Past paper questions

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

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

Factorize

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.

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.

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

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 $B'$.

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 $x=120$, $C=112$.

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.

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

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

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

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

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.

If $3a+1=3(b-2)$, then $b=$

Let $m$ and $n$ be constants. If $m(x-3)^2+n(x+1)^2\equiv x^2-38x+41$, then $m=$

Let $f(x)=x^{4}-x^{3}+x^{2}-x+1$. When $f(x)$ is divided by $x+2$, the remainder is

Let $k$ be a constant. If the quadratic equation $3x^{2}+2kx-k=0$ has equal roots, then $k=$

In the figure, the $x$-intercepts of the straight lines $L_{1}$ and $L_{2}$ are $5$ while the $y$-intercepts of the straight lines $L_{2}$ and $L_{3}$ are $3$. Which of the following are true?

I. The equation of $L_{1}$ is $x=5$.

II. The slope of $L_{2}$ is $\frac{3}{5}$.

III. The point $(2,3)$ lies on $L_{3}$.

The figure shows the graph of $y = ax^2 - 2x + b$, where $a$ and $b$ are constants. Which of the following is/are true?

I. $a > 0$

II. $b < 0$

III. $ab < 1$

The solution of $4x > x - 3$ or $3 - x < x + 7$ is

John buys a vase for \1600$. He then sells the vase to Susan at a profit of$20\%$. At what price should Susan sell the vase in order to have a profit of$20\%$?

If the circumference of a circle is increased by $40\%$, then the area of the circle is increased by

Let $\alpha$ and $\beta$ be non-zero constants. If $(\alpha + \beta) \colon (3\alpha - \beta) = 7 : 3$, then $\alpha : \beta =$

If $z$ varies directly as $x$ and inversely as $y^{2}$, which of the following must be constant?

In the figure, $O$ is the centre of the sector $OABC$. If the area of $\Delta OAC$ is $12\text{ cm}^{2}$, find the area of the segment $ABC$.

The figure shows a right circular cone of height $8$ cm and slant height $17$ cm. Find the volume of the circular cone.

In the figure, $ABCD$ is a rectangle. $E$ is the mid-point of $BC$. $F$ is a point lying on $CD$ such that $DF = 2CF$. If the area of $\triangle CEF$ is $1\mathrm{~cm}^{2}$, then the area of $\triangle AEF$ is

In the figure, $AB = 4\,\text{cm}$, $BC = CD = DE = 8\,\text{cm}$ and $FG = 9\,\text{cm}$. Find the perimeter of $\Delta AEH$.

In the figure, $AB = BC$ and $D$ is a point lying on $BC$ such that $CD = DE$. If $AB \parallel CE$, find $\angle CDE$.

In the figure, $O$ is the centre of the semi-circle $ABCD$. $AC$ and $BD$ intersect at $E$. If $AD \parallel OC$, then $\angle AED =$

In the figure, $O$ is the centre of the circle $ABCD$. If $\overrightarrow{AB} = \overrightarrow{BC} = 2\overrightarrow{CD}$, then $\angle BCD =$

In the figure, $ABCD$ is a square. $F$ is a point lying on $AD$ such that $CF \parallel BE$. If $AB = AE$, find $\angle ABF$ correct to the nearest degree.

For $0^{\circ} \leq \theta \leq 90^{\circ}$, the least value of $\frac{30}{3\sin^{2}\theta+2\sin^{2}(90^{\circ}-\theta)}$ is

Which of the following parallelograms have rotational symmetry and reflectional symmetry?

If the point $(-2,-1)$ is reflected with respect to the straight line $y=-5$, then the coordinates of its image are

The coordinates of the points $A$ and $B$ are $(1,-3)$ and $(-5,7)$ respectively. If $P$ is a point lying on the straight line $y=x+2$ such that $AP=PB$, then the coordinates of $P$ are

The equation of a circle is $2x^{2}+2y^{2}+8x-12y+3=0$. Which of the following are true?

I. The coordinates of the centre of the circle are $(-2,3)$.

II. The radius of the circle is $7$.

III. The point $(2,3)$ lies outside the circle.

Two numbers are randomly drawn at the same time from four cards numbered $2$, $3$, $5$ and $7$ respectively. Find the probability that the sum of the numbers drawn is a multiple of $4$.

The box-and-whisker diagram below shows the distribution of the heights (in cm) of some students. Which of the following is/are true?

II. The inter-quartile range of the distribution is $15\text{ cm}$.
III. Less than half of the students are taller than $170\text{ cm}$.

The figure below shows the cumulative frequency polygons of the test score distributions $X$ and $Y$. Let $m_1$, $r_1$ and $s_1$ be the median, the range and the standard deviation of $X$ respectively while $m_2$, $r_2$ and $s_2$ be the median, the range and the standard deviation of $Y$ respectively. Which of the following are true?

I. $m_1 > m_2$
II. $r_1 > r_2$
III. $s_1 > s_2$

The figure above shows the graph of $y = f(x)$. If $2f(x) = g(x)$, which of the following may represent the graph of $y = g(x)$?