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 $\angle ACB = 42^\circ$.

A box contains $n$ white balls, 5 black balls and 8 red balls. If a ball is randomly drawn from the box, then the probability of drawing a red ball is $\frac{2}{5}$. Find the value of $n$. (3 marks)

The marked price of a vase is $30\%$ above its cost. A loss of \88$is made by selling the vase at a discount of$40\%$ on its marked price. Find the marked price of the vase. (5 marks)

In Figure 1, $ABCDE$ is a circle. It is given that $AB//ED$. $AD$ and $BE$ intersect at the point $F$.

A car travels from city $P$ to city $Q$ at an average speed of $72\text{ km/h}$ and then the car travels from city $Q$ to city $R$ at an average speed of $90\text{ km/h}$. It is given that the car travels $210\text{ km}$ in $161$ minutes for the whole journey. How long does the car take to travel from city $P$ to city $Q$? (5 marks)

The box-and-whisker diagram below shows the distribution of the ages of the clerks in team X of a company. It is given that the range and the inter-quartile range of this distribution are $43$ and $21$ respectively.

The following table shows the distribution of the numbers of children of some families:

The 3rd term and the 4th term of a geometric sequence are $720$ and $864$ respectively.

It is given that $f(x)$ partly varies as $x^{2}$ and partly varies as x. Suppose that $f(2)=60$ and $f(3)=99$.

Simplify $\frac{(m^{5}n^{-2})^{6}}{m^{4}n^{-3}}$ and express your answer with positive indices.

Make $a$ the subject of the formula $\frac{5+b}{1-a}=3b$

Factorize

The cost of a chair is \360$. If the chair is sold at a discount of$20\%$on its marked price, then the percentage profit is$30\%$. Find the marked price of the chair.

The ratio of the capacity of a bottle to that of a cup is $4:3$. The total capacity of 7 bottles and 9 cups is $11$ litres. Find the capacity of a bottle.

In Figure 1, $BD$ is a diameter of the circle $ABCD$. If $AB = AC$ and $\angle BDC = 36^{\circ}$, find $\angle ABD$ (4 marks)

The coordinates of the points A and B are $(-3, 4)$ and $(-2, -5)$ respectively. $A'$ is the reflection image of A with respect to the y-axis. B is rotated anticlockwise about the origin O through $90^{\circ}$ to $B'$.

Let $C$ be the cost of manufacturing a cubical carton of side $x$ cm. It is given that $C$ is partly constant and partly varies as the square of $x$. When $x=20$, $C=42$; when $x=120$, $C=112$.

Figure 2 shows the graphs for Ada and Billy running on the same straight road between town $P$ and town $Q$ during the period 1:00 to 3:00 in an afternoon. Ada runs at a constant speed. It is given that town $P$ and town $Q$ are $16\text{ km}$ apart.