DSE Maths Core · Past Paper Questions

2016 · Paper 1 Q2 Formulae

(a) Make

x

the subject of the formula

Ax = (4x + B)C

. (3 marks)

2016 · Paper 1 Q3 Polynomials

() Simplify

\frac{2}{4x-5}+\frac{3}{1-6x}

. (3 marks)

2016 · Paper 1 Q4 Polynomials

(a)

5m-10n

(b)

m^{2}+mn-6n^{2}

(c)

m^{2}+mn-6n^{2}-5m+10n

(4 marks)

2016 · Paper 1 Q5 Using percentages

In a recreation club, there are 180 members and the number of male members is

40\%

more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)

2016 · Paper 1 Q6 Linear inequalities in one unknown

(a) Solve

(*)

.

(b) Write down the greatest negative integer satisfying

(*)

. (4 marks)

2016 · Paper 1 Q7 Basic properties of circles

(a) Find

\angle AOB

.

(b) Find the perimeter of

\Delta AOB

.

(c) Write down the number of folds of rotational symmetry of

\Delta AOB

. (4 marks)

2016 · Paper 1 Q8 Variations

It is given that

f(x)

is the sum of two parts, one part varies as

x

and the other part varies as

x^{2}

. Suppose that

f(3)=48

and

f(9)=198

.

(a) Find

f(x)

.

(b) Solve the equation

f(x)=90

2016 · Paper 1 Q9 Organisation of data

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

(a) Find

x

,

y

and

z

.

(b) If a plant is randomly selected from the garden, find the probability that the height of the selected plant is less than

1.25\text{ m}

but not less than

0.65\text{ m}

.
(5 marks)

2016 · Paper 1 Q10 Equations of circles

(a) Find the equation of

\Gamma

.

(2 marks)

(b)

\Gamma

intersects the

x

-axis and the

y

-axis at

H

and

K

respectively. Denote the origin by

O

. Let

C

be the circle which passes through

O

,

H

and

K

. Someone claims that the circumference of

C

exceeds

30

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

2016 · Paper 1 Q11 Mensuration

An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is

12\text{ cm }

. Peter then pours

444\pi

cm^{3} of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is

16\text{ cm }

.

(a) Express the final volume of milk in the vessel in terms of

\pi

. (3 marks)

(b) Peter claims that the final area of the wet curved surface of the vessel is at least

800

cm^{2}. Do you agree? Explain your answer. (3 marks)

2016 · Paper 1 Q12 Measures of dispersion

(a) Find

a

and

b

.

(b) Four more children now join the group. It is found that the ages of these four children are all different and the range of the ages of the children in the group remains unchanged. Find

(i) the greatest possible median of the ages of the children in the group,

(ii) the least possible mean of the ages of the children in the group. ( ) Aqv 22d

2016 · Paper 1 Q13 Congruent triangles

(a) Prove that

\Delta ACD \cong \Delta ABE

.

(b) Suppose that

AD=15\text{ cm}

,

BD=7\text{ cm}

and

DE=18\text{ cm}

.

(i) Find

AM

.

(ii) Is

\Delta ABE

a right-angled triangle? Explain your answer. (5 marks)

2016 · Paper 1 Q14 More about polynomials

(a) Find

l

,

m

and

n

.

(b) How many real roots does the equation

p(x) = 0

have? Explain your answer. (5 marks)

2016 · Paper 1 Q15 Permutations and combinations

If

4

boys and

5

girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)

2016 · Paper 1 Q16 Measures of dispersion

In a test, the mean of the distribution of the scores of a class of students is

61

marks. The standard scores of Albert and Mary are

-2.6

and

1.4

respectively. Albert gets

22

marks. A student claims that the range of the distribution is at most

59

marks. Is the claim correct? Explain your answer.

2016 · Paper 1 Q17 Arithmetic and geometric sequences and their summations

The 1st term and the 38th term of an arithmetic sequence are

666

and

555

respectively. Find

(a) the common difference of the sequence, (2 marks)

(b) the greatest value of

n

such that the sum of the first

n

terms of the sequence is positive. (3 marks)

2016 · Paper 1 Q18 Quadratic equations in one unknown

(a) Using the method of completing the square, find the coordinates of the vertex of the graph of

y = f(x)

. (2 marks)

(b) The graph of

y = g(x)

is obtained by translating the graph of

y = f(x)

vertically. If the graph of

y = g(x)

touches the

x

-axis, find

g(x)

. (2 marks)

(c) Under a transformation,

f(x)

is changed to

\frac{-1}{3}x^2 - 12x - 121

. Describe the geometric meaning of the transformation. (2 marks)

2016 · Paper 1 Q19 3-D figures

Figure 2 shows a geometric model

ABCD

in the form of tetrahedron. It is given that

\angle BAD = 86^{\circ}

,

\angle CBD = 43^{\circ}

,

AB = 10

cm,

AC = 6

cm,

BC = 8

cm and

BD = 15

cm.

(a) Find

\angle ABD

and

CD

.

(b) A craftsman claims that the angle between

AB

and the face

BCD

is

\angle ABC

. Do you agree? Explain your answer. (2 marks)

2016 · Paper 1 Q20 Equations of circles

(a) Prove that

OP = PQ

.

(b) A rectangular coordinate system is introduced so that the coordinates of

O

and

Q

are

(0,0)

and

(40,30)

respectively while the

y

-coordinate of

P

is

19

. Let

C

be the circle which passes through

O

,

P

and

Q

.

(i) Find the equation of

C

.

(ii) Let

L_{1}

and

L_{2}

be two tangents to

C

such that the slope of each tangent is

\frac{3}{4}

and the

y

-intercept of

L_{1}

is greater than that of

L_{2}

.

L_{1}

cuts the

x

-axis and the

y

-axis at

S

and

T

respectively while

L_{2}

cuts the

x

-axis and the

y

-axis at

U

and

V

respectively. Someone claims that the area of the trapezium

STUV

exceeds

17000

. Is the claim correct? Explain your answer.