Past paper questions

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.

Town X and town Y are $80\text{ km}$ 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 $72$ cm and height $96$ cm. The height of the vessel is $60$ 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 $28$ cm.

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 $\angle VAB = \angle VDC = 110^\circ$.

Ada and Billy play a game consisting of two rounds. In the first round, Ada and Billy take turns to throw a fair die. The player who first gets a number '3' wins the first round. Ada and Billy play the first round until one of them wins. Ada throws the die first.

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 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?

If z varies inversely as x and directly as the cube of y, which of the following must be constant?

Let $a_n$ be the $n$th term of a sequence. If $a_2 = 7$, $a_4 = 63$ and $a_{n+2} = a_{n+1} + a_n$ for any positive integer $n$, then $a_5 =$

In the figure, AB = AE and $\angle BAE = \angle BCD = \angle CDE = 90^\circ$. If BC = CD = DE = $16\text{ cm }$, then the area of the pentagon ABCDE is

In the figure, $ABCD$ is a square. $BC$ is produced to $G$ such that $\angle CDG = 25^\circ$. $E$ is a point lying on $AB$ such that $AE = CG$. If $F$ is a point lying on $BC$ such that $\angle CDF = 20^\circ$, then $\angle DFE =$

In the figure, B is a point lying on AC such that AB:BC = 3:2. It is given that AE//BD. If the area of $\triangle BCD$ and the area of $\triangle CDE$ are $4\text{ cm}^{2}$ and $8\text{ cm}^{2}$ respectively, then the area of the trapezium ABDE is

In the figure, $\angle ABD = \angle ADC = \angle BCD = 90^\circ$. If $AB = \ell$, then $CD =$

In the figure, $AC$ is a diameter of the circle $ABCDE$. If $\angle ADE = 28^{\circ}$, then $\angle CBE =$

In the figure, O is the centre of the circle $ABCDEF$. $\Delta PQR$ intersects the circle at A, B, C, D, E and F. If $\angle QPR = 38^{\circ}$ and AB = CD = EF, then $\angle QOR =$

If an interior angle of a regular $n$-sided polygon is greater than an exterior angle by $100^{\circ}$, which of the following are true?

I. The value of $n$ is 10.

II. Each exterior angle of the polygon is $40^{\circ}$.

III. The number of axes of reflectional symmetry of the polygon is 9.

The rectangular coordinates of the point $P$ are $(-1, \sqrt{3})$. If $P$ is reflected with respect to the $x$-axis, then the polar coordinates of its image are

The equations of the straight lines $L_{1}$ and $L_{2}$ are $2x + 3y = 5$ and $4x + 6y = 7$ respectively. If $P$ is a moving point in the rectangular coordinate plane such that the perpendicular distance from $P$ to $L_{1}$ is equal to the perpendicular distance from $P$ to $L_{2}$, then the locus of $P$ is a

In the figure, the two straight lines intersect at a point on the positive y-axis. Which of the following are true?

If a diameter of the circle $x^{2}+y^{2}-8x+ky-214=0$ passes through the point $(6,-5)$ and the slope of the diameter is -4, then $k$ =

A box contains $m$ yellow balls and $20$ black balls. If a ball is randomly drawn from the box, then the probability of drawing a yellow ball is $\frac{1}{m}$. Find the value of $m$.

The mean height of $25$ teachers and $140$ students is $150$ cm. If the mean height of the students is $145$ cm, then the mean height of the teachers is

The pie chart below shows the expenditure of John in a certain week. John spends \240$ on clothing that week. Find his expenditure on transportation that week.

The stem-and-leaf diagram below shows the distribution of the ages of the passengers in a bus.

\begin{array}{c|ccccc}{{{\underline{S t e m}\mathrm{(t e n s)}}}}&{{{\underline{L e a f}\mathrm{(u n i t s)}}}}&{{{7}}} \\{{{\hline1}}}&{{{h}}}&{{{4}}}&{{{6}}} \\{{{2}}}&{{{3}}}&{{{3}}}&{{{3}}}&{{{4}}}&{{{6}}}&{{{7}}} \\{{{3}}}&{{{1}}}&{{{2}}}&{{{2}}}&{{{2}}}&{{{6}}}&{{{8}}} \\{{{4}}}&{{{0}}}&{{{k}}} \\\end{array}

If the range of the above distribution is at least $33$, which of the following must be true?

I. $0 \leq h \leq 3$

II. $3 \leq k \leq 9$

III. $3 \leq k - h \leq 5$

The H.C.F. of $3x^{4}y^{2}z$, $4xy^{5}z$ and $6x^{2}y^{3}$ is