Past paper questions

If $\frac{y-1}{c}=\frac{y+1}{d}$, then $y=$

$0.0504545$ =

The solution of $x - \frac{x-1}{2} > 5$ or $1 < x - 11$ is

Let $k$ be a constant. Solve the equation $(x-k)^{2}=4k^{2}$.

The figure shows the graph of $y = -2x^{2} + ax + b$, where $a$ and $b$ are constants. The equation of the axis of symmetry of the graph is

If $a$, $b$ and $c$ are non-zero constants such that $x(x+3a)+a \equiv x^{2}+2(bx+c)$, then $a:b:c=$

Let $f(x)=x^{13}-2x+k$, where $k$ is a constant. If $f(x)$ is divisible by $x+1$, find the remainder when $f(x)$ is divided by $x-1$.

Susan sells two cars for \80,080$each. She gains$30\%$on one and loses$30\%$ on the other. After the two transactions, Susan

A sum of \50\,000$is deposited at an interest rate of$8\%$per annum for$1$ year, compounded monthly. Find the interest correct to the nearest dollar.

The actual area of a playground is $900\text{ m}^{2}$. If the area of the playground on a map is $36\text{ cm}^{2}$, then the scale of the map is

It is given that $z$ varies directly as $x$ and inversely as $\sqrt{y}$. If $y$ is decreased by $64\%$ and $z$ is increased by $25\%$, then $x$

The figure shows the graph of the straight line $x+ay+b=0$. Which of the following are true?

I. $a<0$
II. $b<0$
III. $a<b$

In the figure, the regular octagon is divided into eight identical isosceles triangles and four of the shaded. The number of axes of reflectional symmetry of the octagon is

In the figure, the diameter of the semicircle $ABC$ is $3$ cm. If $AC = 2$ cm, find the area of the shaded region correct to the nearest $0.01$ cm$^2$.

In the figure, the solid consists of a right circular cone and a hemisphere with a common base. The base radius and the height of the circular cone are $3$ cm and $4$ cm respectively. Find the total surface area of the solid.

In the figure, $ABCD$ is a trapezium with $AD \parallel BC$ and $AD: BC = 2: 3$. Let $E$ be the mid-point of $BC$. $AC$ and $DE$ intersect at $F$. If the area of $\triangle CEF$ is $36 \, \text{cm}^2$, then the area of the trapezium $ABCD$ is

In the figure, $ABCD$ is a circle. $AC$ and $BD$ intersect at $E$. If $AB=AD$ and $AD\parallel BC$, then $\angle BAE =$

In the figure, the bearing of $P$ from $O$ is $S86^{\circ}E$ and the bearing of $Q$ from $O$ is $N32^{\circ}E$. If $P$ and $Q$ are equidistant from $O$, then the bearing of $P$ from $Q$ is

If an interior angle of a regular $n$-sided polygon is $4$ times an exterior angle of the polygon, which of the following is/are true?

I. The value of $n$ is $10$.

II. The number of diagonals of the polygon is $10$.

III. The number of folds of rotational symmetry of the polygon is $10$.

In $\triangle ABC$, $AB:BC:AC=8:15:17$. Find $\cos A:\cos C$.

If $0^{\circ}<x<90^{\circ}$, which of the following must be true?

I. $\tan x\tan(90^{\circ}-x)=1$

II. $\sin x - \sin(90^{\circ} - x) < 0$

III. $\cos x + \cos(90^{\circ} - x) > 0$

The coordinates of the points A and B are $(2, 5)$ and $(4, -1)$ respectively. Let P be a moving point in the rectangular coordinate plane such that $AP = BP$. Find the equation of the locus of P.

The equation of the circle C is $2x^{2} + 2y^{2} - 4x + 8y - 5 = 0$. The coordinates of the points P and Q are $(-1, 2)$ and $(4, 0)$ respectively. Which of the following is/are true?

I. The radius of C is 5.

II. The mid-point of PQ lies outside C.

III. If G is the centre of C, then $\angle PGQ$ is an acute angle.

Two numbers are randomly drawn at the same time from seven cards numbered 1, 2, 3, 4, 5, 6 and 7 respectively. Find the probability that the product of the numbers drawn is an odd number.

If the mean and the mode of the nine numbers 14, 6, 4, 5, 7, 5, x, y and z are 8 and 14 respectively, then the median of these nine numbers is

The scatter diagram below shows the relation between $x$ and $y$. Which of the following may represent the relation between $x$ and $y$?

The stem-and-leaf diagram below shows the distribution of the hourly wages (in dollars) of some workers.

\begin{array}{c|ccccccccc}{{{\mathrm{s\!s)}}&{{{\mathrm{Leaf}\,\mathrm{(units)}}}} \\{{{\hline4}}}&{{{0}}}&{{{2}}}&{{{2}}}&{{{2}}}&{{{4}}}&{{{4}}}&{{{4}}}&{{{7}}} \\{{{5}}}&{{{0}}}&{{{0}}}&{{{1}}}&{{{2}}}&{{{2}}}&{{{6}}}&{{{8}}}&{{{9}}} \\{{{6}}}&{{{3}}}&{{{5}}}&{{{5}}}&{{{7}}} \\{{{7}}}&{{{0}}} \\{{{8}}}&{{{2}}}&{{{6}}} \\{{{9}}}&{{{5}}} \\end{array}}

Which of the following box-and-whisker diagrams may represent the distribution of their hourly wages?

The pie charts below show the distributions of the profits of stationery shop X and stationery shop Y from the sales of stationery in a certain month. Which of the following must be true?

Distribution of the profits of stationery shop X

Distribution of the profits of stationery shop Y

The L.C.M. of $a^{2}+4a+4$, $a^{2}-4$ and $a^{3}+8$ is

The figure above shows the graph of $y = ab^{x}$, where $a$ and $b$ are constants. Which of the following graphs may represent the relation between $x$ and $\log_{7} y$?

If $x - \log y = x^2 - \log y^2 - 10 = 2$, then $y =$

If $\alpha \neq \beta$ and $\begin{cases} 3\alpha = \alpha^2 - 5 \\ 3\beta = \beta^2 - 5 \end{cases}$, then $\alpha\beta =$

The real part of $i + 2i^{2} + 3i^{3} + 4i^{4}$ is

Consider the following system of inequalities: $$\begin{cases} x \geq 2 \\ y \geq 0 \\ x + 4y \leq 22 \\ 4x - y \leq 20 \end{cases}$$ Let $D$ be the region which represents the solution of the above system of inequalities. If $(x,y)$ is a point lying in $D$, then the greatest value of $3y-4x+15$ is

The $n$th term of a sequence is $2n-19$. Which of the following is/are true?

I. $25$ is a term of the sequence.

II. The sequence has $10$ negative terms.

III. The sum of the first $n$ terms of the sequence is $n^{2}-18n$.

Let $h$ and $k$ be constants. The figure shows the graph of $y = h + k \tan 2x^\circ$, where $0 \leq x \leq \alpha$. Which of the following are true?

If the height of a regular tetrahedron is $2\text{ cm}$, then the volume of the tetrahedron is

In the figure, $O$ is the centre of the circle $ABC$. $DE$ is the tangent to the circle at $A$. If $AB$ is the angle bisector of $\angle CAE$, then $\angle ACO =$

Find the range of values of $k$ such that the circle $x^2 + y^2 + 2x - 2y - 7 = 0$ and the straight line $3x - 4y + k = 0$ intersect.

Let O be the origin. If the coordinates of the points A and B are $(0, 12)$ and $(30, 12)$ respectively, then the y-coordinate of the circumcentre of $\Delta OAB$ is

If the first three digits and the last five digits of an eight-digit phone number are formed by a permutation of 5, 6, 9 and a permutation of 2, 3, 4, 7, 8 respectively, how many different eight-digit phone numbers can be formed?

If the variance of the five numbers $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ and $x_{5}$ is 13, then the variance of the five numbers $3x_{1}+4$, $3x_{2}+4$, $3x_{3}+4$, $3x_{4}+4$ and $3x_{5}+4$ is

If $5 - 3m = 2n$, then $m =$

Let $p$ and $q$ be constants. If $x^{2}+p(x+5)+q \equiv (x-2)(x+5)$, then $q =$

Let $f(x)=x^{3}+2x^{2}-7x+3$. When $f(x)$ is divided by $x+2$, the remainder is