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 $\angle ACB = 42^\circ$.

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.

The bar chart below shows the distribution of the most favourite fruits of the students in a group. It is given that each student has only one most favourite fruit.

Distribution of the most favourite fruits of the students in the group

If a student is randomly selected from the group, then the probability that the most favourite fruit is apple is $\frac{3}{20}$.

In Figure 3, $OABC$ is a circle. It is given that $AB$ produced and $OC$ produced meet at $D$.

There are 18 boys and 12 girls in a class. From the class, 4 students are randomly selected to form the class committee.

Figure 4 shows a geometric model $ABCD$ in the form of tetrahedron. It is found that $\angle ACB = 60^{\circ}$, $AC = AD = 20$ cm, $BC = BD = 12$ cm and $CD = 14$ cm.