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