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

Factorize

The price of 7 pears and 3 oranges is \47$while the price of 5 pears and 6 oranges is$\$49$. Find the price of a pear. (4 marks)

In a polar coordinate system, $O$ is the pole. The polar coordinates of the points $A$ and $B$ are $(26,10^{\circ})$ and $(26,130^{\circ})$ respectively. Let $L$ be the axis of reflectional symmetry of $\Delta OAB$.

In Figure 1, $ABCD$ is a quadrilateral. The diagonals $AC$ and $BD$ intersect at $E$. It is given that $BE = CE$ and $\angle BAC = \angle BDC$.

A pack of sea salt is termed regular if its weight is measured as $100\text{ g }$ correct to the nearest g.

The bar chart below shows the distribution of the numbers of family members of the employees of company $D$.

Distribution of the numbers of family members of the employees of company D

Figure 1

The weight of a tray of perimeter $\ell$ metres is $W$ grams. It is given that $W$ is the sum of two parts, one part varies directly as $\ell$ and the other part varies directly as $\ell^{2}$. When $\ell=1$, $W=181$ and when $\ell=2$, $W=402$.

Let $f(x)=3x^{3}-7x^{2}+kx-8$, where $k$ is a constant. It is given that $f(x)\equiv(x-2)(ax^{2}+bx+c)$, where $a$, $b$ and $c$ are constants.

(4 marks)

In a workshop, 2 identical solid metal right circular cylinders of base radius $R$ cm are melted and recast into $27$ smaller identical solid right circular cylinders of base radius $r$ cm and height $10$ cm. It is given that the base area of a larger circular cylinder is $9$ times that of a smaller one.

A box contains $5$ white cups and $11$ blue cups. If $6$ cups are randomly drawn from the box at the same time,

If $\frac{y-1}{c}=\frac{y+1}{d}$, then $y=$

$0.0504545$ =

The solution of $x - \frac{x-1}{2} > 5$ or $1 < x - 11$ is

Let $k$ be a constant. Solve the equation $(x-k)^{2}=4k^{2}$.

The figure shows the graph of $y = -2x^{2} + ax + b$, where $a$ and $b$ are constants. The equation of the axis of symmetry of the graph is

If $a$, $b$ and $c$ are non-zero constants such that $x(x+3a)+a \equiv x^{2}+2(bx+c)$, then $a:b:c=$

Let $f(x)=x^{13}-2x+k$, where $k$ is a constant. If $f(x)$ is divisible by $x+1$, find the remainder when $f(x)$ is divided by $x-1$.

Susan sells two cars for \80,080$each. She gains$30\%$on one and loses$30\%$ on the other. After the two transactions, Susan

A sum of \50\,000$is deposited at an interest rate of$8\%$per annum for$1$ year, compounded monthly. Find the interest correct to the nearest dollar.

The actual area of a playground is $900\text{ m}^{2}$. If the area of the playground on a map is $36\text{ cm}^{2}$, then the scale of the map is

It is given that $z$ varies directly as $x$ and inversely as $\sqrt{y}$. If $y$ is decreased by $64\%$ and $z$ is increased by $25\%$, then $x$

The figure shows the graph of the straight line $x+ay+b=0$. Which of the following are true?

I. $a<0$
II. $b<0$
III. $a<b$

In the figure, the regular octagon is divided into eight identical isosceles triangles and four of the shaded. The number of axes of reflectional symmetry of the octagon is

In the figure, the diameter of the semicircle $ABC$ is $3$ cm. If $AC = 2$ cm, find the area of the shaded region correct to the nearest $0.01$ cm$^2$.

In the figure, the solid consists of a right circular cone and a hemisphere with a common base. The base radius and the height of the circular cone are $3$ cm and $4$ cm respectively. Find the total surface area of the solid.