Past paper questions

Let $g(x)=x^{8}+ax^{7}+b$, where $a$ and $b$ are constants. If $g(x)$ is divisible by $x-1$, find the remainder when $g(x)$ is divided by $x+1$.

A sum of $$100\,000$ is deposited at an interest rate of $2\%$ per annum for $3$ years, compounded monthly. Find the interest correct to the nearest dollar.

Let $a$, $b$ and $c$ be non-zero numbers. If $3a = 4b$ and $a:c = 2:5$, then $\frac{a+3b}{b+3c} =$

If $w$ varies directly as the square root of $u$ and inversely as the square of $v$, which of the following must be constant?

Let $a_n$ be the $n$th term of a sequence. If $a_3 = 21$, $a_6 = 89$ and $a_{n+2} = a_n + a_{n+1}$ for any positive integer $n$, then $a_1 =$

The solution of $\frac{1-2x}{3} \ge q x-3$ or $4x+9 < 1$ is

In the figure, ABCDEFGH is an octagon, where all the measurements are correct to the nearest cm. Let $x \, cm^2$ be the actual area of the octagon. Find the range of values of x.

In the figure, the volume of the solid right triangular prism is

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $BC$ such that $BE:EC = 5:3$. $AE$ and $BD$ intersect at the point $F$. If the area of $\triangle ABF$ is $120\ cm^{2}$, then the area of the quadrilateral $CDFE$ is

In the figure, $O$ is the centre of the sector $OABCD$. $AD$ and $OC$ are perpendicular to each other and intersect at the point $E$. $F$ is a point lying on $AD$ such that $BF$ is perpendicular to $AD$. If $AF = 9$ cm, $DF = 39$ cm and $OE = 18$ cm, then the area of the sector $OBC$ is

In the figure, $ABCD$ is a rhombus. $E$ and $F$ are points lying on $AB$ and $AD$ respectively such that $AE = AF$ and $\angle ECF = 42^{\circ}$. If $\angle BAD = 110^{\circ}$, then $\angle BEC =$

In the figure, $ABCDE$ is a regular pentagon. $AD$ and $CE$ intersect at the point $F$. Which of the following are true?
I. $CD = CF$
II. $\Delta ABF \cong \Delta CBF$
III. $\angle AFB + \angle EAF = 90^{\circ}$

In the figure, $ABCD$ is a square. $E$ is a point lying on $AB$ produced such that $BE = 4$ cm. $BC$ and $DE$ intersect at the point $F$. If $EF = 5$ cm, then $DF =$

In the figure, $ABCD$ is a trapezium with $\angle ABC = \angle BAD = 90^{\circ}$. $E$ and $F$ are points lying on $AB$ such that $E$ and $F$ divide $AB$ into three equal parts. Which of the following must be true?

I. $AF\sin\alpha=BE\sin\beta$

II. $CE\cos\alpha=DF\cos\beta$

III. $AD\tan\alpha=BC\tan\beta$

In the figure, $ABCD$ is a circle. $AD$ produced and $BC$ produced meet at the point $E$. It is given that $BD=DE$, $\angle BAC=66^{\circ}$ and $\angle ABD=30^{\circ}$. Find $\angle CED$.

The figure below consists of eight identical squares. The number of folds of rotational symmetry of the figure is

The polar coordinates of the points $C$, $D$ and $E$ are $(16, 127^{\circ})$, $(12, 217^{\circ})$ and $(5, 307^{\circ})$ respectively. Find the perimeter of $\Delta CDE$.

The equations of the straight lines $L_{1}$ and $L_{2}$ are $3x - y + 7 = 0$ and $12x - 4y - 11 = 0$ respectively. Let $P$ be 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 $L_{2}$. Find the equation of the locus of $P$.

The equation of the straight line $L_{1}$ is $4x + 3y - 36 = 0$. The straight line $L_{2}$ is perpendicular to $L_{1}$ and intersects $L_{1}$ at a point lying on the $y$-axis. Find the area of the region bounded by $L_{1}$, $L_{2}$ and the $x$-axis.

The equation of the circle $C$ is $5x^{2}+5y^{2}-30x+10y+6=0$. Which of the following is true?

Two numbers are randomly drawn at the same time from seven cards numbered 1, 1, 1, 2, 2, 3 and 4 respectively. Find the probability that the sum of the numbers drawn is 5.

The mean of the numbers of pages of 10 magazines is 132. If the mean of the numbers of pages of 6 of these 10 magazines is 108, then the mean of the numbers of pages of the remaining 4 magazines is

The stem-and-leaf diagram below shows the distribution of the numbers of books read by 20 students in a year.

$\begin{array}{c|ccccc}{{\underline{S t e m}\textsc{(t e n s)}}}&{{\underline{L e a f}\textsc{(units)}}}&{{{8}}} \\{{{3}}}&{{{a}}}&{{{a}}} \\{{{4}}}&{{{0}}}&{{{2}}}&{{{4}}}&{{{5}}}&{{{5}}}&{{{7}}}&{{{8}}} \\{{{5}}}&{{{3}}} \\{{{6}}}&{{{b}}}&{{{b}}}&{{{9}}}&{{{9}}} \\{{{7}}}&{{{0}}}&{{{8}}} \\\end{array}$

If the inter-quartile range of the above distribution is at most $25$, which of the following must be true

I. $5 \le q a \le q 9$
II. $0 \le q b \le q 4$
III. $1 \le q a - b \le q 6$

Let $f(x)$ be a quadratic function. The figure below may represent the graph of $y = f(x)$ and

The figure shows the graph of $y = \log_a x$ and the graph of $y = \log_b x$ on the same rectangular coordinate system, where $a$ and $b$ are positive constants. If a vertical line cuts the graph of $y = \log_a x$, the graph of $y = \log_b x$ and the x-axis at the points $A$, $B$ and $C$ respectively, which of the following is/are true?

In the figure, the straight line $L$ shows the relation between $\log_{4}x$ and $\log_{4}y$. It is given that $L$ passes through the points $(1,2)$ and $(9,6)$. If $y=kx^{a}$, then $k=$

Consider the following system of inequalities:

$\begin{cases}x-21\leq 0\\x-y-35\leq 0\\x+5y-91\leq 0\\3x+2y\geq 0\end{cases}$

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 least value of $5x + 6y + 234$ is

If the sum of the first $n$ terms of a sequence is $6n^{2}-n$, which of the following is/are true?

I. 22 is a term of the sequence.

II. The 1st term of the sequence is 5.

III. The sequence is a geometric sequence.

If $m \ne q n$ and $2m^{2} + 5m = 2n^{2} + 5n = 14$, then $(m + 2)(n + 2) =$

The real part of $\frac{2i^{12}+3i^{13}+4i^{14}+5i^{15}+6i^{16}}{1-i}$ is

For $0^{\circ} \le q x < 360^{\circ}$, how many roots does the equation $6\cos^{2}x = \cos x + 5$ have?

In the figure, $TA$ is the tangent to the circle $ABCD$ at the point $A$. $CD$ produced and $TA$ produced meet at the point $E$. It is given that $AB = CD$, $\angle BAT = 24^{\circ}$ and $\angle AED = 72^{\circ}$. Find $\angle ABC$.

It is given that $a$ is a positive constant. The straight line $2x + 5y = a$ cuts the $x$-axis and the $y$-axis at the points $P$ and $Q$ respectively. Let $R$ be a point lying on the $y$-axis such that the $x$-coordinate of the orthocentre of $\Delta PQR$ is 10. Find the $y$-coordinate of $R$.

In the figure, $ABCDEFGH$ is a rectangular block. Let X be a point lying on DE such that $DX = 9 \, cm$ and $EX = 4 \, cm$. Denote the angle between $BX$ and the plane $ABGF$ by $\theta$. Find $\cos\theta$.

In a class, there are 14 boys and 15 girls. If 3 students of the same gender are selected from the class to form a team, how many different teams can be formed?

John and Mary take turns to throw a fair die until one of them gets a number ‘1’ or ‘6’. John throws the die first. Find the probability that John gets a number ‘6’.

In a test, the mean of the test scores is 68 marks. Peter gets 46 marks in the test and his standard score is -2.2. If Susan gets 52 marks in the test, then her standard score is

There are 49 terms in an arithmetic sequence. If the variance of the first 7 terms of the sequence is 9, then the variance of the last 7 terms of the sequence is