DSE Maths Core · Past Paper Questions

2013 · Paper 1 Q1 Laws of integral indices

() Simplify

\frac{x^{20}y^{13}}{(x^{5}y)^{6}}

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

2013 · Paper 1 Q2 Formulae

() Make

k

the subject of the formula

\frac{3}{h}-\frac{1}{k}=2

. (3 marks)

2013 · Paper 1 Q3 Polynomials

Factorize

(a)

4m^{2}-25n^{2}

(b)

4m^{2}-25n^{2}+6m-15n

(3 marks)

2013 · Paper 1 Q4 Linear equations in two unknowns

The price of 7 pears and 3 oranges is

\

47

while the price of 5 pears and 6 oranges is

\

49

. Find the price of a pear. (4 marks)

2013 · Paper 1 Q5 Linear inequalities in one unknown

(a) Solve the inequality

\frac{19-7x}{3}>23-5x

(b) Find all integers satisfying both the inequalities

\frac{19-7x}{3}>23-5x

and

18-2x\geq0

.

(4 marks)

2013 · Paper 1 Q6 Rectangular coordinate system

In a polar coordinate system,

O

is the pole. The polar coordinates of the points

A

and

B

are

(26,10^{\circ})

and

(26,130^{\circ})

respectively. Let

L

be the axis of reflectional symmetry of

\Delta OAB

.

(a) Describe the geometric relationship between

L

and

\angle AOB

.

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

L

and

AB

.

(4 marks)

2013 · Paper 1 Q7 Congruent triangles

In Figure 1,

ABCD

is a quadrilateral. The diagonals

AC

and

BD

intersect at

E

. It is given that

BE = CE

and

\angle BAC = \angle BDC

.

(a) Prove that

\Delta ABC \cong \Delta DCB

.

(b) Consider the triangles in Figure 1.

(i) How many pairs of congruent triangles are there?

(ii) How many pairs of similar triangles are there?
(4 marks)

2013 · Paper 1 Q8 Errors in measurement

A pack of sea salt is termed regular if its weight is measured as

100\text{ g }

correct to the nearest g.

(a) Find the least possible weight of a regular pack of sea salt.

(b) Is it possible that the total weight of 32 regular packs of sea salt is measured as

3.1\text{ kg }

correct to the nearest

0.1\text{ kg }

? Explain your answer.

(5 marks)

2013 · Paper 1 Q9 Measures of dispersion

The bar chart below shows the distribution of the numbers of family members of the employees of company

D

.

Distribution of the numbers of family members of the employees of company D

Figure 1

(a) Find the mean, the inter-quartile range and the standard deviation of the above distribution.

(b) An employee leaves company

D

. The number of family members of this employee is

7

. Find the change in the standard deviation of the numbers of family members of the employees of company

D

due to the leaving of this employee. (5 marks)

2013 · Paper 1 Q10 Measures of dispersion

(a) Write down the median and the mode of the ages of the members of Committee A. (2 marks)

(b) The stem-and-leaf diagram below shows the distribution of the ages of the members of Committee B. It is given that the range of this distribution is

47

.

(i) Find

a

and

b

.

(ii) From each committee, a member is randomly selected as the representative of that committee. The two representatives can join a competition when the difference of their ages exceeds

40

. Find the probability that these two representatives can join the competition. (4 marks)

2013 · Paper 1 Q11 Variations

The weight of a tray of perimeter

\ell

metres is

W

grams. It is given that

W

is the sum of two parts, one part varies directly as

\ell

and the other part varies directly as

\ell^{2}

. When

\ell=1

,

W=181

and when

\ell=2

,

W=402

.

(a) Find the weight of a tray of perimeter

1.2

metres. (4 marks)

(b) If the weight of a tray is

594

grams, find the perimeter of the tray. (2 marks)

2013 · Paper 1 Q12 More about polynomials

Let

f(x)=3x^{3}-7x^{2}+kx-8

, where

k

is a constant. It is given that

f(x)\equiv(x-2)(ax^{2}+bx+c)

, where

a

,

b

and

c

are constants.

(4 marks)

(a) Find

a

,

b

and

c

.

(b) Someone claims that all the roots of the equation

f(x)=0

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

2013 · Paper 1 Q13 Mensuration

In a workshop, 2 identical solid metal right circular cylinders of base radius

R

cm are melted and recast into

27

smaller identical solid right circular cylinders of base radius

r

cm and height

10

cm. It is given that the base area of a larger circular cylinder is

9

times that of a smaller one.

(a) Find

(i)

r:R

,

(ii) the height of a larger circular cylinder.

(b) A craftsman claims that a smaller circular cylinder and a larger circular cylinder are similar. Do you agree? Explain your answer. (2 marks)

2013 · Paper 1 Q14 Loci

(a) Write down the coordinates of

R

.

(1 mark)

(b) The equation of the straight line

L

is

4x + 3y + 50 = 0

. It is found that

C

and

L

do not intersect. Let

P

be a point lying on

L

such that

P

is nearest to

R

.

(i) Find the distance between

P

and

R

.

(ii) Let

Q

be a moving point on

C

. When

Q

is nearest to

P

,

(1) describe the geometric relationship between

P

,

Q

and

R

;

(2) find the ratio of the area of

\Delta OPQ

to the area of

\Delta OQR

, where

O

is the origin.

(8 marks)

2013 · Paper 1 Q15 Measures of dispersion

(a) Find the mean of the distribution.

(b) Susan claims that the standard scores of at least half of the students in the test are negative. Do you agree? Explain your answer. (2 marks)

2013 · Paper 1 Q16 More about probability

A box contains

5

white cups and

11

blue cups. If

6

cups are randomly drawn from the box at the same time,

(a) find the probability that at least

4

white cups are drawn; (2 marks)

(b) find the probability that at least

3

blue cups are drawn. (2 marks)

2013 · Paper 1 Q17 Functions and graphs

(a) Let

f(x)=36x-x^{2}

. Using the method of completing the square, find the coordinates of the vertex of the graph of

y=f(x)

. (2 marks)

(b) The length of a piece of string is

108

m. A guard cuts the string into two pieces. One piece is used to enclose a rectangular restricted zone of area

A\text{ m}^{2}

. The other piece of length

x\text{ m}

is used to divide this restricted zone into two rectangular regions as shown in Figure 2.

2013 · Paper 1 Q18 Trigonometry

(a) Figure 3(a) shows a piece of triangular paper card

ABC

with

AB=28\ \text{cm}

,

BC=21\ \text{cm}

and

AC=35\ \text{cm}

. Let

M

be a point lying on

AC

such that

\angle BMC=75^{\circ}

.

(i)

\angle BCM

,

(ii)

CM

.

(3 marks)

(b) Peter folds the triangular paper card described in (a) along

BM

such that

AB

and

BC

lie on the horizontal ground as shown in Figure 3(b). It is given that

\angle AMC = 107^{\circ}

.

(i) Find the distance between

A

and

C

on the horizontal ground.

(ii) Let

N

be a point lying on

BC

such that

MN

is perpendicular to

BC

. Peter claims that the angle between the face

BCM

and the horizontal ground is

\angle ANM

. Do you agree? Explain your answer.

2013 · Paper 1 Q19 Arithmetic and geometric sequences and their summations

(a)

(i) Express, in terms of

r

, the total floor area of all public housing flats at the end of the 2nd year.

(ii) Find

r

.

(b)

(i) Express, in terms of

n

, the total floor area of all public housing flats at the end of the

n^{\text{th}}

year.

(ii) At the end of which year will the total floor area of all public housing flats first exceed

4 \times 10^{7} \text{ m}^{2}

?
(5 marks)

(c) It is assumed that the total floor area of public housing flats needed at the end of the

n

th year is

(a(1.21)^n + b)\text{ m}^2

, where

a

and

b

are constants. Some research results reveal the following information:

[Table]

A research assistant claims that based on the above assumption, the total floor area of all public housing flats will be greater than the total floor area of public housing flats needed at the end of a certain year. Is the claim correct? Explain your answer.
(4 marks)