Past paper questions

Factorize

The number of stickers owned by a boy is $3$ times that owned by a girl. If the boy gives $20$ of his stickers to the girl, then the number of stickers owned by the girl is $2$ times that owned by the boy. Find the total number of stickers owned by the boy and the girl. (4 marks)

The marked price of a shirt is higher than its cost by $
80$. The shirt is sold at a discount of$10\%$on its marked price. After selling the shirt, the percentage profit is$30\%$. Find the marked price of the shirt. (4 marks)

The bar chart below shows the distribution of the numbers of books read by a group of students in a year.

Distribution of the numbers of books read by the group of students in the year

Figure 1

The table below shows the distribution of the numbers of tokens got by a group of children in a game.

A queue is randomly formed by 7 teachers and 3 students.

The straight lines $L_{1}$ and $L_{2}$ are perpendicular to each other. The y-intercept of $L_{1}$ is 3. It is given that $L_{1}$ and $L_{2}$ intersect at the point $(2,6)$. Let $R$ be the region (including the boundary) bounded by $L_{1}$, $L_{2}$ and the x-axis.

Let $A(n)$ be the $n$th term of an arithmetic sequence. It is given that $A(5) = 26$ and $A(12) = 61$.

1. $\frac{(2^n)(8^{3n})}{64^n} =$

2. If $m(m-a)=a(1-m)$, then $a=$

3. $(u + \nu)(u - \nu)(u - 1) =$

If $x=6.24$ (correct to 2 decimal places), find the range of values of $x$.

If $a$, $b$ and $c$ are non-zero constants such that $a(x+3)+b(3x+1)\equiv c(x+2)$, then $a:b=$

Let $f(x)=(x+h)(x-3)+k$, where h and k are constants. If $f(0)=f(8)=1$, find k.

Let $p(x)$ be a polynomial. When $p(x)$ is divided by $x+1$, the remainder is $-2$. If $p(x)$ is divisible by $x-1$, find the remainder when $p(x)$ is divided by $x^{2}-1$.

In a school, $33\%$ of the students are overweight. It is given that $60\%$ of the students in the school are girls and $45\%$ of the girls are overweight. If $x\%$ of the boys in the school are overweight, then $x=$

The solution of $9x + 8 \leq 4(x - 3)$ or $6 - 7x > 20$ is

If $\alpha$ and $\beta$ are non-zero numbers such that $\frac{2\alpha+3\beta}{3\alpha+2\beta}=\frac{7}{10}$, then $\frac{2\alpha+\beta}{\alpha+2\beta}=$

If $w$ varies directly as the square of $x$ and inversely as the cube of $y$, which of the following must be constant?

In the figure, the 1st pattern consists of 3 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $(2n+3)$ dots to the $n$th pattern. Find the number of dots in the 8th pattern.

Let $m$ and $n$ be real constants. Which of the following statements about the graph of $y=(m-x)^{2}+n$ must be true?

I. The graph opens upwards.

II. The $y$-intercept of the graph is positive.

III. The graph passes through the point $(n, m)$.

The base of a solid right prism is a regular 6-sided polygon of side $8$ cm. If the volume of the prism is $288\text{ cm}^{3}$, find the total surface area of the prism correct to the nearest $\text{ cm}^{2}$.

The sum of the total surface areas of two solid hemispheres is $351\pi\text{ cm}^{2}$. If the ratio of the radius of the smaller hemisphere to the radius of the larger hemisphere is $2:3$, then the difference of the volumes of the two hemispheres is

The area of the sector $OAB$ is $\pi\text{ cm}^2$, where $O$ is the centre of the sector $OAB$. If $\angle AOB = 90^{\circ}$, which of the following are true?

In the figure, $AB = BC$ and $AB \parallel CD$. Let $E$ be the point of intersection of $AD$ and $BC$. If $\angle ADC = 28^{\circ}$ and $\angle AEB = 94^{\circ}$, then $\angle CAD =$

In the figure, $ABCD$ is a rectangle. Let $E$ be a point lying on $AC$ such that $BE$ is perpendicular to $AC$. $BE$ is produced to the point $F$ such that $CF = AD$. Denote the point of intersection of $BF$ and $CD$ by $G$. Which of the following are true?

I. $\angle DAE = \angle DGF$
II. $\triangle BCE \sim \triangle CGE$
III. $\triangle BCE \cong \triangle FCE$

In the figure, $ABCD$ is a square. Let $E$ and $F$ be points lying on $AB$ and $BC$ respectively such that $AE = 3BE$ and $\angle DEF = 90^{\circ}$. If the area of $\triangle DEF$ is $25\text{ cm}^{2}$, then the area of $\triangle CDF$ is

If $ABCDEFGH$ is a regular 8-sided polygon, which of the following are true?

I. $AG \parallel BF$

II. $BD = EG$

III. $\angle CAG = 2\angle BDH$

In the figure, $ABCD$ is a circle. If $AC = BD$, $\angle AED = 96^\circ$ and $\angle BDC = 14^\circ$, then $\angle CAD =$

The coordinates of the point $P$ are $(7,5)$. $P$ is reflected with respect to the $y$-axis to the point $Q$. $Q$ is then rotated clockwise about the origin through $90^{\circ}$ to the point $R$. Find the $x$-coordinate of $R$.

In the figure, $\frac{AB}{CD} =$

The coordinates of the points M and N are $(5, 7)$ and $(6, 8)$ respectively. Let P be a moving point in the rectangular coordinate plane such that PM = MN. Find the equation of the locus of P.

The coordinates of the points A, B and C are $(3, 3)$, $(5, 8)$ and $(9, 2)$ respectively. Let P be a point such that AP is a median of $\triangle ABC$. Find the equation of the straight line which passes through A and P.

The slope of the straight line L is 4. It is given that L and the circle $x^{2}+y^{2}-18x-20y+96=0$ intersect at the points P and Q. If the coordinates of the mid-point of PQ are $(s,t)$, which of the following must be true?

The stem-and-leaf diagram below shows the distribution of the weights (in kg) of a group of workers.

$$\begin{array}{c|ccccccc}{\mathrm{Stem~(tens)}}&{\mathrm{Leaf~(units)}}&{}{} \\ {5}&{3}&{6}&{7}&{\vdots}&{\vdots}&{\vdots}&{\vdots}\\ {6}&{1}&{2}&{2}&{2}&{6}&{8}&{\vdots}\\ {7}&{2}&{3}&{4}&{5}&{7}&{7}&{9}\\ {8}&{3}&{4}&{5}&{6}&{6}&{7}&{8} \end{array}$$

If a worker is randomly selected from the group, find the probability that the weight of the selected worker is not less than the lower quartile of the distribution.

The box-and-whisker diagram below shows the distribution of the ages of a group of researchers. Find the inter-quartile range of the distribution.

The mean of $70$ integers is $32$. If the mean of $30$ of these $70$ integers is $24$, then the mean of the remaining $40$ integers is

The H.C.F. and the L.C.M. of three expressions are $x^2 y^2 z$ and $x^3 y^4 z^5$ respectively. If the first expression and the second expression are $x^3 y^2 z^2$ and $x^3 y^3 z^5$ respectively, then the third expression is