Past paper questions

It is given that $\log_{a}y$ is a linear function of $x$, where $0 < a < 1$. The intercepts on the vertical axis and on the horizontal axis of the graph of the linear function are $6$ and $3$ respectively. If $y = mn^{x}$, which of the following is/are true?

If $\log_{4}y = 2x - 1$ and $(\log_{4}y)^{2} = 20x - 31$, then $\log_{2}y =$

Let $z = 4 + 5i^{10} - ki^{15} + 6i^{21} + 2ki^{28}$, where $k$ is a real number. If the real part and the imaginary part of $z$ are equal, then the real part of $z$ is

Consider the following system of inequalities:

$$\begin{cases} 2x + y \geq 8 \\ 2x + 3y \geq 16 \\ 4x + 3y \leq 22 \end{cases}$$

Let $R$ be the region which represents the solution of the above system of inequalities. If $(x, y)$ is a point lying in $R$, then the least value of $7x + 6y$ is

Let $a_n$ be the $n$th term of a geometric sequence. It is given that $a_1 = 8p^2$, $a_2 = 1$ and $a_3 = 27p$, where $p$ is a real number. Find $a_4$.

In the figure, $ABCD$ is a circle. $PA$ and $QB$ are the tangents to the circle at $A$ and $B$ respectively. If $\angle ADC = 79^\circ$, $\angle CBQ = 39^\circ$ and $\angle DAP = 42^\circ$, then $\angle BCD =$

For $0^\circ \leq x < 360^\circ$, how many roots does the equation $\sin^2 x = 6\cos^2 x$ have?

In the figure, $ABCDEFGH$ is a cube. Let $\alpha$ be the angle between $\Delta AFG$ and $\Delta AFH$ while $\beta$ be the angle between $\Delta AFH$ and $\Delta FGH$. Which of the following is true?

Let O be the origin. The coordinates of the points A and B are $(a,0)$ and $(0,b)$ respectively, where $a$ and $b$ are positive numbers. If the circumcentre of $\Delta OAB$ lies on the straight line $4x+16y=17a$, then $a:b=$

If the first five digits and the last two digits of a seven-digit password are formed by a permutation of $1,3,5,7,9$ and a permutation of $2,8$ respectively, how many different seven-digit passwords can be formed?

A box contains 2 white balls, 2 yellow balls and 3 red balls. A boy and a girl take turns to draw one ball randomly from the box with replacement until one of them draws a white ball or a yellow ball. The boy draws a ball first. Find the probability that the girl draws a white ball.

In a test, the median of the test scores of a class of students is 30 marks. All the students fail in the test, so the test score of each student is adjusted such that each score is increased by $50\%$ and then extra 8 marks are added. Let x marks be the median of the test scores of the class of students after the score adjustment. In the test, the standard score of a student before the score adjustment is -2. Denote the standard score of this student after the score adjustment by z. Find x and z.

It is given that d is a real number. Let $S_{1}$ be a group of numbers $\{d-6, d-2, d-1, d+3, d+5, d+7\}$ and $S_{2}$ be another group of numbers $\{d-7, d-5, d-3, d+1, d+2, d+6\}$. Which of the following is/are true?

I. The means of $S_{1}$ and $S_{2}$ are equal.

II. The standard deviations of $S_{1}$ and $S_{2}$ are equal.

III. The inter-quartile ranges of $S_{1}$ and $S_{2}$ are equal.

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.