DSE Maths Core · Past Paper Questions

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 $f(x)$ .

Practise Paper · Paper 1 Q11 Variations

Let $C$ be the cost of manufacturing a cubical carton of side $x$ cm. It is given that $C$ is partly constant and partly varies as the square of $x$ . When $x=20$ , $C=42$ ; when $x=120$ , $C=112$ .

(a)

Find the cost of manufacturing a cubical carton of side $50\text{ cm }$ .

(b)

If the cost of manufacturing a cubical carton is $58$ , find the length of a side of the carton.

(2 marks)

Practise Paper · Paper 1 Q12 Functions and graphs

Figure 2 shows the graphs for Ada and Billy running on the same straight road between town $P$ and town $Q$ during the period 1:00 to 3:00 in an afternoon. Ada runs at a constant speed. It is given that town $P$ and town $Q$ are $16\text{ km}$ apart.

(a)

How long does Billy rest during the period? (2 marks)

(b)

How far from town $P$ do Ada and Billy meet during the period? (3 marks)

(c)

Use average speed during the period to determine who runs faster. Explain your answer. (2 marks)

Practise Paper · Paper 1 Q13 Probability

The bar chart below shows the distribution of the most favourite fruits of the students in a group. It is given that each student has only one most favourite fruit.

Distribution of the most favourite fruits of the students in the group

Figure 1

If a student is randomly selected from the group, then the probability that the most favourite fruit is apple is $\frac{3}{20}$ .

(a)

Find $k$ .

(b)

Suppose that the above distribution is represented by a pie chart.

(i)

Find the angle of the sector representing that the most favourite fruit is orange.

(ii)

Some new students now join the group and the most favourite fruit of each of these students is orange. Will the angle of the sector representing that the most favourite fruit is orange be doubled? Explain your answer. (4 marks)

Practise Paper · Paper 1 Q14 Basic properties of circles

In Figure 3, $OABC$ is a circle. It is given that $AB$ produced and $OC$ produced meet at $D$ .

(a)

Write down a pair of similar triangles in Figure 3. (2 marks)

(b)

Suppose that $\angle AOD = 90^{\circ}$ . A rectangular coordinate system, with $O$ as the origin, is introduced in Figure 3 so that the coordinates of $A$ and $D$ are $(6,0)$ and $(0,12)$ respectively. If the ratio of the area of $\triangle BCD$ to the area of $\triangle OAD$ is $16:45$ , find

(i)

the coordinates of $C$ ,

(ii)

the equation of the circle $OABC$ . (7 marks)

Practise Paper · Paper 1 Q15 Measures of dispersion

The mean score of a class of students in a test is $48$ marks. The scores of Mary and John in the test are $36$ marks and $66$ marks respectively. The standard score of Mary in the test is $-2$ .

(a)

Find the standard score of John in the test.

(b)

A student, David, withdraws from the class and his test score is then deleted. It is given that his test score is $48$ marks. Will there be any change in the standard score of John due to the deletion of the test score of David? Explain your answer. (2 marks)

Practise Paper · Paper 1 Q16 Probability

There are 18 boys and 12 girls in a class. From the class, 4 students are randomly selected to form the class committee.

(a)

Find the probability that the class committee consists of boys only. (2 marks)

(b)

Find the probability that the class committee consists of at least $1$ boy and 1 girl. (2 marks)

Practise Paper · Paper 1 Q17 Quadratic equations in one unknown

(a)

Express $\frac{1}{1+2i}$ in the form of $a+bi$ , where $a$ and $b$ are real numbers. (2 marks)

(b)

The roots of the quadratic equation $x^{2} + px + q = 0$ are $\frac{10}{1+2i}$ and $\frac{10}{1-2i}$ . Find

(i)

$p$ and $q$ ,

(ii)

the range of values of $r$ such that the quadratic equation $x^{2} + px + q = r$ has real roots. (5 marks)

Practise Paper · Paper 1 Q18 3-D figures

Figure 4 shows a geometric model $ABCD$ in the form of tetrahedron. It is found that $\angle ACB = 60^{\circ}$ , $AC = AD = 20$ cm, $BC = BD = 12$ cm and $CD = 14$ cm.

(a)

Find the length of $AB$ .

(2 marks)

(b)

Find the angle between the plane $ABC$ and the plane $ABD$ .

(4 marks)

(c)

Let $P$ be a movable point on the slant edge $AB$ . Describe how $\angle CPD$ varies as $P$ moves from $A$ to $B$ . Explain your answer.

(2 marks)

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

The amount of investment of a commercial firm in the 1st year is $$4000000 $. The amount of investment in each successive year is$ r% $less than the previous year. The amount of investment in the 4th year is$ $1048576$.

(a)

Find $r$ . (2 marks)

(b)

The revenue made by the firm in the 1st year is $$2000000 $. The revenue made in each successive year is$ 20%$ less than the previous year.

(i)

Find the least number of years needed for the total revenue made by the firm to exceed $$9000000$ .

(ii)

Will the total revenue made by the firm exceed $$10000000$? Explain your answer.

(iii)

The manager of the firm claims that the total revenue made by the firm will exceed the total amount of investment. Do you agree? Explain your answer. (10 marks)