Past paper questions

If $m$ and $n$ are constants such that $(x+3)^{2}+mx\equiv(x-n)(x+1)+2n$, then $m=$

Let $c$ be a constant. Solve the equation $(x-c)(x-4c)=(3c-x)(x-4c)$.

If $\frac{2}{u} + \frac{3}{v} = 4$, then $u =$

It is given that $x$ is a real number. If $x$ is rounded down to 3 significant figures, then the result is 345. Find the range of values of $x$.

The solution of $3y-5<5y+1\leq 11$ is

Let $f(x)=x^2-x+1$. If $k$ is a constant, which of the following must be true?

Let $g(x)=x^{2}+ax+b$, where $a$ and $b$ are constants. If $g(x)$ is divisible by $x+2a$, find the remainder when $g(x)$ is divided by $x-2a$.

Let $h$ and $k$ be real constants such that $hk<0$. Which of the following statements about the graph of $y=(h-x)(k-x)$ are true?

A sum of \88\,000$is deposited at an interest rate of$6\%$per annum for$4$ years, compounded monthly. Find the interest correct to the nearest dollar.

Let $x$, $y$ and $z$ be non-zero numbers. If $x:y=8:5$ and $2x=4z-3y$, then $y:z=$

If $u$ varies directly as the square root of $v$ and inversely as $w$, which of the following are true?

I. $u^{2}$ varies directly as $v$ and inversely as the square of $w$.

II. $v$ varies directly as $w$ and inversely as the square root of $u$.

III. $w$ varies directly as the square root of $v$ and inversely as $u$.

In the figure, the 1st pattern consists of 8 dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $(2n+6)$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

The radius of a solid hemisphere and the base radius of a solid right circular cylinder are equal. If the height of the circular cylinder is equal to its base diameter, then the ratio of the total surface area of the hemisphere to the total surface area of the circular cylinder is

The diameter of a circle is $10\text{ cm}$. The circle is divided into a major segment and a minor segment by a chord of length $8\text{ cm}$. Find the area of the major segment correct to the nearest $\text{ cm}^{2}$.

In the figure, $M$ and $N$ are points lying on $PQ$ and $QR$ respectively such that $PM:MQ=5:6$ and $QN:NR=3:4$. If the area of the quadrilateral $MNRP$ is $59\text{ cm}^{2}$, then the area of $\Delta MNQ$ is

In the figure, the perimeter of the rectangle $ABCD$ is $170\text{ cm}$. It is given that $EBF$ is a straight line and $\angle AEB = \angle BFC = 90^{\circ}$. If $AE = 24\text{ cm}$ and $BC = 34\text{ cm}$, then $EF =$

In the figure, $ABC$ is an equilateral triangle. Let $D$ and $E$ be points lying on $AC$ and $BC$ respectively such that $AD=CE$. If $\angle CBD=38^{\circ}$, then $\angle AEB=$

The figure shows a parallelogram. Which of the following must be true?

I. $a + b = 180^{\circ}$
II. $b + c = 360^{\circ}$
III. $c + d = 540^{\circ}$

In the figure, $O$ is the centre of the circle $ABC$. If $\angle ABO = 36^{\circ}$, then $\angle ACB =$

In the figure, $ABC$ is a right-angled triangle with $\angle ABC = 90^\circ$. Let $D$ and $E$ be points lying on $AC$ and $BC$ respectively such that $ABED$ is a cyclic quadrilateral. If $AB = 660 \, \text{cm}$, $AD = 572 \, \text{cm}$ and $BE = 275 \, \text{cm}$, then $CD =$

It is given that $PQR$S is a rhombus. Let $T$ be the point of intersection of $PR$ and $QS$. If $\angle QRT = \theta$, then $\frac{PQ}{ST} =$

The figure shows the graph of the straight line $mx + ny = 3$. Which of the following are true?

The rectangular coordinates of the point Q are $(4\sqrt{3}, -4)$. If Q is rotated clockwise about the origin through $90^{\circ}$, then the polar coordinates of its image are

The straight line $12x - 5y = 60$ cuts the x-axis and the y-axis at the points A and B respectively. Let P be a moving point in the rectangular coordinate plane such that $AP = BP$. Find the equation of the locus of P.

The coordinates of the points $P$ and $Q$ are $(10, -24)$ and $(17, -7)$ respectively. Let $C$ be the circle which passes through the origin, $P$ and $Q$. Which of the following is true?

$5\diamond 2$ is a 3-digit number, where $\diamond$ is an integer from 0 to 9 inclusive. Find the probability that the 3-digit number is divisible by 7.

The mean weight of 60 actors and 40 actresses is $57$ kg. If the mean weight of the actors is $63$ kg, then the mean weight of the actresses is

Consider the following positive integers:

If both the mean and the median of the above positive integers are 6, which of the following must be true?

I. The mode of the above positive integers is 6.

II. The least possible range of the above positive integers is 6.

III. The greatest possible variance of the above positive integers is 6.

Which of the following is the least?

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.