Past paper questions

Simplify $\frac{3}{k-9} + \frac{2}{5k+6}$

A fan is sold at a discount of $30\%$ on its marked price. After selling the fan, the profit is \78$and the percentage profit is$26\%$. Find the marked price of the fan. (4 marks)

Consider the compound inequality

$-2(3x+2) > x+10$ or $2x \leq -8$ $\ldots$ $(*)$.

The coordinates of the points $S$ and $T$ are $(12, -5)$ and $(-3, -7)$ respectively. $S$ is rotated anticlockwise about $O$ through $90^{\circ}$ to $S'$, where $O$ is the origin. $T'$ is the reflection image of $T$ with respect to the $x$-axis.

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the times taken to complete a $3\text{ km }$ race by a group of students.

The stem-and-leaf diagram below shows the distribution of the ages of the players of a football team.

Stem (tens) | Leaf (units)
1 | 7 8 9
2 | 0 a a 8 8 9
3 | b b 5 5 6 6 6 6 7 8
4 | 3

The inter-quartile range and the median of the distribution are $14$ and $31$ respectively.

The equation of the circle $C$ is $x^{2}+y^{2}-154x-128y+224=0$. Denote the centre of $C$ by $G$. The coordinates of the point $H$ are $(65, 48)$.

There are two solid metal spheres. The ratio of the surface area of the smaller sphere to the surface area of the larger sphere is $4:9$. The radius of the larger sphere is $9\text{ cm}$.

Let $p(x)=2x^{3}+ax^{2}+bx-20$, where $a$ and $b$ are constants. When $p(x)$ is divided by $x^{2}-2x+3$, the remainder is $x+13$.

Let $g(x) = 3x^{2} + 12kx + 16k^{2} + 8$, where $k$ is a non-zero real constant.

The centre of the circle $C$ is the point $G(83,112)$. It is found that the point $A(158,12)$ lies outside $C$. $AP$ and $AQ$ are the tangents to $C$ at the points $P$ and $Q$ respectively. It is given that $C$ passes through the point $(23,67)$.

If $m$ and $n$ are constants such that $(x+3)^{2}+mx\equiv(x-n)(x+1)+2n$, then $m=$

Let $c$ be a constant. Solve the equation $(x-c)(x-4c)=(3c-x)(x-4c)$.

If $\frac{2}{u} + \frac{3}{v} = 4$, then $u =$

It is given that $x$ is a real number. If $x$ is rounded down to 3 significant figures, then the result is 345. Find the range of values of $x$.

The solution of $3y-5<5y+1\leq 11$ is

Let $f(x)=x^2-x+1$. If $k$ is a constant, which of the following must be true?

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

Let $h$ and $k$ be real constants such that $hk<0$. Which of the following statements about the graph of $y=(h-x)(k-x)$ are true?

A sum of \88\,000$is deposited at an interest rate of$6\%$per annum for$4$ years, compounded monthly. Find the interest correct to the nearest dollar.

Let $x$, $y$ and $z$ be non-zero numbers. If $x:y=8:5$ and $2x=4z-3y$, then $y:z=$

If $u$ varies directly as the square root of $v$ and inversely as $w$, which of the following are true?

I. $u^{2}$ varies directly as $v$ and inversely as the square of $w$.

II. $v$ varies directly as $w$ and inversely as the square root of $u$.

III. $w$ varies directly as the square root of $v$ and inversely as $u$.

In the figure, the 1st pattern consists of 8 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $(2n+6)$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

The radius of a solid hemisphere and the base radius of a solid right circular cylinder are equal. If the height of the circular cylinder is equal to its base diameter, then the ratio of the total surface area of the hemisphere to the total surface area of the circular cylinder is

The diameter of a circle is $10\text{ cm}$. The circle is divided into a major segment and a minor segment by a chord of length $8\text{ cm}$. Find the area of the major segment correct to the nearest $\text{ cm}^{2}$.

In the figure, $M$ and $N$ are points lying on $PQ$ and $QR$ respectively such that $PM:MQ=5:6$ and $QN:NR=3:4$. If the area of the quadrilateral $MNRP$ is $59\text{ cm}^{2}$, then the area of $\Delta MNQ$ is

In the figure, the perimeter of the rectangle $ABCD$ is $170\text{ cm}$. It is given that $EBF$ is a straight line and $\angle AEB = \angle BFC = 90^{\circ}$. If $AE = 24\text{ cm}$ and $BC = 34\text{ cm}$, then $EF =$

In the figure, $ABC$ is an equilateral triangle. Let $D$ and $E$ be points lying on $AC$ and $BC$ respectively such that $AD=CE$. If $\angle CBD=38^{\circ}$, then $\angle AEB=$

The figure shows a parallelogram. Which of the following must be true?

I. $a + b = 180^{\circ}$
II. $b + c = 360^{\circ}$
III. $c + d = 540^{\circ}$

In the figure, $O$ is the centre of the circle $ABC$. If $\angle ABO = 36^{\circ}$, then $\angle ACB =$

In the figure, $ABC$ is a right-angled triangle with $\angle ABC = 90^\circ$. Let $D$ and $E$ be points lying on $AC$ and $BC$ respectively such that $ABED$ is a cyclic quadrilateral. If $AB = 660 \, \text{cm}$, $AD = 572 \, \text{cm}$ and $BE = 275 \, \text{cm}$, then $CD =$

It is given that $PQR$S is a rhombus. Let $T$ be the point of intersection of $PR$ and $QS$. If $\angle QRT = \theta$, then $\frac{PQ}{ST} =$

The figure shows the graph of the straight line $mx + ny = 3$. Which of the following are true?

The rectangular coordinates of the point Q are $(4\sqrt{3}, -4)$. If Q is rotated clockwise about the origin through $90^{\circ}$, then the polar coordinates of its image are

The straight line $12x - 5y = 60$ cuts the x-axis and the y-axis at the points A and B respectively. Let P be a moving point in the rectangular coordinate plane such that $AP = BP$. Find the equation of the locus of P.

The coordinates of the points $P$ and $Q$ are $(10, -24)$ and $(17, -7)$ respectively. Let $C$ be the circle which passes through the origin, $P$ and $Q$. Which of the following is true?

$5\diamond 2$ is a 3-digit number, where $\diamond$ is an integer from 0 to 9 inclusive. Find the probability that the 3-digit number is divisible by 7.

The mean weight of 60 actors and 40 actresses is $57$ kg. If the mean weight of the actors is $63$ kg, then the mean weight of the actresses is

Consider the following positive integers:

If both the mean and the median of the above positive integers are 6, which of the following must be true?

I. The mode of the above positive integers is 6.

II. The least possible range of the above positive integers is 6.

III. The greatest possible variance of the above positive integers is 6.

Which of the following is the least?