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.

$\frac{(2x^{4})^{3}}{2x^{5}} =$

$(4x+y)^{2}-(4x-y)^{2}=$

If $p$ and $q$ are constants such that $x^{2}+p\equiv(x+2)(x+q)+10$, then $p=$

If $k$ is a constant such that $x^{3}+4x^{2}+kx-12$ is divisible by $x+3$, then $k=$

If $m+2n+6=2m-n=7$, then $n=$

The figure shows the graph of $y = a(x + b)^2$, where $a$ and $b$ are constants. Which of the following is true?

The solution of $15+4x<3$ or $9-2x>1$ is

In a company, $37.5\%$ of the employees are female. If $60\%$ of the male employees and $80\%$ of the female employees are married, then the percentage of married employees in the company is

If $x$ and $y$ are non-zero numbers such that $\frac{6x+5y}{3y-2x}=7$, then $x:y=$

It is given that $y$ partly varies directly as $x^{2}$ and partly varies inversely as $x$. When $x=1$, $y=-4$ and when $x=2$, $y=5$. When $x=-2$, $y=$

Mary performs a typing task for 7 hours. Her average typing speeds for the first 3 hours and the last 4 hours are 56 words per minute and 56 words per minute respectively. Find her average typing speed for the 7 hours.

In the figure, the 1st pattern consists of 1 dot. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $n$ dots to the $n$th pattern. Find the number of dots in the 8th pattern.

$0.0322515 =$

The length of a piece of thin string is measured as $25\text{ m}$ correct to the nearest m. If the string is cut into $n$ pieces such that the length of each piece is measured as $5\text{ cm}$ correct to the nearest cm, find the greatest possible value of $n$.

In the figure, the area of quadrilateral $ABCD$ is

In the figure, $OAB$ and $OCD$ are sectors with centre $O$. If $\widehat{AB}=12\pi$ cm, $\widehat{CD}=16\pi$ cm and $OA=30$ cm, then $AC=$

In the figure, $ABCD$ is a parallelogram. $E$ and $F$ are points lying on $AB$ and $CD$ respectively. $AD$ produced and $EF$ produced meet at $G$. It is given that $DF:FC=3:4$ and $AD:DG=1:1$. If the area of $\triangle DFG$ is $3\mathrm{~cm}^{2}$, then the area of the parallelogram $ABCD$ is

In the figure, $D$ is a point lying on $AC$ such that $BD$ is perpendicular to $AC$. If $BC = \ell$, then $AB=$

$\frac{\cos 60^{\circ}}{1-\cos(90^{\circ}-\theta)}+\frac{\cos 240^{\circ}}{1-\cos(270^{\circ}-\theta)}=$

In the figure, $O$ is the centre of the circle $ABCD$. If $\angle BAO = 28^{\circ}$, $\angle BCD = 114^{\circ}$ and $\angle CDO = 42^{\circ}$, then $\angle ABC =$

In the figure, $AB$ is a diameter of the circle $ABCD$. If $AB=12\text{cm}$ and $CD=6\text{cm}$, then the area of the shaded region is

Which of the following statements about a regular 12-sided polygon are true?

I. Each exterior angle is $30^{\circ}$.

II. Each interior angle is $150^{\circ}$.

III. The number of axes of reflectional symmetry is 6.

The rectangular coordinates of the point $P$ are $(-3, -3\sqrt{3})$. If $P$ is rotated anticlockwise about the origin through $90^{\circ}$, then the polar coordinates of its image are

If $P$ is a moving point in the rectangular coordinate plane such that the distance between $P$ and the point $(20,12)$ is equal to 5, then the locus of $P$ is a

In the figure, the equations of the straight lines $L_{1}$ and $L_{2}$ are $ax + y = b$ and $cx + y = d$ respectively. Which of the following are true?

I. $a < 0$

II. $a < c$

III. $b > d$

IV. $ad > bc$

In the figure, the radius of the circle and the coordinates of the centre are $r$ and $(h, k)$ respectively. Which of the following are true?

I. $h + k > 0$

II. $r - h > 0$

III. $r - k > 0$

$9$☆◇ is a $3$-digit number, where ☆ and ◇ are integers from $0$ to $9$ inclusive. Find the probability that the $3$-digit number is divisible by $5$.

The stem-and-leaf diagram below shows the distribution of the ages of a group of members in a recreational centre.

$\begin{array}{c|ccccccccc}{\mathrm{Stem~(tens)}}&{\underline{{\mathrm{Leaf~(units)}}}}&{}\\ \hline {5}&{0}&{5}&{6}&{6}&{8}&{\cdots}&{\cdots}&{\cdots}\\ {6}&{1}&{4}&{5}&{5}&{7}&{8}&{8}&{9}\\ {7}&{3}&{4}&{4}&{6}&{7}&{9}&{\cdots}&{\cdots}\\ {8}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\ {9}&{1}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots} \end{array}$

A member is randomly selected from the group. Find the probability that the selected member is not under age of 74.

The bar chart below shows the distribution of the numbers of rings owned by the girls in a group. Find the standard deviation of the distribution correct to 2 decimal places.

Consider the following data:

[Table]

If both the mean and the median of the above data are 14, which of the following are true?

I. $m \ge q 14$

II. $n \le q 16$

III. $m + n = 30$

The H.C.F. and the L.C.M. of three expressions are $ab^{2}$ and $4a^{4}b^{5}c^{6}$ respectively. If the first expression and the second expression are $2a^{2}b^{4}c$ and $4a^{4}b^{2}c^{6}$ respectively, then the third expression is