar. Is the claim correct? Explain your answer. (4 marks)

2022 · Paper 1 Q14 More about polynomials

Let

p(x)=2x^{3}+ax^{2}+bx-20

, where

a

and

b

are constants. When

p(x)

is divided by

x^{2}-2x+3

, the remainder is

x+13

.

(a) Find

a

and

b

.

(b) Is

x-5

a factor of

p(x)

? Explain your answer.

(c) Someone claims that the equation

p(x)=0

has two irrational roots. Do you agree? Explain your answer. (3 marks)

2022 · Paper 1 Q15 More about probability

(a) find the probability that there are

2

boys and

2

girls in the committee; (2 marks)

(b) find the probability that the number of boys and the number of girls in the committee are different. (2 marks)

2022 · Paper 1 Q16 Functions and graphs

Let

g(x) = 3x^{2} + 12kx + 16k^{2} + 8

, where

k

is a non-zero real constant.

(a) Using the method of completing the square, express, in terms of

k

, the coordinates of the vertex of the graph of

y = g(x)

. (2 marks)

(b) On the same rectangular coordinate system, denote the vertex of the graph of

y = g(x)

and the vertex of the graph of

y = 2g(-x)

by

A

and

B

respectively. Let

M

be a point lying on

AB

such that the area of

riangle OBM

is the triple of the area of

riangle OAM

, where

O

is the origin. Express, in terms of

k

, the coordinates of

M

. (3 marks)

2022 · Paper 1 Q17 Arithmetic and geometric sequences and their summations

(a) Express

\alpha^{2}+\beta^{2}

in terms of

c

.

(3 marks)

(b) The 1st term, the 2nd term and the 3rd term of an arithmetic sequence are

c^{2}

,

\alpha^{2} + \beta^{2}

and

85

respectively. Find the least value of

n

such that the sum of the first

n

terms of the sequence is greater than

2 \times 10^{6}

. (4 marks)

2022 · Paper 1 Q18 Trigonometry

(a) Find

(i) the length of

QR

(ii)

\angle PQR

. (4 marks)

(b) Let

M

be the mid-point of

QR

. A craftsman finds that the angle between

PR

and the horizontal ground is

70^{\circ}

. The craftsman claims that the angle between

PM

and the horizontal ground exceeds

40^{\circ}

. Is the claim correct? Explain your answer. (3 marks)

2022 · Paper 1 Q19 Equations of circles

The centre of the circle

C

is the point

G(83,112)

. It is found that the point

A(158,12)

lies outside

C

.

AP

and

AQ

are the tangents to

C

at the points

P

and

Q

respectively. It is given that

C

passes through the point

(23,67)

.

(a) Find the equation of the straight line passing through

A

and

G

. (2 marks)

(b) Find the coordinates of the point of intersection of

AG

and

PQ

. (3 marks)

(c) Find the equation of the inscribed circle of

\Delta APQ

. (4 marks)

(d) Someone claims that the ratio of the area of the inscribed circle to the area of the circumcircle of

\Delta APQ

is

1:4

. Do you agree? Explain your answer. (3 marks)