Past paper questions

Simplify $\frac{m^{-12}n^{8}}{n^{3}}$ and express your answer with positive indices.

Make $a$ the subject of the formula $\frac{3a+b}{8}=b-1$

The daily wage of Ada is $20\%$ higher than that of Billy while the daily wage of Billy is $20\%$ lower than that of Christine. It is given that the daily wage of Billy is $$480$.

There are 132 guards in an exhibition centre consisting of 6 zones. Each zone has the same number of guards. In each zone, there are 4 more female guards than male guards. Find the number of male guards in the exhibition centre. (4 marks)

The box-and-whisker diagram below shows the distribution of the times taken by a large group of students of an athletic club to finish a $100\text{ m}$ race:

The inter-quartile range and the range of the distribution are $3.2$ s and $6.8$ s respectively.

In Figure 1, $AB$, $BC$, $CD$ and $AD$ are chords of the circle. $AC$ and $BD$ intersect at $E$. It is given that $BE = 8\text{ cm}$, $CE = 20\text{ cm}$ and $DE = 15\text{ cm}$.

In Figure 2, the volume of the solid right prism $ABCDEFGH$ is $1020\text{ cm}^{3}$. The base $ABCD$ of the prism is a trapezium, where $AD$ is parallel to $BC$. It is given that $\angle BAD = 90^{\circ}$, $AB = 12\text{ cm}$, $BC = 6\text{ cm}$ and $DE = 10\text{ cm}$.

Tom conducts a survey on the numbers of hours spent on doing homework in a week by secondary students. Questionnaires are sent out and twenty of them are returned. The stem-and-leaf diagram below shows the numbers of hours recorded in the twenty questionnaires:

$\begin{array}{c|cccccccccccc} \text{Stem (tens)} & {\text{Leaf (units)}} \\ \hline 1 & 0 & 0 & 1 & 1 & 2 & 3 & 4 & 5 & 5 & 6 & 6 & 7 \\ 2 & 0 & 0 & 0 & 5 & 8 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 3 & 4 & 6 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \end{array}$

Let $C$ be the cost of painting a can of surface area $A\ m^2$. It is given that $C$ is the sum of two parts, one part is a constant and the other part varies as $A$. When $A=2$, $C=62$; when $A=6$, $C=74$.

The standard deviation of the test scores obtained by a class of students in a Mathematics test is $10$ marks. All the students fail in the test, so the test score of each student is adjusted such that each score is increased by $20\%$ and then extra $5$ marks are added.

There are 8 departments in a company. To form a task group of 16 members, 2 representatives are nominated by each department. From the task group, 4 members are randomly selected.

The coordinates of the centre of the circle $C$ are $(6,10)$. It is given that the $x$-axis is a tangent to $C$.

Figure 5(a) shows a right pyramid $VABCD$ with a square base, where $\angle VAB = 72^{\circ}$. The length of a side of the base is $20\text{ cm}$. Let $P$ and $Q$ be the points lying on $VA$ and $VD$ respectively such that $PQ$ is parallel to $BC$ and $\angle PBA = 60^{\circ}$. A geometric model is made by cutting off the pyramid $VPBCQ$ from $VABCD$ as shown in Figure 5(b).

In a city, the air cargo terminal X of an airport handles goods of weight $A(n)$ tonnes in the nth year since the start of its operation, where n is a positive integer. It is given that $A(n) = ab^{2n}$, where a and b are positive constants. It is found that the weights of the goods handled by X in the 1st year and the 2nd year since the start of its operation are 254100 tonnes and 307461 tonnes respectively.

Simplify $\frac{(xy^{-2})^{3}}{y^{4}}$ and express your answer with positive indices. (3 marks)

The table below shows the distribution of the numbers of calculators owned by some students.

Consider the formula $2(3m+n)=m+7$.

The marked price of a toy is $$255$. The toy is now sold at a discount of $40\%$ on its marked price.

The coordinates of the points $P$ and $Q$ are $(-3,5)$ and $(2,-7)$ respectively. $P$ is rotated anticlockwise about the origin $O$ through $270^{\circ}$ to $P^{\prime}$. $Q$ is translated leftwards by 21 units to $Q^{\prime}$.

In Figure 1, $D$ is a point lying on $AC$ such that $\angle BAC = \angle CBD$.

Town X and town Y are $80$ km apart. Figure 2 shows the graphs for car $A$ and car $B$ travelling on the same straight road between town $X$ and town $Y$ during the period 7:30 to 9:30 in a morning. Car $A$ travels at a constant speed during the period. Car $B$ comes to rest at 8:15 in the morning.

There are 33 paintings in an art gallery. The box-and-whisker diagram below shows the distribution of the prices (in thousand dollars) of the paintings in the art gallery. It is given that the mean of this distribution is $53$ thousand dollars.

The circle $C$ passes through the point $A(6,11)$ and the centre of $C$ is the point $G(0,3)$.

It is given that $f(x)$ is the sum of two parts, one part varies as $x^{2}$ and the other part is a constant. Suppose that $f(2)=59$ and $f(7)=-121$.

Figure 3 shows a vessel in the form of a frustum which is made by cutting off the lower part of an inverted right circular cone of base radius $72\text{ cm}$ and height $96\text{ cm}$. The height of the vessel is $60\text{ cm}$. The vessel is placed on a horizontal table. Some water is now poured into the vessel. John finds that the depth of water in the vessel is $28\text{ cm}$.

The graph in Figure 4 shows the linear relation between $\log_{4}x$ and $\log_{8}y$. The slope and the intercept on the horizontal axis of the graph are $\frac{-1}{3}$ and 3 respectively. Express the relation between $x$ and $y$ in the form $y=Ax^{k}$, where $A$ and $k$ are constants. (3 marks)

In Figure 5, the 1st pattern consists of 3 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding 2 dots to the $n$th pattern. Find the least value of $m$ such that the total number of dots in the first $m$ patterns exceeds $6888$. (4 marks)

Figure 6(a) shows a solid pyramid VABCD with a rectangular base, where $AB = 18\text{ cm}$, $BC = 10\text{ cm}$, $VB = VC = 30\text{ cm}$ and $\angle VAB = \angle VDC = 110^\circ$.

$(2n^{3})^{-5}=$

$u^{2}-v^{2}-5u+5v=$

If p and q are constants such that $px(x-1)+x^{2} \equiv qx(x-2)+4x$, then p =

Let $a$ be a constant. If the quadratic equation $x^{2}+ax+a=1$ has equal roots, then $a=$

The figure shows the graph of $y = mx^2 + x + n$, where $m$ and $n$ are constants. Which of the following is true?

If $a>b$ and $k<0$, which of the following must be true?

I. $a^{2}>b^{2}$

II. $a+k>b+k$

III. $\frac{a}{k^{2}}>\frac{b}{k^{2}}$

The solution of $-3x<6<2x$ is

The price of 2 bowls and 3 cups is $$506$. If the price of 5 bowls and the price of 4 cups are the same, then the price of a bowl is

There are 792 workers in a factory. If the number of male workers is $20\%$ less than that of female workers, then the number of male workers is

If the angle and the radius of a sector are decreased by $x\%$ and $50\%$ respectively so that its area is decreased by $90\%$, then $x=$

The width and the length of a thin rectangular metal sheet are measured as $8\text{ cm}$ and $10\text{ cm}$ correct to the nearest $\text{cm}$ respectively. Let $x\text{ cm}^{2}$ be the actual area of the metal sheet. Find the range of values of $x$.

It is given that $\frac{4}{5a} = \frac{5}{7b} = \frac{7}{9c}$, where $a$, $b$ and $c$ are positive numbers. Which of the following is true?