Past paper questions

The graph in the figure shows the linear relation between $x$ and $\log_{3} y$. If $y = mn^{x}$, then $n=$

Let $f(x)$ be a quadratic function. If the coordinates of the vertex of the graph of $y=f(x)$ are $(3,-4)$, which of the following must be true?

The figure shows a shaded region (including the boundary). If $(h, k)$ is a point lying in the shaded region, which of the following are true?

I. $k \geq 3$

II. $h - k \geq -3$

III. $2h + k \leq 6$

Let $a_{n}$ be the $n$th term of an arithmetic sequence. If $a_{18}=26$ and $a_{23}=61$, which of the following are true?

I. $a_{14}<0$

II. $a_{1}-a_{2}<0$

III. $a_{1}+a_{2}+a_{3}+\cdots+a_{27}>0$

Which of the following may represent the graph of $y = f(x)$ and the graph of $y = f(x - 2) + 1$ on the same rectangular coordinate system?

The figure shows

The figure shows a regular tetrahedron $ABCD$. Find the angle between the plane $ABC$ and the plane $BCD$ correct to the nearest degree.

In the figure, $PQ$ is the tangent to the circle $ABC$ at $O$, where $O$ is the centre of the semicircle $PBQ$. It is given that $BCP$ is a straight line. If $\angle BPQ = 12^{\circ}$, then $\angle BAC =$

Find the range of values of $k$ such that the circle $x^{2}+y^{2}+2x-4y-13=0$ and the straight line $x-y+k=0$ intersect at two distinct points.

A drama club is formed by 12 boys and 8 girls. If a team of 5 students is selected from the club to participate in a competition and the team consists of at least one girl, how many different teams can be formed?

A box contains six balls numbered $7$, $8$, $9$, $9$ and $9$ respectively. John repeats drawing one ball at a time randomly from the box without replacement until the number drawn is $9$. Find the probability that he needs exactly three draws.

Let $m_{1}$, $r_{1}$ and $v_{1}$ be the mean, the range and the variance of a group of numbers $\{x_{1}, x_{2}, x_{3}, \ldots, x_{100}\}$ respectively. If $m_{2}$, $r_{2}$ and $v_{2}$ are the mean, the range and the variance of the group of numbers $\{x_{1}, x_{2}, x_{3}, \ldots, x_{100}, m_{1}\}$ respectively, which of the following must be true?

I. $m_{1}=m_{2}$

II. $r_{1}=r_{2}$

III. $v_{1}=v_{2}$

In a recreation club, there are 180 members and the number of male members is $40\%$ more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)

It is given that $f(x)$ is the sum of two parts, one part varies as $x$ and the other part varies as $x^{2}$. Suppose that $f(3)=48$ and $f(9)=198$.

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is $12\text{ cm }$. Peter then pours $444\pi$ cm^{3} of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm }$.

If $4$ boys and $5$ girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)

In a test, the mean of the distribution of the scores of a class of students is $61$ marks. The standard scores of Albert and Mary are $-2.6$ and $1.4$ respectively. Albert gets $22$ marks. A student claims that the range of the distribution is at most $59$ marks. Is the claim correct? Explain your answer.

The 1st term and the 38th term of an arithmetic sequence are $666$ and $555$ respectively. Find

Figure 2 shows a geometric model $ABCD$ in the form of tetrahedron. It is given that $\angle BAD = 86^{\circ}$, $\angle CBD = 43^{\circ}$, $AB = 10$ cm, $AC = 6$ cm, $BC = 8$ cm and $BD = 15$ cm.

If $\frac{a}{x} \div \frac{b}{y} = 3$, then $x =$

If $4\alpha + \beta = 7\alpha + 3\beta = 5$, then $\beta =$

Let $f(x)=4x^{3}+kx+3$, where $k$ is a constant. If $f(x)$ is divisible by $2x+1$, find the remainder when $f(x)$ is divided by $x+1$.

The solution of $-5x > 21 - 2x$ and $6x - 18 < 0$ is

If $k$ is a constant such that the quadratic equation $x^{2}+kx+8k+36=0$ has equal roots, then $k=$

If $-1 < a < 0$, which of the following may represent the graph of $y = (ax + 1)^2 + a$?

The monthly salary of Donald is $25\%$ higher than that of Peter while the monthly salary of Peter is $25\%$ lower than that of Teresa. It is given that the monthly salary of Donald is \33\,360$. The monthly salary of Teresa is

If $x$ and $y$ are non-zero numbers such that $(3y-4x):(2x+y)=5:6$, then $x:y=$

It is given that $z$ varies directly as $\sqrt{x}$ and inversely as $y$. If $x$ is decreased by $36\%$ and $y$ is increased by $60\%$, then $z$

The cost of flour of brand X is \42/\text{kg}$. If$3\text{ kg}$of flour of brand X and$2\text{ kg}$of flour of brand Y are mixed so that the cost of the mixture is$\$36/\text{kg}$, find the cost of flour of brand Y.

In the figure, the 1st pattern consists of $9$ dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $5$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

According to the figure, which of the following must be true?

I. $a + c = 180^{\circ}$

II. $a + b - c = 180^{\circ}$

III. $b + c = 360^{\circ}$

In the figure, $ABC$ is a straight line. If $AB = 24\ cm$, $AD = 40\ cm$, $BD = 32\ cm$ and $CD = 68\ cm$, then $BC =$

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $CD$ such that $BE = CE$. If $\angle ADC = 114^\circ$, then $\angle ABE =$