Past paper questions

Simplify $\frac{x^{20}y^{13}}{(x^{5}y)^{6}}$ and express your answer with positive indices. (3 marks)

Make $k$ the subject of the formula $\frac{3}{h}-\frac{1}{k}=2$. (3 marks)

The price of 7 pears and 3 oranges is $47 while the price of 5 pears and 6 oranges is$49. Find the price of a pear. (4 marks)

In a polar coordinate system, $O$ is the pole. The polar coordinates of the points $A$ and $B$ are $(26,10^{\circ})$ and $(26,130^{\circ})$ respectively. Let $L$ be the axis of reflectional symmetry of $\Delta OAB$.

In Figure 1, $ABCD$ is a quadrilateral. The diagonals $AC$ and $BD$ intersect at $E$. It is given that $BE = CE$ and $\angle BAC = \angle BDC$.

A pack of sea salt is termed regular if its weight is measured as $100\text{ g}$ correct to the nearest g.

The bar chart below shows the distribution of the numbers of family members of the employees of company $D$.

The ages of the members of Committee A are shown as follows:

The weight of a tray of perimeter $\ell$ metres is $W$ grams. It is given that $W$ is the sum of two parts, one part varies directly as $\ell$ and the other part varies directly as $\ell^{2}$. When $\ell=1$, $W=181$ and when $\ell=2$, $W=402$.

Let $f(x)=3x^{3}-7x^{2}+kx-8$, where $k$ is a constant. It is given that $f(x)\equiv(x-2)(ax^{2}+bx+c)$, where $a$, $b$ and $c$ are constants.

In a workshop, 2 identical solid metal right circular cylinders of base radius $R$ cm are melted and recast into 27 smaller identical solid right circular cylinders of base radius $r$ cm and height $10$ cm. It is given that the base area of a larger circular cylinder is 9 times that of a smaller one.

The equation of the circle $C$ is $x^{2}+y^{2}-12x-34y+225=0$. Denote the centre of $C$ by $R$.

The box-and-whisker diagram below shows the distribution of the scores (in marks) of the students of a class in a test. Susan gets the highest score while Tom gets $65$ marks in the test. The standard scores of Susan and Tom in the test are $3$ and $0.5$ respectively.

A box contains 5 white cups and 11 blue cups. If 6 cups are randomly drawn from the box at the same time,

The development of public housing in a city is under study. It is given that the total floor area of all public housing flats at the end of the 1st year is $9 \times 10^6 \text{ m}^2$ and in subsequent years, the total floor area of public housing flats built each year is $r\%$ of the total floor area of all public housing flats at the end of the previous year, where $r$ is a constant, and the total floor area of public housing flats pulled down each year is $3 \times 10^5 \text{ m}^2$. It is found that the total floor area of all public housing flats at the end of the 3rd year is $1.026 \times 10^7 \text{ m}^2$.

If $\frac{y-1}{c}=\frac{y+1}{d}$, then y=

The solution of $x - \frac{x-1}{2} > 5$ or $1 < x - 11$ is

Let $k$ be a constant. Solve the equation $(x-k)^{2}=4k^{2}$.

The figure shows the graph of $y = -2x^{2} + ax + b$, where $a$ and $b$ are constants. The equation of the axis of symmetry of the graph is

If $a$, $b$ and $c$ are non-zero constants such that $x(x+3a)+a \equiv x^{2}+2(bx+c)$, then $a:b:c=$

Let $f(x)=x^{13}-2x+k$, where $k$ is a constant. If $f(x)$ is divisible by $x+1$, find the remainder when $f(x)$ is divided by $x-1$.

Susan sells two cars for \80\,080$each. She gains$30\%$on one and loses$30\%$ on the other. After the two transactions, Susan

A sum of \50\,000$is deposited at an interest rate of$8\%$ per annum for 1 year, compounded monthly. Find the interest correct to the nearest dollar.

The actual area of a playground is $900 ext{ m}^{2}$. If the area of the playground on a map is $36 ext{ cm}^{2}$, then the scale of the map is

It is given that $z$ varies directly as $x$ and inversely as $\sqrt{y}$. If $y$ is decreased by $64\%$ and $z$ is increased by $25\%$, then $x$

The figure shows the graph of the straight line $x + ay + b = 0$. Which of the following are true?

I. $a < 0$

II. $b < 0$

III. $a < b$

In the figure, the regular octagon is divided into eight identical isosceles triangles and four of the shaded. The number of axes of reflectional symmetry of the octagon is

In the figure, the diameter of the semicircle $ABC$ is $3\text{ cm }$. If $AC = 2$ cm, find the area of the shaded region correct to the nearest $0.01$ cm$^2$.

In the figure, the solid consists of a right circular cone and a hemisphere with a common base. The base radius and the height of the circular cone are $3\text{ cm }$ and $4\text{ cm }$ respectively. Find the total surface area of the solid.

In the figure, $ABCD$ is a trapezium with $AD \parallel BC$ and $AD:BC = 2:3$. Let $E$ be the mid-point of $BC$. $AC$ and $DE$ intersect at $F$. If the area of $\triangle CEF$ is $36\text{ cm}^{2}$, then the area of the trapezium $ABCD$ is

In the figure, $ABCD$ is a circle. $AC$ and $BD$ intersect at $E$. If $AB = AD$ and $AD \parallel BC$, then $\angle BAE =$

In the figure, the bearing of P from O is $S86^{\circ}E$ and the bearing of Q from O is $N32^{\circ}E$. If P and Q are equidistant from O, then the bearing of P from Q is

If an interior angle of a regular n-sided polygon is 4 times an exterior angle of the polygon, which of the following is/are true?

I. The value of n is 10.

II. The number of diagonals of the polygon is 10.

III. The number of folds of rotational symmetry of the polygon is 10.

In $\triangle ABC$, $AB:BC:AC=8:15:17$. Find $\cos A:\cos C$.

If $0^{\circ}<x<90^{\circ}$, which of the following must be true?

I. $\tan x\tan(90^{\circ}-x)=1$

II. $\sin x - \sin(90^{\circ} - x) < 0$

III. $\cos x + \cos(90^{\circ} - x) > 0$

The coordinates of the points $A$ and $B$ are $(2, 5)$ and $(4, -1)$ respectively. Let $P$ be a moving point in the rectangular coordinate plane such that $AP = BP$. Find the equation of the locus of $P$.

The equation of the circle $C$ is $2x^{2} + 2y^{2} - 4x + 8y - 5 = 0$. The coordinates of the points $P$ and $Q$ are $(-1, 2)$ and $(4, 0)$ respectively. Which of the following is/are true?

I. The radius of $C$ is 5.

II. The mid-point of $PQ$ lies outside $C$.

III. If $G$ is the centre of $C$, then $\angle PGQ$ is an acute angle.

Two numbers are randomly drawn at the same time from seven cards numbered 1, 2, 3, 4, 5, 6 and 7 respectively. Find the probability that the product of the numbers drawn is an odd number.

If the mean and the mode of the nine numbers 14, 6, 4, 5, 7, 5, x, y and z are 8 and 14 respectively, then the median of these nine numbers is

The scatter diagram below shows the relation between x and y. Which of the following may represent the relation between x and y?

The stem-and-leaf diagram below shows the distribution of the hourly wages (in dollars) of some workers.

\begin{array}{c|ccccccccc}{{\mathrm{s}\!s)}}&{{{\mathrm{Leaf}\,\mathrm{(units)}}}} \\{{{\hline4}}}&{{{0}}}&{{{2}}}&{{{2}}}&{{{2}}}&{{{4}}}&{{{4}}}&{{{4}}}&{{{7}}} \\{{{5}}}&{{{0}}}&{{{0}}}&{{{1}}}&{{{2}}}&{{{2}}}&{{{6}}}&{{{8}}}&{{{9}}} \\{{{6}}}&{{{3}}}&{{{5}}}&{{{5}}}&{{{7}}} \\{{{7}}}&{{{0}}} \\{{{8}}}&{{{2}}}&{{{6}}} \\{{{9}}}&{{{5}}} \\ \end{array}

Which of the following box-and-whisker diagrams may represent the distribution of their hourly wages?

Figure 1
Figure 2
Figure 3
Figure 4

The pie charts below show the distributions of the profits of stationery shop X and stationery shop Y from the sales of stationery in a certain month. Which of the following must be true?

Distribution of the profits of stationery shop X

Distribution of the profits of stationery shop Y

[Figure 5]

[Figure 6]

The L.C.M. of $a^{2}+4a+4$, $a^{2}-4$ and $a^{3}+8$ is