Past paper questions

$(x+1)(x^{2}+x+1)=$

$\frac{(3y^6)^4}{3y^2} =$

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

$0.0023456789 =$

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 $5 ext{ kg}$ 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 $10 ext{ g}$ 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 $DN = 6 ext{ cm}$ and $EN = 5 ext{ cm}$, then the area of $\Delta ABC$ is

The height and the base radius of a right circular cone are $12\text{ cm}$ and $9\text{ cm}$ 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 $8\text{ cm}$. 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, $\frac{AD}{AB} =$

$\frac{\cos 180^{\circ}}{1+\sin(90^{\circ}+\theta)}+\frac{\cos 360^{\circ}}{1+\sin(270^{\circ}+\theta)}=$

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 = $24\text{ cm }$, 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 $3 \le q m \le q 5$, which of the following must be true?

I. $p > q$

II. $p > r$

III. $q > r$

$\frac{1}{x^2 - 2x + 1} - \frac{1}{x^2 + x - 2} =$

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

$11 + 2^{6} + 2^{10} + 2^{11} =$

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?

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. $x_2 + x_4 + x_6 + \cdot s + x_{2n} < 2015$

For $0^{\circ} \le q x < 360^{\circ}$, 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. $y_{1}\ge q y_{2}$

III. $z_{1}\le q z_{2}$

There are only two kinds of admission tickets for a theatre: regular tickets and concessionary tickets. The prices of a regular ticket and a concessionary ticket are $$126$ and $$78$ respectively. On a certain day, the number of regular tickets sold is 5 times the number of concessionary tickets sold and the sum of money for the admission tickets sold is $$50,976$. Find the total number of admission tickets sold that day. (4 marks)

The coordinates of the points $A$ and $B$ are $(-3, 4)$ and $(9, -9)$ respectively. $A$ is rotated anticlockwise about the origin through $90^{\circ}$ to $A'$. $B'$ is the reflection image of $B$ with respect to the $x$-axis.