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.

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?

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 $CD =$

$\left(\cos(90^{\circ}+\theta)+1\right)\left(\sin(360^{\circ}-\theta)-1\right)=$

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 $150\text{ cm}$. If the mean height of the students is $145\text{ cm}$, 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.

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

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

I. $0 \le q h \le q 3$

II. $3 \le q k \le q 9$

III. $3 \le q k - h \le q 5$

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