Past paper questions

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

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 $\frac{1}{\nu}$.

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?

[Figure 1]

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 $0^{\circ} \le q \theta \le q 360^{\circ}$, 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 $CQ = 9\,\text{cm}$ and $\angle PH = 15\,\text{cm}$. 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)}}}}}}&{{{8}}} \\{{{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.

There are only two kinds of admission tickets for a theatre: regular tickets and concessionary tickets. The prices of a regular ticket and a concessionary ticket are \126$and$\$78$ respectively. On a certain day, the number of regular tickets sold is 5 times the number of concessionary tickets sold and the sum of money for the admission tickets sold is \50,976$. Find the total number of admission tickets sold that day. (4 marks)

The coordinates of the points $A$ and $B$ are $(-3, 4)$ and $(9, -9)$ respectively. $A$ is rotated anticlockwise about the origin through $90^{\circ}$ to $A'$. $B'$ is the reflection image of $B$ with respect to the $x$-axis.

The pie chart below shows the distribution of the seasons of birth of the students in a school.

Distribution of the seasons of birth of the students in the school

If a student is randomly selected from the school, then the probability that the selected student was born in spring is $\frac{1}{9}$.

It is given that $y$ varies inversely as $\sqrt{x}$. When $x=144$, $y=81$.

A bottle is termed standard if its capacity is measured as $200\text{ mL}$ correct to the nearest $10\text{ mL}$.

In Figure 1, $OPQR$ is a quadrilateral such that $OP = OQ = OR$. $OQ$ and $PR$ intersect at the point $S$. $S$ is the mid-point of $PR$.

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

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

It is given that the mean and the range of the above distribution are \70$and$\$22$ respectively.

A solid metal right prism of base area $84\text{ cm}^{2}$ and height $20\text{ cm}$ is melted and recast into two similar solid right pyramids. The bases of the two pyramids are squares. The ratio of the base area of the smaller pyramid to the base area of the larger pyramid is $4:9$.

The coordinates of the points $E$, $F$ and $G$ are $(-6,5)$, $(-3,11)$ and $(2,-1)$ respectively. The circle $C$ passes through $E$ and the centre of $C$ is $G$.

Let $f(x)=6x^{3}-13x^{2}-46x+34$. When $f(x)$ is divided by $2x^{2}+ax+4$, the quotient and the remainder are $3x+7$ and $bx+c$ respectively, where $a$, $b$ and $c$ are constants.

Let $a$ and $b$ be constants. Denote the graph of $y = a + \log_{b}x$ by $G$. The $x$-intercept of $G$ is $9$ and $G$ passes through the point $(243, 3)$.

Express $x$ in terms of $y$. (4 marks)

A city adopts a plan to import water from another city. It is given that the volume of water imported in the 1st year since the start of the plan is $1.5 \times 10^7 \, \text{m}^3$ and in subsequent years, the volume of water imported each year is $10\%$ less than the volume of water imported in the previous year.

In a bag, there are 4 green pens, 7 blue pens and 8 black pens. If 5 pens are randomly drawn from the bag at the same time,

The equation of the parabola $\Gamma$ is $y=2x^{2}-2kx+2x-3k+8$, where $k$ is a real constant. Denote the straight line $y=19$ by $L$.

$ABC$ is a thin triangular metal sheet, where $BC = 24\text{ cm}$, $\angle BAC = 30^\circ$ and $\angle ACB = 42^\circ$.

If $\frac{a+4b}{2a}=2+\frac{b}{a}$, then $a=$

The solution of $6-x<2x-3$ or $7-3x>1$ is

Let $k$ be a constant. If $f(x)=2x^{2}-5x+k$, then $f(2)-f(-2)=$

Let $p(x)=2x^{2}-11x+c$, where $c$ is a constant. If $p(x)$ is divisible by $x-7$, find the remainder when $p(x)$ is divided by $2x+1$.

If $m$ and $n$ are constants such that $4x^{2}+m(x+1)+28=mx(x+3)+n(x-4)$, then $n=$

The figure shows the graph of $y=(px+5)^{2}+q$, where $p$ and $q$ are constants. Which of the following is true?

A sum of \2\,000$is deposited at an interest rate of$5\%$per annum for$4$ years, compounded half-yearly. Find the interest correct to the nearest dollar.

The scale of a map is $1:20\,000$. If the area of a zoo on the map is $4\text{ cm}^{2}$, then the actual area of the zoo is

It is given that $y$ is the sum of two parts, one part is a constant and the other part varies as $x^{2}$. When $x=1$, $y=7$ and when $x=2$, $y=13$. If $x=3$, then $y=$

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

In the figure, $D$ is a point lying on $AC$ such that $BD$ is perpendicular to $AC$. It is given that $AC=14\mathrm{~cm}$ and $BD=12\mathrm{~cm}$. If the area of $\triangle ABD$ is greater than the area of $\triangle BCD$ by $24\mathrm{~cm}^{2}$, then the perimeter of $\triangle ABC$ is

The base radius of a right circular cone is 2 times the base radius of a right circular cylinder while the height of the circular cylinder is 3 times the height of the circular cone. If the volume of the circular cone is $36\pi\text{ cm}^{3}$, then the volume of the circular cylinder is

In the figure, $ABCD$ and $BEDF$ are parallelograms. $E$ is a point lying on $BC$ such that $BE:EC=2:3$. $AC$ cuts $BF$ and $DE$ at $G$ and $H$ respectively. If the area of $\triangle ABG$ is $135\text{ cm}^{2}$, then the area of the quadrilateral $DFGH$ is