Past paper questions

Let $a$, $b$ and $c$ be non-zero numbers such that $\frac{a}{b}=\frac{6}{7}$ and $3a=4c$. Find $\frac{b+2c}{a+2b}$. (3 marks)

In a recruitment exercise, the number of male applicants is $28\%$ more than the number of female applicants. The difference of the number of male applicants and the number of female applicants is $91$. Find the number of male applicants in the recruitment exercise. (4 marks)

Consider the compound inequality

$$3-x>\frac{7-x}{2}\quad or\quad5+x>4$$

Let $p(x)=4x^{2}+12x+c$, where c is a constant. The equation $p(x)=0$ has equal roots. Find

In Figure 1, $B$ and $D$ are points lying on $AC$ and $AE$ respectively. $BE$ and $CD$ intersect at the point $F$. It is given that $AB = BE$, $BD \parallel CE$, $\angle CAE = 30^\circ$ and $\angle ADB = 42^\circ$.

The table below shows the distribution of the numbers of subjects taken by a class of students.

[Table]

The price of a brand X souvenir of height $h$ cm is \P$.$\$P$ is partly constant and partly varies as $h^{3}$. When $h=3$, \$P = 59$$and when$$h = 7$$,$\$P=691$.

The stem-and-leaf diagram below shows the distribution of the weights (in grams) of the letters in a bag.

The height and the base radius of a solid right circular cone are $36\text{ cm}$ and $15\text{ cm}$ respectively. The circular cone is divided into three parts by two planes which are parallel to its base. The heights of the three parts are equal. Express, in terms of $\pi$,

The cubic polynomial $f(x)$ is divisible by $x-1$. When $f(x)$ is divided by $x^{2}-1$, the remainder is $kx+8$, where $k$ is a constant.

The coordinates of the points $A$ and $B$ are $(-10,0)$ and $(30,0)$ respectively. The circle $C$ passes through $A$ and $B$. Denote the centre of $C$ by $G$. It is given that the $y$-coordinate of $G$ is $-15$.

In a box, there are 3 blue plates, 7 green plates and 9 purple plates. If 4 plates are randomly selected from the box at the same time, find

The 3rd term and the 6th term of a geometric sequence are 144 and 486 respectively.

Let $g(x) = x^2 - 2kx + 2k^2 + 4$, where $k$ is a real constant.

In Figure 2, $U$, $V$ and $W$ are points lying on a circle. Denote the circle by $C$. $TU$ is the tangent to $C$ at $U$ such that $TVW$ is a straight line.

PQRS is a quadrilateral paper card, where $PQ = 60$ cm, $PS = 40$ cm, $\angle PQR = 30^{\circ}$, $\angle PRQ = 55^{\circ}$ and $\angle QPS = 120^{\circ}$. The paper card is held with QR lying on the horizontal ground as shown in Figure 3.

$a(a+b)=2(b-a)$, then $b=$

Let $f(x)=3x^{2}-x-2$. If $\beta$ is a constant, then $f(1+\beta)-f(1-\beta)=$

Let $g(x)=ax^{3}+4ax^{2}-24$, where a is a constant. If $x+2$ is a factor of $g(x)$, then $g(2)=$

If $h$ and $k$ are constants such that $(x+h)(x+6) \equiv (x+4)^{2}+k$, then $k =$

In the figure, the equations of the straight lines $L_{1}$ and $L_{2}$ are $x + ay + b = 0$ and $bx + y + c = 0$ respectively. Which of the following are true?

I. $c < 0$
II. $ab < 1$
III. $ac < b$

The cost of a toy is $x\%$ lower than its selling price. After selling the toy, the percentage profit is $25\%$. Find $x$.

The actual area of a golf course is $0.75\text{ km}^{2}$. If the area of the course on a map is $100\text{ cm}^{2}$, then the scale of the map is

It is given that $w$ varies as the cube of $u$ and the square root of $v$. When $u=2$ and $v=4$, $w=8$. When $u=4$ and $v=9$, $w=$

In the figure, the 1st pattern consists of 3 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $(2n+1)$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

The solution of $5-4x<9$ and $\frac{2x-3}{7}>1$ is

In the figure, $P_{QRST}$ is a pentagon, where all the measurements are correct to the nearest cm. Let $A \text{ cm}^2$ be the actual area of the pentagon. Find the range of values of $A$.

The angle of a sector is decreased by $60\%$ but its radius is increased by $k\%$. If the arc length of the sector remains unchanged, find the value of $k$.

If the volume of a right circular cylinder of base radius $5a\text{ cm}$ and height $7b\text{ cm}$ is $525\text{ cm}^{3}$, then the volume of a right circular cone of base radius $7a\text{ cm}$ and height $5b\text{ cm}$ is

In the figure, P and Q are points lying on OR while U and T are points lying on OS such that $OP = PQ = QR$ and $PU \parallel QT \parallel RS$. The ratio of the area of the trapezium $QRST$ is

In the figure, $ABCD$ is a parallelogram. Let $E$ be a point lying on $AD$ such that $AE:ED=2:5$. $CB$ is produced to the point $F$ such that $BF=DE$. Denote the point of intersection of $AB$ and $EF$ by $G$. It is given that $BD$ and $CG$ intersect at the point $H$. If the area of $\triangle AEG$ is $48\mathrm{~cm}^{2}$, then the area of $\triangle CDH$ is

According to the figure, which of the following must be true?

I. $u - v + w = 0^{\circ}$

II. $u + v - w = 180^{\circ}$

III. $u + v + w = 450^{\circ}$

In the figure, $ABC$ is an equilateral triangle and $CDE$ is an isosceles triangle with $CD = CE$. If $\angle DCE = 78^{\circ}$ and $\angle ADC = \angle CAD = 40^{\circ}$, then $\angle CBE =$

In the figure, $ABCD$ is a rectangle. Let $E$ be a point lying on $AD$ such that $BE = 8\ \text{cm}$ and $CE = 15\ \text{cm}$. If $BC = 17\ \text{cm}$, find the area of the rectangle $ABCD$.

In the figure, $ABCDE$ is a circle. If $AB=10\text{ cm}$, $BC=5\text{ cm}$, $\angle ABC=90^{\circ}$ and $\angle CED=40^{\circ}$ find $CD$ correct to the nearest $\text{cm}$.

A ship is $50\text{ km}$ due west of a lighthouse. If the ship moves in the direction $S60^{\circ}E$, find the shortest distance between the ship and the lighthouse.

The point $P$ is translated leftwards by 4 units to the point $Q$. If the coordinates of the reflection image of $Q$ with respect to the $y$-axis are $(5,-1)$, then the polar coordinates of $P$ are

Let $A$ be the point of intersection of the straight lines $9x+4y-7=0$ and $9x-4y+7=0$. If $P$ is a moving point in the rectangular coordinate plane such that the distance between $P$ and $A$ is $8$, then the locus of $P$ is a

The equation of the straight line $L$ is $kx+4y-2k=0$, where $k$ is a constant. If $L$ is perpendicular to the straight line $6x-9y+4=0$, find the $y$-intercept of $L$.

The equations of the circles $C_{1}$ and $C_{2}$ are $2x^{2}+2y^{2}+4x+8y-149=0$ and $x^{2}+y^{2}-8x-20y-53=0$ respectively. Which of the following is/are true?
I. The centre of $C_{1}$ lies on $C_{2}$.
II. The radii of $C_{1}$ and $C_{2}$ are equal.
III. $C_{1}$ and $C_{2}$ intersect at two distinct points.

Two numbers are randomly drawn at the same time from four cards numbered 3, 5, 7 and 9 respectively. Find the probability that the product of the numbers drawn is greater than $35$.

The bar chart below shows the distribution of the numbers of pens owned by some students. Find the inter-quartile range of the distribution.

Consider the following integers:

$3$ $3$ $8$ $8$ $10$ $12$ $m$ $n$

Let $x$, $y$ and $z$ be the median, the mean and the mode of the above integers respectively. If the range of the above integers is $9$, which of the following must be true?

I. $x=8$

II. $y=8$

III. $z=8$

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