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 $\frac{(xy)^2}{x^{-5}y^6}$ and express your answer with positive indices.

Make $b$ the subject of the formula $a(b+7)=a+b$

Factorize

The marked price of a handbag is \560$. It is given that the marked price of the handbag is$40\%$ higher than the cost.

In a football league, each team gains 3 points for a win, 1 point for a draw and 0 point for a loss. The champion of the league plays 36 games and gains a total of 84 points. Given that the champion does not lose any games, find the number of games that the champion wins.

In Figure 2, $O$ is the centre of the semicircle $ABCD$. If $AB \parallel OC$ and $\angle BAD = 38^{\circ}$, find $\angle BDC$.

In Figure 3, the coordinates of the point $A$ are $(-2,5)$. $A$ is rotated clockwise about the origin $O$ through $90^{\circ}$ to $A'$. $A''$ is the reflection image of $A$ with respect to the $y$-axis.

In a factory, the production cost of a carpet of perimeter $s$ metres is $C$. It is given that $C$ is a sum of two parts, one part varies as $s$ and the other part varies as the square of $s$. When $s=2$, $C=356$; when $s=5$, $C=1250$.

In Figure 6, the straight line $L_{1}$: $4x-3y+12=0$ and the straight line $L_{2}$ are perpendicular to each other and intersect at $A$. It is given that $L_{1}$ cuts the $y$-axis at $B$ and $L_{2}$ passes through the point $(4,9)$.

The data below show the percentages of customers who bought newspaper A from a magazine stall in city H for five days randomly selected in a certain week:

$62\%$ $63\%$ $55\%$ $62\%$ $58\%$

(a) Find the median and the mean of the above data. (2 marks)

(b) Let $a\%$ and $b\%$ be the percentages of customers who bought newspaper A from the stall for the other two days in that week. The two percentages are combined with the above data to form a set of seven data.

(i) Write down the least possible value of the median of the combined set of seven data.

(ii) It is known that the median and the mean of the combined set of seven data are the same as that found in (a). Write down one pair of possible values of $a$ and $b$.

(c) The stall-keeper claims that since the median and the mean found in (a) exceed $50\%$, newspaper A has the largest market share among the newspapers in city H. Do you agree? Explain your answer. (2 marks)

A committee consists of 5 teachers from school A and 4 teachers from school B. Four teachers are randomly selected from the committee.

A researcher defined Scale A and Scale B to represent the magnitude of an explosion as shown in the following table:

In Figure 8(a), $ABC$ is a triangular paper card. $D$ is a point lying on $AB$ such that $CD$ is perpendicular to $AB$. It is given that $AC=20$ cm, $\angle CAD=45^{\circ}$ and $\angle CBD=30^{\circ}$.

In Figure 9, the circle passes through four points A, B, C and D. PQ is the tangent to the circle at C and is parallel to BD. AC and BD intersect at E. It is given that AB = AD.

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

Let $a$ be a constant. Solve the equation $(x-a)(x-a-1)=(x-a)$.

Find the range of values of $k$ such that the quadratic equation $x^{2}-6x=2-k$ has no real roots.

In the figure, the quadratic graph of $y=f(x)$ intersects the straight line $L$ at $A(1,k)$ and $B(7,k)$. Which of the following are true?

I. The solution of the inequality $f(x)>k$ is $x<1$ or $x>7$.

II. The roots of the equation $f(x)=k$ are 1 and 7.

III. The equation of the axis of symmetry of the quadratic graph of $y=f(x)$ is $x=3$.

The solution of $5-2x<3$ and $4x+8>0$ is

Mary sold two bags for \240$each. She gained$20\%$on one and lost$20\%$ on the other. After the two transactions, Mary

Let $a_n$ be the $n$th term of a sequence. If $a_1 = 4$, $a_2 = 5$ and $a_{n+2} = a_n + a_{n+1}$ for any positive integer $n$, then $a_{10} =$

If the length and the width of a rectangle are increased by $20\%$ and $x\%$ respectively so that its area is increased by $50\%$, then $x =$

If $x$, $y$ and $z$ are non-zero numbers such that $2x = 3y$ and $x = 2z$, then $(x + z) \colon (x + y) =$

It is given that $z$ varies directly as $x$ and inversely as $y$. When $x=3$ and $y=4$, $z=18$. When $x=2$ and $z=8$, $y=$

The lengths of the three sides of a triangle are measured as $15$ cm, $24$ cm and $25$ cm respectively. If the three measurements are correct to the nearest cm, find the percentage error in calculating the perimeter of the triangle correct to the nearest $0.1\%$.

In the figure, $O$ is the centre of the circle. $C$ and $D$ are points lying on the circle. $OBC$ and $BAD$ straight lines. If $OC = 20$ cm and $OA = AB = 10$ cm, find the area of the shaded region $BCD$ correct the nearest $\text{cm}^{2}$.

The figure shows a right circular cylinder, a hemisphere and a right circular cone with equal base radii. Their curved surface areas are $a\text{ cm}^2$, $b\text{ cm}^2$ and $c\text{ cm}^2$ respectively.