Past paper questions

If $\frac{a}{x} \div \frac{b}{y} = 3$, then $x =$

If $4\alpha + \beta = 7\alpha + 3\beta = 5$, then $\beta =$

Let $f(x)=4x^{3}+kx+3$, where $k$ is a constant. If $f(x)$ is divisible by $2x+1$, find the remainder when $f(x)$ is divided by $x+1$.

The solution of $-5x > 21 - 2x$ and $6x - 18 < 0$ is

If $k$ is a constant such that the quadratic equation $x^{2}+kx+8k+36=0$ has equal roots, then $k=$

If $-1 < a < 0$, which of the following may represent the graph of $y = (ax + 1)^2 + a$?

The monthly salary of Donald is $25\%$ higher than that of Peter while the monthly salary of Peter is $25\%$ lower than that of Teresa. It is given that the monthly salary of Donald is \33\,360$. The monthly salary of Teresa is

If $x$ and $y$ are non-zero numbers such that $(3y-4x):(2x+y)=5:6$, then $x:y=$

It is given that $z$ varies directly as $\sqrt{x}$ and inversely as $y$. If $x$ is decreased by $36\%$ and $y$ is increased by $60\%$, then $z$

The cost of flour of brand X is \42/\text{kg}$. If$3\text{ kg}$of flour of brand X and$2\text{ kg}$of flour of brand Y are mixed so that the cost of the mixture is$\$36/\text{kg}$, find the cost of flour of brand Y.

In the figure, the 1st pattern consists of $9$ dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $5$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

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

I. $a + c = 180^{\circ}$

II. $a + b - c = 180^{\circ}$

III. $b + c = 360^{\circ}$

In the figure, $ABC$ is a straight line. If $AB = 24\ cm$, $AD = 40\ cm$, $BD = 32\ cm$ and $CD = 68\ cm$, then $BC =$

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $CD$ such that $BE = CE$. If $\angle ADC = 114^\circ$, then $\angle ABE =$

The figure shows a right prism. Find the volume of the prism.

In the figure, $OAB$ and $OCD$ are sectors with centre $O$, where $OA=33\text{ cm}$ and $OC=39\text{ cm}$. The area of the shaded region $ABDC$ is $72\pi\text{cm}^{2}$. Which of the following is/are true?
I. The angle of the sector $OAB$ is $60^{\circ}$.
II. The area of the sector $OAB$ is $11\pi\text{cm}^{2}$.
III. The perimeter of the sector $OCD$ is $13\pi\text{cm}$.

In the figure, $ABCD$, $CDEF$ and $EFGH$ are squares. $AG$ cuts $CD$ and $EF$ at $P$ and $Q$ respectively. Find the ratio of the area of quadrilateral $DEQP$ to the area of quadrilateral $ABCP$.

In the figure, $AD =$

In the figure, $ABCD$ is a rhombus. $C$ is the centre of the circle $BDE$ and $ADE$ is a straight line. $BE$ and $CD$ intersect at $F$. If $\angle ADC = 118^\circ$, then $\angle DFE =$

The figure below consists of regular hexagons. The number of axes of reflectional symmetry of the figure is

If the sum of the interior angles of a regular $n$-sided polygon is $3240^{\circ}$, which of the following is true?

If the straight lines $hx + ky + 15 = 0$ and $4x + 3y - 5 = 0$ are perpendicular to each other and intersect at a point on the $x$-axis, then $k =$

The coordinates of the points $A$ and $B$ are $(9,-2)$ and $(-1,8)$ respectively. If $C$ is a point lying on the straight line $x-2y=0$ such that $AC=BC$, then the $x$-coordinate of $C$ is

The equation of the circle $C$ is $3x^{2}+3y^{2}-12x+30y+65=0$. Which of the following are true?

I. The radius of $C$ is $14$.
II. The origin lies outside $C$.
III. The coordinates of the centre of $C$ are $(2, -5)$.

Christine has one $1$ coin, one $2$ coin, one $5$ coin and one $10$ coin in her pocket. If Christine takes out three coins randomly from her pocket, find the probability that she gets at least $13$.

A bag contains $1$ red ball, $3$ yellow balls and $6$ white balls. In a lucky draw, a ball is randomly drawn from the bag and a certain number of tokens will be got according to the following table:

[Table]

the expected number of tokens got in the lucky draw.

Consider the following data:

$32$ $68$ $79$ $86$ $88$ $98$ $98$ $a$ $b$ $c$

If the mean and the mode of the above data are $77$ and $68$ respectively, then the median of the above data is

The L.C.M. of $9a^{2}b$, $12a^{4}b^{3}$ and $15a^{6}$ is

The graph in the figure shows the linear relation between $x$ and $\log_{9} y$. If $y = ab^{x}$, then $b =$

34. Let $u=\frac{7}{a+i}$ and $\nu=\frac{7}{a-i}$, where $a$ is a real number. Which of the following must be true?

I. $uv$ is a rational number.

II. The real part of $u$ is equal to the real part of $\nu$.

III. The imaginary part of $\frac{1}{u}$ is equal to the imaginary part of $\frac{1}{\nu}$.

35. In the figure, $PQ$ and $SR$ are parallel to the $x$-axis. If $(x,y)$ is a point lying in the shaded region $PQRS$ (including the boundary), at which point does $7y - 5x + 3$ attain its greatest value?

36. Let $a_n$ be the $n$th term of a geometric sequence. If $a_3=21$ and $a_7=189$, which of the following must be true?

I. The common ratio of the sequence is less than $1$.

II. Some of the terms of the sequence are irrational numbers.

III. The sum of the first $99$ terms of the sequence is greater than $3 \times 10^{24}$.

Let $a$ and $b$ be constants. If the figure shows the graph of $y = a\cos 2x^\circ$, then

For $0^{\circ} \leq \theta \leq 360^{\circ}$, how many roots does the equation $5\sin^{2}\theta + \sin\theta - 4 = 0$ have?

In the figure, $ABCDEFGH$ is a rectangular block. $AC$ and $BD$ intersect at $P$. $Q$ is a point lying on $CH$ such that $CQ = 9\,cm$ and $\angle PH = 15\,cm$. Find $\sin \angle PFQ$.

In the figure, $AC$ is a diameter of the circle $ABCD$. $PB$ and $PD$ are tangents to the circle. $AD$ produced and $BC$ produced meet at $Q$. If $\angle BPD = 68^\circ$, then $\angle AQB =$

The straight line $2x - y - 6 = 0$ and the circle $x^{2} + y^{2} - 8y - 14 = 0$ intersect at $P$ and $Q$. Find the $y$-coordinate of the mid-point of $PQ$.

There are $9$ cans of coffee and $3$ cans of tea in a box. If $4$ cans are chosen from the box, find the probability that at least $2$ cans of tea are chosen.

There are 20 boys and 15 girls in a class. If 6 students are selected from the class to form a committee consisting of at most $2$ girls, how many different committees can be formed?

The stem-and-leaf diagram below shows the distribution of the scores (in marks) of a group of students in a test. Ada gets the highest score in the test.

$$\begin{array}{c|ccccc}{{\underline{\mathrm{Stem(tens)}}}}&{{\underline{\mathrm{Leaf(units)}}}}\\{{4}}&{{5}}&{{6}}&{{7}}&{{8}}\\{{5}}&{{5}}&{{5}}&{{6}}&{{8}}\\{{6}}&{{3}}&{{5}}&{{5}}&{{6}}&{{9}}\\{{7}}&{{0}}&{{0}}&{{1}}\\{{8}}&{{0}}&{{2}}&{{5}}\end{array}$$

Which of the following is/are true?

I. The upper quartile of the distribution is $55$ marks.

II. The standard score of Ada in the test is lower than $2$.

III. The standard deviation of the distribution is greater than $12$ marks.

The variance of a set of numbers is $49$. Each number of the set is multiplied by $4$ and then $9$ is added to each resulting number to form a new set of numbers. Find the variance of the new set of numbers.