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)

2022 · Paper 1 Q1 Laws of integral indices

(a)

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

(a)

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 marks)

2022 · Paper 1 Q4 More about polynomials

Factorize

(a)

$9c^{2}-6c+1$

(b)

$(4c+d)^{2}-9c^{2}+6c-1$ (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 Inequalities and linear programming

Consider the compound inequality

$-2(3x+2) > x+10$ or $2x \le q -8$ $\ldots$ $(*)$ .

(a)

Solve (*). (4 marks)

(b)

Write down the greatest integer satisfying (*).

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 $S'T'$

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 Variations

It is given that $f(x)$ partly varies as $x^{2}$ and partly varies as $x$ . Suppose that $f(4)=96$ and $f(-5)=15$ .

(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 Organisation of data

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.

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.

(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 $9$ 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 $10$ cm and $6$ cm respectively. A student finds that the base radius of $B$ is $12$ 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 Probability

There are 10 boys and 12 girls in a class. If 4 students are randomly selected from the class to form a committee,

(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 $\Delta OBM$ is the triple of the area of $\Delta OAM$ , 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

Let $c$ be a real constant. The roots of the equation $x^{2}+cx-9=0$ are $\alpha$ and $\beta$ .

(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

In Figure 2, the triangular paper card $PQR$ is held such that $PQ$ lies on the horizontal ground. It is given that $PQ = 30$ cm, $PR = 25$ cm and $\angle QPR = 95^\circ$ .

(a)

(i)

the length of $QR$ ,

(ii)

$\angle PQR$ .

(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.

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)