Past paper questions

Factorize

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 pie chart below shows the distribution of the seasons of birth of the students in a school.

Distribution of the seasons of birth of the students in the school

If a student is randomly selected from the school, then the probability that the selected student was born in spring is $\frac{1}{9}$.

It is given that $y$ varies inversely as $\sqrt{x}$. When $x=144$, $y=81$.

In Figure 1, $OPQR$ is a quadrilateral such that $OP = OQ = OR$. $OQ$ and $PR$ intersect at the point $S$. $S$ is the mid-point of $PR$.

The stem-and-leaf diagram below shows the distribution of the hourly wages (in dollars) of the workers in a group.

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

It is given that the mean and the range of the above distribution are $70 and$22 respectively.

A solid metal right prism of base area $84 ext{ cm}^{2}$ and height $20 ext{ cm}$ is melted and recast into two similar solid right pyramids. The bases of the two pyramids are squares. The ratio of the base area of the smaller pyramid to the base area of the larger pyramid is $4:9$.

The coordinates of the points $E$, $F$ and $G$ are $(-6,5)$, $(-3,11)$ and $(2,-1)$ respectively. The circle $C$ passes through $E$ and the centre of $C$ is $G$.

Let $a$ and $b$ be constants. Denote the graph of $y = a + \log_{b}x$ by $G$. The $x$-intercept of $G$ is 9 and $G$ passes through the point $(243, 3)$.

A city adopts a plan to import water from another city. It is given that the volume of water imported in the 1st year since the start of the plan is $1.5 \times 10^7 \, \text{m}^3$ and in subsequent years, the volume of water imported each year is $10\%$ less than the volume of water imported in the previous year.

The equation of the parabola $\Gamma$ is $y=2x^{2}-2kx+2x-3k+8$, where $k$ is a real constant. Denote the straight line $y=19$ by $L$.

ABC is a thin triangular metal sheet, where $BC = 24\text{ cm}$, $\angle BAC = 30^\circ$ and $\angle ACB = 42^\circ$.

If $\frac{a+4b}{2a}=2+\frac{b}{a}$, then $a=$

The solution of $6-x<2x-3$ or $7-3x>1$ is

Let $k$ be a constant. If $f(x)=2x^{2}-5x+k$, then $f(2)-f(-2)=$

Let $p(x)=2x^{2}-11x+c$, where $c$ is a constant. If $p(x)$ is divisible by $x-7$, find the remainder when $p(x)$ is divided by $2x+1$.

If $m$ and $n$ are constants such that $4x^{2}+m(x+1)+28=mx(x+3)+n(x-4)$, then $n=$

The figure shows the graph of $y=(px+5)^{2}+q$, where $p$ and $q$ are constants. Which of the following is true?

A sum of \2\,000$is deposited at an interest rate of$5\%$per annum for$4$ years, compounded half-yearly. Find the interest correct to the nearest dollar.

The scale of a map is $1:20\,000$. If the area of a zoo on the map is $4\text{ cm}^{2}$, then the actual area of the zoo is

It is given that $y$ is the sum of two parts, one part is a constant and the other part varies as $x^{2}$. When $x=1$, $y=7$ and when $x=2$, $y=13$. If $x=3$, then $y=$

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

In the figure, $D$ is a point lying on $AC$ such that $BD$ is perpendicular to $AC$. It is given that $AC=14\text{ cm}$ and $BD=12\text{ cm}$. If the area of $\triangle ABD$ is greater than the area of $\triangle BCD$ by $24\text{ cm}^{2}$, then the perimeter of $\triangle ABC$ is

The base radius of a right circular cone is 2 times the base radius of a right circular cylinder while the height of the circular cylinder is 3 times the height of the circular cone. If the volume of the circular cone is $36\pi\ \text{cm}^{3}$, then the volume of the circular cylinder is

In the figure, $ABCD$ and $BEDF$ are parallelograms. $E$ is a point lying on $BC$ such that $BE:EC=2:3$. $AC$ cuts $BF$ and $DE$ at $G$ and $H$ respectively. If the area of $\triangle ABG$ is $135\mathrm{~cm}^{2}$, then the area of the quadrilateral $DFGH$ is

In the figure, $ABC$ is an equilateral triangle of side $16\mathrm{~cm}$. $D$ and $E$ are points lying on $AB$ and $BC$ respectively such that $AD=4\mathrm{~cm}$ and $\angle CDE=60^{\circ}$. Find $CE$.

In the figure, $AB = BC$ and $D$ is a point lying on $AE$ such that $AC = AD$. If $AE \parallel BC$, then $\angle ABC =$

In the figure, the length of the line segment joining A and H is

$ABCD$ is a parallelogram. Let $E$ be the mid-point of $AD$. If $\angle ABE = \angle CBD = \angle DBE$, which of the following are true?

In the figure, $AD$ is a diameter of the circle $ABCDE$. If $BC = CD$ and $\angle ABC = 110^{\circ}$, then $\angle BED =$

In the figure, $ABCD$ is a rectangle. If $E$ is a point lying on $CD$ such that $\angle CBE=40^{\circ}$, find $\angle AED$ correct to the nearest degree.

In the figure, the equations of the straight lines $L_{1}$ and $L_{2}$ are $x + my = n$ and $x + py = q$ respectively. Which of the following are true?

The straight line L is perpendicular to the straight line $9x - 5y + 45 = 0$. If the $x$-intercept of L is $-3$, then the equation of L is

The polar coordinates of the points $P$, $Q$ and $R$ are $(3,160^{\circ})$, $(4,280^{\circ})$ and $(6,340^{\circ})$ respectively. The perpendicular distance from $Q$ to $PR$ is

The equations of the circles $C_{1}$ and $C_{2}$ are $x^{2}+y^{2}+8x-4y-5=0$ and $2x^{2}+2y^{2}+8x-4y-5=0$ 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 is/are true?

I. $G_{1}$, $G_{2}$ and $O$ are collinear.

II. The radii of $C_{1}$ and $C_{2}$ are equal.

III. $O$ is equidistant from $G_{1}$ and $G_{2}$.

It is given that $A$ and $B$ are two distinct points lying on the circle $x^{2}+y^{2}-6x-4y-87=0$. Let $P$ be a moving point in the rectangular coordinate plane such that $AP=BP$. The equation of the locus of $P$ is $x+2y+k=0$, where $k$ is a constant. Find $k$.

The bar chart below shows the distribution of the numbers of tokens got by a group of children in a game. If a child is randomly selected from the group, find the probability that the selected child gets fewer than 5 tokens in the game.

The box-and-whisker diagram below shows the distribution of the numbers of online hours spent by a class of students in a certain week. Find the lower quartile of the distribution.

Consider the following positive integers:

$2$ $3$ $4$ $6$ $7$ $9$ $10$

Let $a$, $b$ and $c$ be the modes, the median and the range of the above positive integers respectively. If the mean of the above positive integers is $5$, which of the following must be true?

I. $a=2$

II. $b=4$

III. $c=8$

The figure above shows the graph of $y = f(x)$. If $g(x) = f\left(\frac{x}{2}\right)$, which of the following may represent the graph of $y = g(x)$?