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.