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.