Past paper questions

Make $h$ the subject of the formula $\frac{5}{h+k} = \frac{k}{h-3}$.

Simplify $\frac{x^{-8}y}{(x^7y^9)^{-6}}$ and express your answer with positive indices.

A packet of cheese is termed regular if its weight is measured as $220$ g correct to the nearest $10$ g. Someone claims that the total weight of $250$ regular packets of cheese can be measured as $53.6$ kg correct to the nearest $0.1$ kg. Is the claim correct? Explain your answer.

Consider the compound inequality

$3x + 2 > \frac{4x - 5}{2}$ and $3x - 2 < 7$ ..... (*).

On a ferry, the number of female passengers is $40\%$ more than the number of male passengers. If $24$ female passengers leave the ferry, then the number of male passengers is $40\%$ more than the number of female passengers. Find the number of male passengers on the ferry. (4 marks)

Let $a$, $b$ and $c$ be non-zero numbers such that $7a=6b$ and $\frac{4a-3c}{2b-c}=9$. Find

The stem-and-leaf diagram below shows the distribution of the numbers of working hours of a group of workers in a week.

The base radius and the curved surface area of a solid metal right circular cone are $14$ cm and $700\pi\text{ cm}^2$ respectively.

If $\frac{a+5b}{7a+2b}=\frac{1}{b+3}$，then $a=$

If $c$ and $d$ are constants such that $(x+2)(x+c)+12\equiv x(x+d)+6c(x+1)$, then $d=$

The solution of $x-3<-5$ or $\frac{6-x}{4}<2$ is

If $y = 73.8$ (correct to 3 significant figures), find the range of values of $y$.

Let $g(x)=13-5x^{2}$. If $\alpha$ is a constant, find $g(1-3\alpha)$.

Let $h(x)=ax^{6}+16x^{3}+b$, where $a$ and $b$ are constants. If $h(x)$ is divisible by $2x-3$, find the remainder when $h(x)$ is divided by $2x+3$.

Which of the following statements about the graph of $y = 5 + (x - 3)^2$ is true?

The marked price of a jacket is $60\%$ above its cost. A profit of \104$is made by selling the jacket at a discount of$25\%$ on its marked price. Find the cost of the jacket.

The scale of a map is $1:50\,000$. If the actual area of an airport is $10\text{ km}^{2}$, then the area of this airport on the map is ___.

It is given that $z$ varies as the square of $x$ and the cube root of $y$. When $x=12$ and $y=64$, $z=36$. When $x=16$ and $y=729$, $z=$

Let $a_n$ be the $n$th term of a sequence. If $a_6=23$, $a_8=60$ and $a_{n+2}=a_{n+1}+a_n$ for any positive integer $n$, then $a_3=$

The length of a side of a solid cube is $60$ cm. The volume of a solid right circular cylinder is equal to the volume of the cube while the curved surface area of the circular cylinder is equal to the total surface area of the cube. Find the base radius of the circular cylinder.

In the figure, $AC$ is a diameter of the circle $ABCD$ while $BD$ and $EF$ are diameters of the circle $BEDF$. It is given that $C$ and $E$ lie on $AF$. Let $G$ be the point of intersection of $AF$ and $BD$. If $AG = 30\text{ cm}$ and $CG = 10\text{ cm}$, find the area of the shaded region correct to the nearest $\text{cm}^{2}$.

In the figure, $PQRS$ is a parallelogram. Let $X$ be a point lying on $PQ$. Denote the point of intersection of $PR$ and $SX$ by $Y$. If the area of $\Delta PXY$ and the area of the quadrilateral $QRYX$ are $32\text{ cm}^{2}$ and $58\text{ cm}^{2}$ respectively, then the area of $\Delta RSY$ is

It is given that $ABCD$ is a rhombus. Denote the point of intersection of $AC$ and $BD$ by $E$. Which of the following must be true?

I. $AE=BE$

II. $\frac{AE}{AC}=\frac{BE}{BD}$

III. $AE^{2}+BE^{2}=CD^{2}$

The figure shows the square $ABCD$, the regular pentagon $ADEFG$ and the regular hexagon $AGHIJK$. Find $\angle ABK$.

In the figure, $PQRS$ is a rectangle. Let $T$ be a point lying on $QR$ such that $\angle PTS=90^{\circ}$. $PQ$ produced and $ST$ produced meet at the point $U$. $PT$ is produced to the point $V$ such that $RT=RV$. Which of the following must be true?

The figure shows the cyclic quadrilateral $RSTU$, where $ST = TU$. $RS$ produced and $UT$ produced meet at the point $V$ while $RU$ produced and $ST$ produced meet at the point $W$. If $\angle RWS = 32^{\circ}$ and $\angle RVU = 48^{\circ}$, then $\angle RSU =$

In the figure, $ABCD$ is a trapezium with $AD//BC$. Let $E$ be the mid-point of $AD$. It is given that $\angle ABE = \angle BCE = 90^\circ$. Find $\frac{CE}{DE}$.

The rectangular coordinates of the point $P$ are $(\sqrt{2}, -\sqrt{2})$. If $P$ is rotated anticlockwise about the origin through $90^{\circ}$, then the polar coordinates of its image are

Find the constant $a$ such that the straight lines $2x+(a+3)y-5=0$ and $ax-4y+1=0$ are perpendicular to each other.

The equations of the straight lines $\ell$ and $L$ are $9x+12y-37=0$ and $12x+16y+85=0$ respectively. $\ell$ cuts the x-axis at the point $A$ while $L$ cuts the y-axis at the point $B$. Let $P$ be a moving point in the rectangular coordinate plane such that the perpendicular distance from $P$ to $\ell$ is equal to the perpendicular distance from $P$ to $L$. Denote the locus of $P$ by $\Gamma$. Which of the following are true?

I. $\Gamma$ is parallel to $L$.

II. $\Gamma$ is perpendicular to $AB$.

III. $\Gamma$ passes through the mid-point of $AB$.

The equations of the circles $C_{1}$ and $C_{2}$ are $x^{2}+y^{2}+7x-4y+15=0$ and $2x^{2}+2y^{2}-2x-16y-17=0$ respectively. Let $G_{1}$ and $G_{2}$ be the centres of $C_{1}$ and $C_{2}$ respectively. Denote the origin by O. Which of the following are true?

I. $\Delta OG_{1}G_{2}$ is an equilateral triangle.

II. The line segment $OG_{1}$ lies inside $C_{2}$.

III. $C_{1}$ and $C_{2}$ intersect at two distinct points.

A box contains five cards numbered 1, 2, 3, 4 and 5 respectively while another box contains four cards numbered 6, 7, 8 and 9 respectively. If a number is randomly drawn from each box, find the probability that the product of the two numbers drawn is divisible by 4.