DSE Maths Core · Past Paper Questions

2017 · Paper 1 Q1 Formulae

(a)

Make $y$ the subject of the formula $k=\frac{3x-y}{y}$ . (3 marks)

2017 · Paper 1 Q2 Laws of integral indices

(a)

Simplify $\frac{(m^{4}n^{-1})^{3}}{(m^{-2})^{5}}$ and express your answer with positive indices.

2017 · Paper 1 Q3 More about polynomials

(a)

(i)

$x^{2}-4x y+3y^{2}$

(ii)

$x^{2}-4x y+3y^{2}+11x-33y\quad.$ (3 marks)

2017 · Paper 1 Q4 Linear equations in one unknown

There are only two kinds of admission tickets for a theatre: regular tickets and concessionary tickets. The prices of a regular ticket and a concessionary ticket are $$126$ and $$78$ respectively. On a certain day, the number of regular tickets sold is 5 times the number of concessionary tickets sold and the sum of money for the admission tickets sold is $$50,976$ . Find the total number of admission tickets sold that day. (4 marks)

2017 · Paper 1 Q5 Inequalities and linear programming

(a)

Find the range of values of $x$ which satisfy both $7(x-2)\le q\frac{11x+8}{3}$ and $6-x<5$ .

(b)

How many integers satisfy both inequalities in (a)? (4 marks)

2017 · Paper 1 Q6 Rectangular coordinate system

The coordinates of the points $A$ and $B$ are $(-3, 4)$ and $(9, -9)$ respectively. $A$ is rotated anticlockwise about the origin through $90^{\circ}$ to $A'$ . $B'$ is the reflection image of $B$ with respect to the $x$ -axis.

(a)

Write down the coordinates of $A'$ and $B'$ .

(b)

Prove that $AB$ is perpendicular to $A'B'$ . (4 marks)

2017 · Paper 1 Q7 Probability

The pie chart below shows the distribution of the seasons of birth of the students in a school.

Distribution of the seasons of birth of the students in the school

If a student is randomly selected from the school, then the probability that the selected student was born in spring is $\frac{1}{9}$ .

(a)

Find $x$ .

(b)

In the school, there are 180 students born in winter. Find the number of students in the school. (4 marks)

2017 · Paper 1 Q8 Variations

It is given that $y$ varies inversely as $\sqrt{x}$ . When $x=144$ , $y=81$ .

(a)

Express $y$ in terms of $x$ .

(b)

If the value of $x$ is increased from $144$ to $324$ , find the change in the value of $y$ . (5 marks)

2017 · Paper 1 Q9 Errors in measurement

A bottle is termed standard if its capacity is measured as $200\text{ mL}$ correct to the nearest $10\text{ mL}$ .

(a)

Find the least possible capacity of a standard bottle.

(b)

Someone claims that the total capacity of $120$ standard bottles can be measured as $23.3\text{ L}$ correct to the nearest $0.1\text{ L}$ . Do you agree? Explain your answer. (5 marks)

2017 · Paper 1 Q10 Basic properties of circles

In Figure 1, $OPQR$ is a quadrilateral such that $OP = OQ = OR$ . $OQ$ and $PR$ intersect at the point $S$ . $S$ is the mid-point of $PR$ .

(a)

Prove that $\Delta OPS \cong \Delta ORS$ .

(b)

It is given that $O$ is the centre of the circle which passes through $P$ , $Q$ and $R$ . If $OQ=6$ cm and $\angle PRQ=10^{\circ}$ , find the area of the sector $OPQR$ in terms of $\pi$ . (4 marks)

2017 · Paper 1 Q11 Measures of dispersion

The stem-and-leaf diagram below shows the distribution of the hourly wages (in dollars) of the workers in a group.

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

It is given that the mean and the range of the above distribution are $$70$ and $$22$ respectively.

(a)

Find the median and the standard deviation of the above distribution. (5 marks)

(b)

If a worker is randomly selected from the group, find the probability that the hourly wage of the selected worker exceeds $$70$ . (2 marks)

2017 · Paper 1 Q12 Mensuration

A solid metal right prism of base area $84\text{ cm}^{2}$ and height $20\text{ cm}$ is melted and recast into two similar solid right pyramids. The bases of the two pyramids are squares. The ratio of the base area of the smaller pyramid to the base area of the larger pyramid is $4:9$ .

(a)

Find the volume of the larger pyramid.

(3 marks)

(b)

If the height of the larger pyramid is $12\text{ cm}$ , find the total surface area of the smaller pyramid. (4 marks)

2017 · Paper 1 Q13 Equations of circles

The coordinates of the points $E$ , $F$ and $G$ are $(-6,5)$ , $(-3,11)$ and $(2,-1)$ respectively. The circle $C$ passes through $E$ and the centre of $C$ is $G$ .

(a)

Find the equation of $C$ .

(b)

Prove that $F$ lies outside $C$ . (2 marks)

(c)

Let $H$ be a moving point on $C$ . When $H$ is farthest from $F$ ,

(i)

describe the geometric relationship between $F$ , $G$ and $H$ ;

(ii)

find the equation of the straight line which passes through $F$ and $H$ .

2017 · Paper 1 Q14 More about polynomials

Let $f(x)=6x^{3}-13x^{2}-46x+34$ . When $f(x)$ is divided by $2x^{2}+ax+4$ , the quotient and the remainder are $3x+7$ and $bx+c$ respectively, where $a$ , $b$ and $c$ are constants.

(a)

Find $a$ . (3 marks)

(b)

Let $g(x)$ be a quadratic polynomial such that when $g(x)$ is divided by $2x^{2}+ax+4$ , the remainder is $bx+c$ .

(i)

Prove that $f(x)-g(x)$ is divisible by $2x^{2}+ax+4$ .

(ii)

Someone claims that all the roots of the equation $f(x)-g(x)=0$ are integers. Do you agree? Explain your answer. (5 marks)

2017 · Paper 1 Q15 Exponential and logarithmic functions

Let $a$ and $b$ be constants. Denote the graph of $y = a + \log_{b}x$ by $G$ . The $x$ -intercept of $G$ is $9$ and $G$ passes through the point $(243, 3)$ .

2017 · Paper 1 Q15 Exponential and logarithmic functions

Express $x$ in terms of $y$ . (4 marks)

2017 · Paper 1 Q16 Arithmetic and geometric sequences and their summations

A city adopts a plan to import water from another city. It is given that the volume of water imported in the 1st year since the start of the plan is $1.5 \times 10^7 \, \text{m}^3$ and in subsequent years, the volume of water imported each year is $10\%$ less than the volume of water imported in the previous year.

(a)

Find the total volume of water imported in the first 20 years since the start of the plan. (2 marks)

(b)

Someone claims that the total volume of water imported since the start of the plan will not exceed $1.6 \times 10^{8}$ m $^3$ . Do you agree? Explain your answer. (2 marks)

2017 · Paper 1 Q17 Probability

In a bag, there are 4 green pens, 7 blue pens and 8 black pens. If 5 pens are randomly drawn from the bag at the same time,

(a)

find the probability that exactly 4 green pens are drawn; (2 marks)

(b)

find the probability that exactly 3 green pens are drawn; (2 marks)

(c)

find the probability that not more than 2 green pens are drawn. (2 marks)

2017 · Paper 1 Q18 Quadratic equations in one unknown

The equation of the parabola $\Gamma$ is $y=2x^{2}-2kx+2x-3k+8$ , where $k$ is a real constant. Denote the straight line $y=19$ by $L$ .

(a)

Prove that $L$ and $\Gamma$ intersect at two distinct points.
(3 marks)

(b)

The points of intersection of $L$ and $\Gamma$ are $A$ and $B$ .

(i)

Let $a$ and $b$ be the $x$ -coordinates of $A$ and $B$ respectively. Prove that $(a-b)^{2}=k^{2}+4k+23$ .

(ii)

Is it possible that the distance between $A$ and $B$ is less than $4$ ? Explain your answer.
(5 marks)

2017 · Paper 1 Q19 Trigonometry

$ABC$ is a thin triangular metal sheet, where $BC = 24\text{ cm}$ , $\angle BAC = 30^\circ$ and $\angle ACB = 42^\circ$ .

(a)

Find the length of $AC$ . (2 marks)

(b)

In Figure 2, the thin metal sheet $ABC$ is held such that only the vertex $B$ lies on the horizontal ground. $D$ and $E$ are points lying on the horizontal ground vertically below the vertices $A$ and $C$ respectively. $AC$ produced meets the horizontal ground at the point $F$ . A craftsman finds that $AD=10\text{ cm}$ and $CE=2\text{ cm}$ .

Figure 2

(i)

Find the distance between $C$ and $F$ .

(ii)

Find the area of $\Delta ABF$ .

(iii)

Find the inclination of the thin metal sheet $ABC$ to the horizontal ground.

(iv)

The craftsman claims that the area of $\Delta BDF$ is greater than $460\text{ cm}^{2}$ . Do you agree? Explain your answer. (11 marks)

2018 · Paper 1 Q1 Formulae

(a)

Make $b$ the subject of the formula $\frac{a+4}{3}=\frac{b+1}{2}$ .

(b)

(3 marks)

2018 · Paper 1 Q2 Laws of integral indices

(a)

Simplify $\frac{xy^7}{(x^{-2}y^3)^4}$ and express your answer with positive indices.

(b)

(3 marks)

2018 · Paper 1 Q3 Approximate values and numerical estimation

(a)

Round up $265.473$ to the nearest integer.

(b)

Round down $265.473$ to $1$ decimal place.

(c)

Round off $265.473$ to $2$ significant figures. (3 marks)

2018 · Paper 1 Q4 Probability

A box contains $n$ white balls, $5$ black balls and $8$ red balls. If a ball is randomly drawn from the box, then the probability of drawing a red ball is $\frac{2}{5}$ . Find the value of $n$ . (3 marks)

2018 · Paper 1 Q5 Polynomials

Factorize

(a)

$9r^{3}-18r^{2}s$

(b)

$9 r^{3}-18 r^{2}s-r s^{2}+2 s^{3}.$ (4 marks)

2018 · Paper 1 Q6 Inequalities and linear programming

(a)

Find the range of values of $x$ which satisfy both $\frac{3-x}{2}>2x+7$ and $x+8\geq 0$ .

(b)

Write down the greatest integer satisfying both inequalities in (a). (4 marks)

2018 · Paper 1 Q7 Using percentages

The marked price of a vase is $30\%$ above its cost. A loss of $$88$ is made by selling the vase at a discount of $40\%$ on its marked price. Find the marked price of the vase. (5 marks)

2018 · Paper 1 Q8 Basic properties of circles

In Figure 1, $ABCDE$ is a circle. It is given that $AB//ED$ . $AD$ and $BE$ intersect at the point $F$ .

(a)

Express $x$ and $y$ in terms of $\theta$ . (5 marks)

2018 · Paper 1 Q9 Rates, ratios and proportions

A car travels from city $P$ to city $Q$ at an average speed of $72\text{ km/h}$ and then the car travels from city $Q$ to city $R$ at an average speed of $90\text{ km/h}$ . It is given that the car travels $210\text{ km}$ in $161$ minutes for the whole journey. How long does the car take to travel from city $P$ to city $Q$ ? (5 marks)

2018 · Paper 1 Q10 Measures of dispersion

The box-and-whisker diagram below shows the distribution of the ages of the clerks in team X of a company. It is given that the range and the inter-quartile range of this distribution are $43$ and $21$ respectively.

(a)

Find $a$ and $b$ .

(b)

There are five clerks in team Y of the company and three of them are of age $50$ or is given that the range of the ages of the clerks in team Y is $20$ . Team X and team Y are now combined to form a section. The manager of the company claims that the range of the ages of the clerks in the section and the range of the ages of the clerks in team X must be the same. Do you agree? Explain your answer. (2 marks)

2018 · Paper 1 Q11 Organisation of data

The following table shows the distribution of the numbers of children of some families:

[Table]

It is given that $k$ is a positive integer.

(a)

If the mode of the distribution is 2, write down

(i)

the least possible value of $k$ ;

(ii)

the greatest possible value of $k$ .

(b)

If the median of the distribution is 2, write down

(i)

the least possible value of $k$ ;

(ii)

the greatest possible value of $k$ .

(c)

If the mean of the distribution is 2, find the value of $k$ . (2 marks)

2018 · Paper 1 Q12 More about polynomials

Let $f(x)=4x(x+1)^{2}+ax+b$ , where $a$ and $b$ are constants. It is given that $x-3$ is a factor of $f(x)$ . When $f(x)$ is divided by $x+2$ , the remainder is $2b+165$ .

(a)

Find $a$ and $b$ . (3 marks)

(b)

Someone claims that the equation $f(x)=0$ has at least one irrational root. Do you agree? Explain your answer. (4 marks)

2018 · Paper 1 Q13 Similar triangles

In Figure 2, $ABCD$ is a trapezium with $\angle ABC = 90^\circ$ and $AB \parallel DC$ . $E$ is a point lying on $BC$ such that $\angle AED = 90^\circ$ .

(a)

Prove that $\triangle ABE \sim \triangle ECD$ . (2 marks)

(b)

It is given that $AB = 15\text{ cm}$ , $AE = 25\text{ cm}$ and $CE = 36\text{ cm}$ .

(i)

Find the length of $CD$ .

(ii)

Find the area of $\triangle ADE$ .

(iii)

Is there a point $F$ lying on $AD$ such that the distance between $E$ and $F$ is less than $23\text{ cm}$ ? Explain your answer. (6 marks)

2018 · Paper 1 Q14 Mensuration

A right circular cylindrical container of base radius $8\text{ cm}$ and height $64\text{ cm}$ and an inverted right circular conical vessel of base radius $20\text{ cm}$ and height $60\text{ cm}$ are held vertically. The container is fully filled with water. The water in the container is now poured into the vessel.

(a)

Find the volume of water in the vessel in terms of $\pi$ . (2 marks)

(b)

Find the depth of water in the vessel. (4 marks)

(c)

If a solid metal sphere of radius $14\text{ cm}$ is then put into the vessel and the sphere is totally immersed in the water, will the water overflow? Explain your answer. (3 marks)

2018 · Paper 1 Q14 Equations of straight lines

The coordinates of the points $A$ , $B$ and $C$ are $(4, -8)$ , $(-1, -2)$ and $(-2, 2)$ respectively. $D$ is a point lying on the line segment $BC$ such that $AD$ is perpendicular to $BC$ . Let $E$ be the intersection point of $AD$ and the $y$ -axis. Suppose that $F$ is a point lying on the line segment $AB$ such that the area of $\Delta AEF$ is $\frac{1}{3}$ of the area of $\Delta ABD$ .

(a)

Find the coordinates of $D$ and $E$ .

(b)

Find the equation of the circle passing through $A$ , $E$ and $F$ .

2018 · Paper 1 Q15 Permutations and combinations

An eight-digit phone number is formed by a permutation of 2, 3, 4, 5, 6, 7, 8 and 9.

(a)

How many different eight-digit phone numbers can be formed? (1 mark)

(b)

If the first digit and the last digit of an eight-digit phone number are odd numbers, how many different eight-digit phone numbers can be formed? (2 marks)

2018 · Paper 1 Q16 Arithmetic and geometric sequences and their summations

The 3rd term and the 4th term of a geometric sequence are 720 and 864 respectively.

(a)

Find the $1^{st}$ term of the sequence. (2 marks)

(b)

Find the greatest value of $n$ such that the sum of the $(n+1)$ th term and the $(2n+1)$ th term is less than $5 \times 10^{14}$ . (3 marks)

2018 · Paper 1 Q17 Trigonometry

(a)

In Figure 3(a), $ABCD$ is a paper card in the shape of a parallelogram. It is given that $AB = 60 \text{ cm}$ , $\angle ABD = 20^{\circ}$ and $\angle BAD = 120^{\circ}$ . Find the length of $AD$ .

(b)

The paper card in Figure 3(a) is folded along $BD$ such that the distance between $A$ and $C$ is $40 \text{ cm}$ (see Figure 3(b)).

(i)

Find $\angle ABC$ .

(ii)

Find the angle between the plane $ABD$ and the plane $BCD$ . (5 marks)

2018 · Paper 1 Q18 Variations

It is given that $f(x)$ partly varies as $x^{2}$ and partly varies as $x$ . Suppose that $f(2)=60$ and $f(3)=99$ .

(a)

Find $f(x)$ . (3 marks)

(b)

Let Q be the vertex of the graph of $y = f(x)$ and R be the vertex of the graph of $y = 27 - f(x)$ .

(i)

Using the method of completing the square, find the coordinates of Q.

(ii)

Write down the coordinates of R.

(iii)

The coordinates of the point $S$ are $(56, 0)$ . Let $P$ be the circumcentre of $\Delta QRS$ . Describe the geometric relationship between $P$ , $Q$ and $R$ . Explain your answer. (5 marks)

2018 · Paper 1 Q19 Equations of circles

(a)

Find the equation of $C$ in terms of $r$ . Hence, express $r^{2}$ in terms of $k$ . (4 marks)

(b)

$L$ passes through the point $D(18,39)$ .

(i)

Find $r$ .

(ii)

It is given that $L$ cuts the $y$ -axis at the point $E$ . Let $F$ be a point such that $C$ is the inscribed circle of $\Delta DEF$ . Is $\Delta DEF$ an obtuse-angled triangle? Explain your answer. (8 marks)

Practise Paper · Paper 1 Q1 Laws of integral indices

(a)

Simplify $\frac{(m^{5}n^{-2})^{6}}{m^{4}n^{-3}}$ and express your answer with positive indices. (3 marks)

Practise Paper · Paper 1 Q2 Formulae

(a)

Make a the subject of the formula $\frac{5+b}{1-a}=3b$ (3 marks)

Practise Paper · Paper 1 Q3 More about polynomials

(a)

Factorize

$9x^{2}-42x y+49y^{2},$

$9x^{2}-42x y+49y^{2}-6x+14y\quad.$ (3 marks)

Practise Paper · Paper 1 Q4 Using percentages

The cost of a chair is $$360 $. If the chair is sold at a discount of$ 20% $on its marked price, then the percentage profit is$ 30%$. Find the marked price of the chair. (4 marks)

Practise Paper · Paper 1 Q5 Rates, ratios and proportions

The ratio of the capacity of a bottle to that of a cup is $4:3$ . The total capacity of 7 bottles and 9 cups is 11 litres. Find the capacity of a bottle. (4 marks)

Practise Paper · Paper 1 Q6 Trigonometry

In a polar coordinate system, the polar coordinates of the points $A$ , $B$ and $C$ are $(13,157^{\circ})$ , $(14,247^{\circ})$ and $(15,337^{\circ})$ respectively.

(a)

Let $O$ be the pole. Are $A$ , $O$ and $C$ collinear? Explain your answer.

(b)

Find the area of $\Delta ABC$ . (4 marks)

Practise Paper · Paper 1 Q7 Basic properties of circles

In Figure 1, $BD$ is a diameter of the circle $ABCD$ . If $AB=AC$ and $\angle BDC=36^{\circ}$ , find $\angle ABD$ (4 marks)

Practise Paper · Paper 1 Q8 Loci

The coordinates of the points $A$ and $B$ are $(-3, 4)$ and $(-2, -5)$ respectively. $A'$ is the reflection image of $A$ with respect to the $y$ -axis. $B$ is rotated anticlockwise about the origin $O$ through $90^{\circ}$ to $B'$ .

(a)

Write down the coordinates of $A'$ and $B'$ .

(b)

Let $P$ be a moving point in the rectangular coordinate plane such that $P$ is equidistant from $A'$ and $B'$ . Find the equation of the locus of $P$ . (5 marks)

Practise Paper · Paper 1 Q9 Measures of dispersion

(a)

Find the least possible value and the greatest possible value of the inter-quartile range of the distribution.

(b)

If $r=9$ and the median of the distribution is 3, how many possible values of $s$ are there? Explain your answer.

Practise Paper · Paper 1 Q10 More about polynomials

Let $f(x)$ be a polynomial. When $f(x)$ is divided by $x-1$ , the quotient is $6x^{2}+17x-2$ . It is given that $f(1)=4$ .

(a)

Find $f(-3)$ .

(b)

Factorize $f(x)$ .