Past paper questions

If $p+3q=4$ and $5p+9q=2$, then p=

If $m$ and $n$ are constants such that $x^{2}+mx+n\equiv(x+4)(x-m)+6$, then $n=$

The solution of $18+7x>4$ or $5-2x<3$ is

If $\beta$ is a root of the equation $4x^{2}-5x-1=0$, then $7+10\beta-8\beta^{2}=$

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

If the price of a souvenir is increased by $70\%$ and then decreased by $60\%$, find the percentage change in the price of the souvenir.

A sum of \50\,000$is deposited at an interest rate of$6\%$per annum for$3$ years, compounded quarterly. Find the amount correct to the nearest dollar.

Let $a$, $b$ and $c$ be non-zero numbers. If $a:c=5:3$ and $b:c=3:2$, then $(a+c):(b+c)=$

It is given that $z$ varies as $x^{3}$ and $y^{2}$. When $x=2$ and $y=1$, $z=14$. When $x=3$ and $y=-2$, $z=$

In the figure, the 1st pattern consists of 5 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 6th pattern.

There is a bag of white sugar. The weight of white sugar in the bag is measured as $5$ kg correct to the nearest kg. If the bag of white sugar is packed into $n$ packets such that the weight of white sugar in each packet is measured as $10$ g correct to the nearest g, find the greatest possible value of $n$.

In the figure, $N$ is a point lying on $AC$ and $E$ is a point lying on $DN$. If $DN = 6$ cm and $EN = 5$ cm, then the area of $riangle ABC$ is

The height and the base radius of a right circular cone are $12$ cm and $9$ cm respectively. The figure shows a frustum which is made by cutting off the upper part of the circular cone. The height of the frustum is $8$ cm. Find the volume of the frustum.

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $CD$ such that $DE:EC=2:3$. $AD$ produced and $BE$ produced meet at $F$ while $AE$ produced and $BC$ produced meet at $G$. If the area of $\Delta DEF$ is $8\text{ cm}^{2}$, then the area of $\Delta CEG$ is

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

In the figure, $AD$ is a diameter of the circle $ABCDE$. If $\angle BAD=58^{\circ}$ and $BC=CD$, then $\angle AEC=$

The diameters $AC$ and $BD$ of the circle $ABCD$ intersect at the point $E$. If $\angle AEB = 90^{\circ}$ and $AC = 24$ cm, then the area of $\triangle AEB$ is

If an interior angle of a regular polygon is 5 times an exterior angle of the polygon, which of the following is/are true?

I. Each interior angle of the polygon is $150^{\circ}$.

II. The number of diagonals of the polygon is 6.

III. The number of folds of rotational symmetry of the polygon is 6.

The rectangular coordinates of the point $A$ are $(\sqrt{3}, -1)$. If $A$ is reflected with respect to the $y$-axis, then the polar coordinates of its image are

The coordinates of the points A and B are $(2, 0)$ and $(1, 5)$ respectively. If P is a moving point in the rectangular coordinate plane such that P is equidistant from A and B, then the locus of P is

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

I. $a < 0$

II. $a < b$

III. $c > 0$

A circle $C$ passes through the point $(0, 3)$. If the coordinates of the centre of $C$ are $(-4, 3)$, then the equation of $C$ is

Two fair dice are thrown in a game. If the sum of the two numbers thrown is $7$, \36$will be gained; otherwise,$\$6$ will be gained. Find the expected gain of the game.

The bar chart below shows the distribution of the numbers of keys owned by the students in a class. Find the probability that a randomly selected student from the class owns $3$ keys.

The box-and-whisker diagram below shows the distribution of the numbers of books read by some teachers in a term. Find the inter-quartile range of the distribution.

Consider the following integers:

Let $p$, $q$ and $r$ be the mean, the median and the mode of the above integers respectively. If $3 \leq m \leq 5$, which of the following must be true?

I. $p > q$

II. $p > r$

III. $q > r$

The graph in the figure shows the linear relation between $\log_3 x$ and $\log_3 y$. Which of the following must be true?

Let $k$ be a constant. If the roots of the quadratic equation $x^{2}+kx-2=0$ are $\alpha$ and $\beta$, then $\alpha^{2}+\beta^{2}=$

Let $z = (a+5)i^6 + (a-3)i^7$, where $a$ is a real number. If $z$ is a real number, then $a =$

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

I. $a \leq 4$
II. $a \geq b - 5$
III. $a \geq 10 - 2b$

Let $x_n$ be the $n$th term of a geometric sequence. If $x_6 = 216$ and $x_8 = 96$, which of the following must be true?

I. $x_3 = 729$

II. $\frac{x_5}{x_7} > 1$

III. $x_2 + x_4 + x_6 + \cdots + x_{2n} < 2015$

For $0^{\circ} \leq x < 360^{\circ}$, how many roots does the equation $\cos^{2}x - \sin x = 1$ have?

Let $k$ be a positive constant and $-180^{\circ} < \theta < 180^{\circ}$. If the figure shows the graph of $y = \sin(kx^{\circ} + \theta)$ then

Find the constant $k$ such that the circle $x^{2}+y^{2}+2x-6y+k=0$ and the straight line $x+y+4=0$ intersect at only one point.

Let $O$ be the origin. The coordinates of the points $P$ and $Q$ are $(0,60)$ and $(96,48)$ respectively. The $x$-coordinate of the orthocentre of $\Delta OPQ$ is

A queue is formed by 6 boys and 2 girls. If no girls are next to each other, how many different queues can be formed?

Bag $P$ contains $2$ red balls and $4$ green balls while bag $Q$ contains $1$ red ball and $3$ green balls. If a bag is randomly chosen and then a ball is randomly drawn from the bag, find the probability that a green ball is drawn.

Let $x_{1}$, $y_{1}$ and $z_{1}$ be the mean, the median and the variance of a group of numbers $\{a_{1}, a_{2}, a_{3}, \ldots, a_{50}\}$ respectively while $x_{2}$, $y_{2}$ and $z_{2}$ be the mean, the median and the variance of the group of numbers $\{a_{1}, a_{2}, a_{3}, \ldots, a_{49}\}$ respectively. If $x_{1} = a_{50}$, which of the following must be true?

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

II. $y_{1}\geq y_{2}$

III. $z_{1}\leq z_{2}$

Simplify $\frac{(mn^{-2})^{5}}{m^{-4}}$ and express your answer with positive indices.

Factorize

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

In a recruitment exercise, the number of male applicants is $28\%$ more than the number of female applicants. The difference of the number of male applicants and the number of female applicants is $91$. Find the number of male applicants in the recruitment exercise. (4 marks)

Consider the compound inequality