Past paper questions

If $p$ and $q$ are constants such that $x^{2}+p\equiv(x+2)(x+q)+10$, then $p=$

If $k$ is a constant such that $x^{3}+4x^{2}+kx-12$ is divisible by $x+3$, then $k=$

If $m+2n+6=2m-n=7$, then $n=$

The figure shows the graph of $y=a(x+b)^{2}$, where $a$ and $b$ are constants. Which of the following is true?

The solution of $15+4x<3$ or $9-2x>1$ is

In a company, $37.5\%$ of the employees are female. If $60\%$ of the male employees and $80\%$ of the female employees are married, then the percentage of married employees in the company is

If $x$ and $y$ are non-zero numbers such that $\frac{6x+5y}{3y-2x}=7$, then $x:y=$

It is given that $y$ partly varies directly as $x^{2}$ and partly varies inversely as $x$. When $x=1$, $y=-4$ and when $x=2$, $y=5$. When $x=-2$, $y=$

Mary performs a typing task for $7$ hours. Her average typing speeds for the first $3$ hours and the last $4$ hours are $56$ words per minute and $56$ words per minute respectively. Find her average typing speed for the $7$ hours.

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

The length of a piece of thin string is measured as $25$ m correct to the nearest m. If the string is cut into $n$ pieces such that the length of each piece is measured as $5$ cm correct to the nearest cm, find the greatest possible value of $n$.

In the figure, the area of quadrilateral $ABCD$ is

In the figure, $OAB$ and $OCD$ are sectors with centre $O$. If $\widehat{AB}=12\pi$ cm, $\widehat{CD}=16\pi$ cm and $OA=30$ cm, then $AC=$

In the figure, $ABCD$ is a parallelogram. $E$ and $F$ are points lying on $AB$ and $CD$ respectively. $AD$ produced and $EF$ produced meet at $G$. It is given that $DF:FC=3:4$ and $AD:DG=1:1$. If the area of $\triangle DFG$ is $3\mathrm{~cm}^{2}$, then the area of the parallelogram $ABCD$ is

In the figure, $D$ is a point lying on $AC$ such that $BD$ is perpendicular to $AC$. If $BC = \ell$, then $AB =$

In the figure, $O$ is the centre of the circle $ABCD$. If $\angle BAO = 28^{\circ}$, $\angle BCD = 114^{\circ}$ and $\angle CDO = 42^{\circ}$, then $\angle ABC =$

In the figure, $AB$ is a diameter of the circle $ABCD$. If $AB=12\text{cm}$ and $CD=6\text{cm}$, then the area of the shaded region is

Which of the following statements about a regular 12-sided polygon are true?

I. Each exterior angle is $30^{\circ}$.

II. Each interior angle is $150^{\circ}$.

III. The number of axes of reflectional symmetry is 6.

The rectangular coordinates of the point P are $(-3, -3\sqrt{3})$. If P is rotated anticlockwise about the origin through $90^{\circ}$, then the polar coordinates of its image are

If $P$ is a moving point in the rectangular coordinate plane such that the distance between $P$ and the point $(20,12)$ is equal to 5, then the locus of $P$ is a

In the figure, the equations of the straight lines $L_{1}$ and $L_{2}$ are $ax+y=b$ and $cx+y=d$ respectively. Which of the following are true?

In the figure, the radius of the circle and the coordinates of the centre are $r$ and $(h, k)$ respectively. Which of the following are true?

$9\star\Diamond$ is a 3-digit number, where $\star$ and $\Diamond$ are integers from $0$ to $9$ inclusive. Find the probability that the 3-digit number is divisible by $5$.

The stem-and-leaf diagram below shows the distribution of the ages of a group of members in a recreational centre.

$$\begin{array}{c|ccccccccc}{\mathrm{Stem~(tens)}}&{\underline{{\mathrm{Leaf~(units)}}}}&{}\\ \hline {5}&{0}&{5}&{6}&{6}&{8}&{⋰}&{⋰}&{⋰}\\ {6}&{1}&{4}&{5}&{5}&{7}&{8}&{8}&{9}\\ {7}&{3}&{4}&{4}&{6}&{7}&{9}&{⋰}&{⋰}\\ {8}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}\\ {9}&{1}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰}&{⋰} \end{array}$$

A member is randomly selected from the group. Find the probability that the selected member is not under age of 74.

The bar chart below shows the distribution of the numbers of rings owned by the girls in a group. Find the standard deviation of the distribution correct to 2 decimal places.

Consider the following data:

19

10

12

12

13

13

14

15

16

m

n

If both the mean and the median of the above data are 14, which of the following are true?

I. $m \geq 14$

II. $n \leq 16$

III. $m + n = 30$

The H.C.F. and the L.C.M. of three expressions are $ab^{2}$ and $4a^{4}b^{5}c^{6}$ respectively. If the first expression and the second expression are $2a^{2}b^{4}c$ and $4a^{4}b^{2}c^{6}$ respectively, then the third expression is

The graph in the figure shows the linear relation between $x$ and $\log_{3} y$. If $y = mn^{x}$, then $n=$

Let $f(x)$ be a quadratic function. If the coordinates of the vertex of the graph of $y=f(x)$ are $(3,-4)$, which of the following must be true?

The figure shows a shaded region (including the boundary). If $(h, k)$ is a point lying in the shaded region, which of the following are true?

I. $k \geq 3$

II. $h - k \geq -3$

III. $2h + k \leq 6$

Let $a_{n}$ be the $n$th term of an arithmetic sequence. If $a_{18}=26$ and $a_{23}=61$, which of the following are true?

I. $a_{14}<0$

II. $a_{1}-a_{2}<0$

III. $a_{1}+a_{2}+a_{3}+\cdots+a_{27}>0$

Which of the following may represent the graph of $y = f(x)$ and the graph of $y = f(x - 2) + 1$ on the same rectangular coordinate system?

The figure shows

The figure shows a regular tetrahedron $ABCD$. Find the angle between the plane $ABC$ and the plane $BCD$ correct to the nearest degree.

In the figure, $PQ$ is the tangent to the circle $ABC$ at $O$, where $O$ is the centre of the semicircle $PBQ$. It is given that $BCP$ is a straight line. If $\angle BPQ = 12^{\circ}$, then $\angle BAC =$

Find the range of values of $k$ such that the circle $x^{2}+y^{2}+2x-4y-13=0$ and the straight line $x-y+k=0$ intersect at two distinct points.

A drama club is formed by 12 boys and 8 girls. If a team of 5 students is selected from the club to participate in a competition and the team consists of at least one girl, how many different teams can be formed?

A box contains six balls numbered $7$, $8$, $9$, $9$ and $9$ respectively. John repeats drawing one ball at a time randomly from the box without replacement until the number drawn is $9$. Find the probability that he needs exactly three draws.

Let $m_{1}$, $r_{1}$ and $v_{1}$ be the mean, the range and the variance of a group of numbers $\{x_{1}, x_{2}, x_{3}, \ldots, x_{100}\}$ respectively. If $m_{2}$, $r_{2}$ and $v_{2}$ are the mean, the range and the variance of the group of numbers $\{x_{1}, x_{2}, x_{3}, \ldots, x_{100}, m_{1}\}$ respectively, which of the following must be true?

I. $m_{1}=m_{2}$

II. $r_{1}=r_{2}$

III. $v_{1}=v_{2}$

If $\frac{a}{x} \div \frac{b}{y} = 3$, then $x =$

If $4\alpha + \beta = 7\alpha + 3\beta = 5$, then $\beta =$