Past paper questions

Simplify $\frac{(mn^{-2})^{5}}{m^{-4}}$ and express your answer with positive indices.

Factorize

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

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

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.

PQRS is a quadrilateral paper card, where $PQ = 60$ cm, PS = $40\text{ cm }$, $\angle PQR = 30^\circ$, $\angle PRQ = 55^\circ$ 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?

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 $0.75\text{ km}^{2}$ on a 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.

$\bullet \bullet \bullet \quad \Longrightarrow$ $\bullet \quad \bullet \quad \rightarrow$ $\bullet \quad \bullet \quad \rightarrow$ $\bullet \quad \bullet \quad \rightarrow$ $\bullet \quad \bullet \quad \rightarrow$ $\bullet \quad \bullet \quad \rightarrow$ $\bullet \quad \bullet \quad \rightarrow$

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 $A\, cm^2$ 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 $5a \, cm$ and height $7b \, cm$ is $525 \, cm^{3}$, then the volume of a right circular cone of base radius $7a \, cm$ and height $5b \, cm$ 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 $50$ 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?

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 $12\text{ m }$ 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. $z = 8$