Past paper questions

In a recreation club, there are 180 members and the number of male members is $40\%$ more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)

It is given that $f(x)$ is the sum of two parts, one part varies as $x$ and the other part varies as $x^{2}$. Suppose that $f(3)=48$ and $f(9)=198$.

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is $12\text{ cm }$. Peter then pours $444\pi$ cm^{3} of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm }$.

If $4$ boys and $5$ girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)

In a test, the mean of the distribution of the scores of a class of students is $61$ marks. The standard scores of Albert and Mary are $-2.6$ and $1.4$ respectively. Albert gets $22$ marks. A student claims that the range of the distribution is at most $59$ marks. Is the claim correct? Explain your answer.

The 1st term and the 38th term of an arithmetic sequence are $666$ and $555$ respectively. Find

Figure 2 shows a geometric model $ABCD$ in the form of tetrahedron. It is given that $\angle BAD = 86^{\circ}$, $\angle CBD = 43^{\circ}$, $AB = 10$ cm, $AC = 6$ cm, $BC = 8$ cm and $BD = 15$ cm.

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 $\frac{(xy)^2}{x^{-5}y^6}$ and express your answer with positive indices.

Make $b$ the subject of the formula $a(b+7)=a+b$

Factorize

The marked price of a handbag is \560$. It is given that the marked price of the handbag is$40\%$ higher than the cost.

In a football league, each team gains 3 points for a win, 1 point for a draw and 0 point for a loss. The champion of the league plays 36 games and gains a total of 84 points. Given that the champion does not lose any games, find the number of games that the champion wins.

In Figure 2, $O$ is the centre of the semicircle $ABCD$. If $AB \parallel OC$ and $\angle BAD = 38^{\circ}$, find $\angle BDC$.

In Figure 3, the coordinates of the point $A$ are $(-2,5)$. $A$ is rotated clockwise about the origin $O$ through $90^{\circ}$ to $A'$. $A''$ is the reflection image of $A$ with respect to the $y$-axis.

In a factory, the production cost of a carpet of perimeter $s$ metres is $C$. It is given that $C$ is a sum of two parts, one part varies as $s$ and the other part varies as the square of $s$. When $s=2$, $C=356$; when $s=5$, $C=1250$.