Past paper questions

Make $h$ the subject of the formula $9(h+6k)=7h+8$.

Simplify $\frac{3}{7x-6}-\frac{2}{5x-4}$.

The length and the breadth of a rectangle are $24\text{ cm}$ and $(13+r)\text{ cm}$ respectively. If the length of a diagonal of the rectangle is $(17-3r)\text{ cm}$, find $r$.

In a playground, the ratio of the number of adults to the number of children is $13:6$. If $9$ adults and $24$ children enter the playground, then the ratio of the number of adults to the number of children is $8:7$. Find the original number of adults in the playground. (4 marks)

Let $p(x)$ be a cubic polynomial. When $p(x)$ is divided by $x-1$, the remainder is $50$. When $p(x)$ is divided by $x+2$, the remainder is $-52$. It is given that $p(x)$ is divisible by $2x^{2}+9x+14$.

Let $f(x)=\frac{1}{1+k}\left(x^{2}+(6k-2)x+(9k+25)\right)$, where $k$ is a positive constant. Denote the point $(4,33)$ by $F$.

If $6x - 7y = 40 = 2x + 11y$, then $y =$

If $\alpha$ and $\beta$ are constants such that $(x-8)(x+\alpha)-6\equiv(x-9)^2+\beta$, then $\beta=$

If $h = 3 - \frac{5}{k + 4}$, then $k =$

If $0.06557 < x < 0.06564$, which of the following is true?

The least integer satisfying the compound inequality $-2(x-5)+5<21$ or $\frac{3x-5}{7}>1$ is

Let $c$ be a constant. If $f(x) = x^3 + cx^2 + c$, then $f(c) + f(-c) =$

Let $k$ be a constant such that $2x^{4} + kx^{3} - 4x - 16$ is divisible by $2x + k$. Find $k$.

Which of the following statements about the graph of $y=(3-x)(x+2)+6$ is/are true?

I. The graph opens downwards.

II. The graph passes through the point $(1,10)$.

III. The $x$-intercepts of the graph are $-2$ and $3$.

A. I only

B. II only

C. I and III only

D. II and III only

A sum of \65\,000$is deposited at an interest rate of$7\%$per annum for$8$ years, compounded quarterly. Find the amount correct to the nearest dollar.

The costs of tea of brand A and brand B are \140/\text{kg}$and$\$315/\text{kg}$ respectively. If $x$ kg of tea of brand A and $y$ kg of tea of brand B are mixed so that the cost of the mixture is \210/\text{kg}$, then$x:y=$

It is given that $z$ varies directly as the square of $x$ and inversely as the square root of $y$. If $x$ is decreased by $40\%$ and $y$ is increased by $44\%$, then $z$

In the figure, the 1st pattern consists of 6 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding 4 dots to the $n$th pattern. Find the number of dots in the 9th pattern.

The base of a solid right pyramid is a square of side $18\text{ cm}$. If the height of the pyramid is $12\text{ cm}$, then the total surface area of the pyramid is

In the figure, $ABCD$ is a parallelogram and $AEFG$ is a square. It is given that $BE:EF:FC = 2:7:3$. $BD$ cuts $AE$ and $FG$ at the points $X$ and $Y$ respectively. If the area of $\Delta ABX$ is $24\text{ cm}^{2}$, then the area of the quadrilateral $CDYF$ is

In the figure, $ABC$ and $ADE$ are straight lines. It is given that $AB = BD$ and $BC = CD$. If $\angle CDE = 66^{\circ}$, then $\angle ACD =$

In the figure, $ABC$ is an isosceles triangle with $AB = AC$. $D$ and $E$ are points lying on $AB$ such that $AD = DE = 2EB$ while $F$ is a point lying on $AC$ such that $DF // EC$. If $\angle ADF = 90^{\circ}$ and $CE = 60$ cm, then $EF =$

In the figure, $ABCD$ is a trapezium with $AB//DC$ and $\angle ABD=90^{\circ}$. If $AB=18\text{cm}$, $BC=26\text{cm}$ and $AD=30\text{cm}$, find the area of the trapezium $ABCD$.

In the figure, $ABCD$ is a rhombus. $ABE$ and $BCF$ are straight lines such that $BE = EF$. If $\angle BEF = 56^{\circ}$, then $\angle BDC =$

In the figure, O is the centre of the semi-circle $ABCD$. If $AC = BD$ and $\angle COD = 48^{\circ}$, then $\angle ABD =$

In the figure, $ABCD$ is a rectangle. $E$ is a point lying on $AD$. Find $\frac{CE}{AC}$

In the figure, the equation of the straight line $L$ is $ax + by + 15 = 0$. Which of the following are true?

I. $a>b$

II. $a>-3$

III. $b>-5$

Find the constant $k$ such that the straight lines $3x + 2y + k = 0$ and $kx + 12y - 6 = 0$ are perpendicular to each other.

The coordinates of the point $A$ are $(-5,-2)$ . $A$ is translated rightwards by 9 units to the point $B$ . $B$ is then rotated anticlockwise about the origin through $90^{\circ}$ to the point $C$ . Find the $y$-coordinate of $C$.

The equation of the straight line $L$ is $5x - 7y - 14 = 0$. If $P$ is a moving point in the rectangular coordinate plane such that the perpendicular distance from $P$ to $L$ is equal to 3, then the locus of $P$ is

Denote the circle $2x^{2}+2y^{2}+4x-12y+15=0$ by $C$. Which of the following is/are true?

Two numbers are randomly drawn at the same time from nine balls numbered 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. Find the probability that the two numbers drawn are consecutive integers.

Which of the following can be obtained from any box-and-whisker diagram?

I. Range

II. Standard deviation

III. Inter-quartile range

The table below shows the distribution of the numbers of merits obtained by some students in a year.

[Table]

It is given that $\log_{9} y$ is a linear function of $\log_{3} x$. The intercepts on the vertical axis and on the horizontal axis of the graph of the linear function are $7$ and $8$ respectively. Which of the following must be true?

If $\frac{3}{3\log x-2}+7=\frac{2}{2\log x+1}$, then $\log\frac{1}{x}=$