Past paper questions

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.

Simplify $\frac{3}{k-9} + \frac{2}{5k+6}$

A fan is sold at a discount of $30\%$ on its marked price. After selling the fan, the profit is \78$and the percentage profit is$26\%$. Find the marked price of the fan. (4 marks)

Consider the compound inequality

$-2(3x+2) > x+10$ or $2x \leq -8$ $\ldots$ $(*)$.

The coordinates of the points $S$ and $T$ are $(12, -5)$ and $(-3, -7)$ respectively. $S$ is rotated anticlockwise about $O$ through $90^{\circ}$ to $S'$, where $O$ is the origin. $T'$ is the reflection image of $T$ with respect to the $x$-axis.

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the times taken to complete a $3\text{ km }$ race by a group of students.

The stem-and-leaf diagram below shows the distribution of the ages of the players of a football team.

Stem (tens) | Leaf (units)
1 | 7 8 9
2 | 0 a a 8 8 9
3 | b b 5 5 6 6 6 6 7 8
4 | 3

The inter-quartile range and the median of the distribution are $14$ and $31$ respectively.

The equation of the circle $C$ is $x^{2}+y^{2}-154x-128y+224=0$. Denote the centre of $C$ by $G$. The coordinates of the point $H$ are $(65, 48)$.

There are two solid metal spheres. The ratio of the surface area of the smaller sphere to the surface area of the larger sphere is $4:9$. The radius of the larger sphere is $9\text{ cm}$.

Let $p(x)=2x^{3}+ax^{2}+bx-20$, where $a$ and $b$ are constants. When $p(x)$ is divided by $x^{2}-2x+3$, the remainder is $x+13$.

Let $g(x) = 3x^{2} + 12kx + 16k^{2} + 8$, where $k$ is a non-zero real constant.

The centre of the circle $C$ is the point $G(83,112)$. It is found that the point $A(158,12)$ lies outside $C$. $AP$ and $AQ$ are the tangents to $C$ at the points $P$ and $Q$ respectively. It is given that $C$ passes through the point $(23,67)$.

If $m$ and $n$ are constants such that $(x+3)^{2}+mx\equiv(x-n)(x+1)+2n$, then $m=$

Let $c$ be a constant. Solve the equation $(x-c)(x-4c)=(3c-x)(x-4c)$.

If $\frac{2}{u} + \frac{3}{v} = 4$, then $u =$

It is given that $x$ is a real number. If $x$ is rounded down to 3 significant figures, then the result is 345. Find the range of values of $x$.

The solution of $3y-5<5y+1\leq 11$ is

Let $f(x)=x^2-x+1$. If $k$ is a constant, which of the following must be true?

Let $g(x)=x^{2}+ax+b$, where $a$ and $b$ are constants. If $g(x)$ is divisible by $x+2a$, find the remainder when $g(x)$ is divided by $x-2a$.

Let $h$ and $k$ be real constants such that $hk<0$. Which of the following statements about the graph of $y=(h-x)(k-x)$ are true?

A sum of \88\,000$is deposited at an interest rate of$6\%$per annum for$4$ years, compounded monthly. Find the interest correct to the nearest dollar.

Let $x$, $y$ and $z$ be non-zero numbers. If $x:y=8:5$ and $2x=4z-3y$, then $y:z=$

If $u$ varies directly as the square root of $v$ and inversely as $w$, which of the following are true?

I. $u^{2}$ varies directly as $v$ and inversely as the square of $w$.

II. $v$ varies directly as $w$ and inversely as the square root of $u$.

III. $w$ varies directly as the square root of $v$ and inversely as $u$.

In the figure, the 1st pattern consists of 8 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $(2n+6)$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

The radius of a solid hemisphere and the base radius of a solid right circular cylinder are equal. If the height of the circular cylinder is equal to its base diameter, then the ratio of the total surface area of the hemisphere to the total surface area of the circular cylinder is

The diameter of a circle is $10\text{ cm}$. The circle is divided into a major segment and a minor segment by a chord of length $8\text{ cm}$. Find the area of the major segment correct to the nearest $\text{ cm}^{2}$.

In the figure, $M$ and $N$ are points lying on $PQ$ and $QR$ respectively such that $PM:MQ=5:6$ and $QN:NR=3:4$. If the area of the quadrilateral $MNRP$ is $59\text{ cm}^{2}$, then the area of $\Delta MNQ$ is