Past paper questions

It is given that $f(x)$ is the sum of two parts, one part varies as $x$ and the other part varies as $x^{2}$. Suppose that $f(3)=48$ and $f(9)=198$.

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

[Table]

[Table]

The coordinates of the points $A$ and $B$ are $(5,7)$ and $(13,1)$ respectively. Let $P$ be a moving point in the rectangular coordinate plane such that $P$ is equidistant from $A$ and $B$. Denote the locus of $P$ by $\Gamma$.

An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is $12\text{ cm}$. Peter then pours $444\pi\text{ cm}^{3}$ of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm}$.

The bar chart below shows the distribution of the ages of the children in a group, where $a>11$ and $4<b<10$. The median of the ages of the children in the group is $7.5$.

Distribution of the ages of the children in the group

In Figure 1, $ABC$ is a triangle. $D$, $E$ and $M$ are points lying on $BC$ such that $BD=CE$, $\angle ADC=\angle AEB$ and $DM=EM$.

Let $p(x)=6x^{4}+7x^{3}+ax^{2}+bx+c$, where $a$, $b$ and $c$ are constants. When $p(x)$ is divided by $x+2$ and when $p(x)$ is divided by $x-2$, the two remainders are equal. It is given that $p(x)=(lx^{2}+5x+8)(2x^{2}+mx+n)$, where $l$, $m$ and $n$ are constants.

If $4$ boys and $5$ girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)

In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of Albert and Mary are $-2.6$ and $1.4$ respectively. Albert gets 22 marks. A student claims that the range of the distribution is at most $59$ marks. Is the claim correct? Explain your answer. (3 marks)

The 1st term and the 38th term of an arithmetic sequence are $666$ and $555$ respectively. Find

Let $f(x)=\frac{-1}{3}x^{2}+12x-121$.

Figure 2 shows a geometric model $ABCD$ in the form of tetrahedron. It is given that $\angle BAD = 86^{\circ}$, $\angle CBD = 43^{\circ}$, $AB = 10$ cm, $AC = 6$ cm, $BC = 8$ cm and $BD = 15$ cm.

$\Delta OPQ$ is an obtuse-angled triangle. Denote the in-centre and the circumcentre of $\Delta OPQ$ by $I$ and $J$ respectively. It is given that $P$, $I$ and $J$ are collinear.

If $\frac{a}{x} \div \frac{b}{y} = 3$, then $x =$

If $4\alpha + \beta = 7\alpha + 3\beta = 5$, then $\beta =$

Let $f(x)=4x^{3}+kx+3$, where $k$ is a constant. If $f(x)$ is divisible by $2x+1$, find the remainder when $f(x)$ is divided by $x+1$.

The solution of $-5x > 21 - 2x$ and $6x - 18 < 0$ is

If $k$ is a constant such that the quadratic equation $x^{2}+kx+8k+36=0$ has equal roots, then $k=$

If $-1 < a < 0$, which of the following may represent the graph of $y = (ax + 1)^2 + a$?

The monthly salary of Donald is $25\%$ higher than that of Peter while the monthly salary of Peter is $25\%$ lower than that of Teresa. It is given that the monthly salary of Donald is \33\,360$. The monthly salary of Teresa is

If $x$ and $y$ are non-zero numbers such that $(3y-4x):(2x+y)=5:6$, then $x:y=$

It is given that $z$ varies directly as $\sqrt{x}$ and inversely as $y$. If $x$ is decreased by $36\%$ and $y$ is increased by $60\%$, then $z$

The cost of flour of brand X is \42/kg$. If$3$kg of flour of brand X and$2$kg of flour of brand Y are mixed so that the cost of the mixture is$\$36/kg$, find the cost of flour of brand Y.

In the figure, the 1st pattern consists of $9$ dots. For any positive integer $n$, the $(n+1)$th pattern is formed by adding $5$ dots to the $n$th pattern. Find the number of dots in the 7th pattern.

According to the figure, which of the following must be true?

In the figure, $ABC$ is a straight line. If $AB = 24\text{ cm}$, $AD = 40\text{ cm}$, $BD = 32\text{ cm}$ and $CD = 68\text{ cm}$, then $BC =$

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $CD$ such that $BE = CE$. If $\angle ADC = 114^\circ$, then $\angle ABE =$

The figure shows a right prism. Find the volume of the prism.

In the figure, $OAB$ and $OCD$ are sectors with centre $O$, where $OA=33$ cm and $OC=39$ cm. The area of the shaded region $ABDC$ is $72\pi\text{ cm}^{2}$. Which of the following is/are true?

I. The angle of the sector $OAB$ is $60^{\circ}$.
II. The area of the sector $OAB$ is $11\pi\text{ cm}^{2}$.
III. The perimeter of the sector $OCD$ is $13\pi\text{ cm}$.

In the figure, $ABCD$, $CDEF$ and $EFGH$ are squares. $AG$ cuts $CD$ and $EF$ at $P$ and $Q$ respectively. Find the ratio of the area of quadrilateral $DEQP$ to the area of quadrilateral $ABCP$.

In the figure, $AD =$

In the figure, $ABCD$ is a rhombus. $C$ is the centre of the circle $BDE$ and $ADE$ is a straight line. $BE$ and $CD$ intersect at $F$. If $\angle ADC = 118^{\circ}$, then $\angle DFE =$

The figure below consists of eight regular hexagons. The number of axes of reflectional symmetry of the figure is

If the sum of the interior angles of a regular $n$-sided polygon is $3240^{\circ}$, which of the following is true?

If the straight lines $hx + ky + 15 = 0$ and $4x + 3y - 5 = 0$ are perpendicular to each other and intersect at a point on the $x$-axis, then $k =$

The coordinates of the points $A$ and $B$ are $(9,-2)$ and $(-1,8)$ respectively. If $C$ is a point lying on the straight line $x-2y=0$ such that $AC=BC$, then the $x$-coordinate of $C$ is

The equation of the circle $C$ is $3x^{2}+3y^{2}-12x+30y+65=0$. Which of the following are true?

Christine has one $1$ coin, one $2$ coin, one $5$ coin and one $10$ coin in her pocket. If Christine takes out three coins randomly from her pocket, find the probability that she gets at least $13$.

A bag contains $1$ red ball, $3$ yellow balls and $6$ white balls. In a lucky draw, a ball is randomly drawn from the bag and a certain number of tokens will be got according to the following table:

[Table]

the expected number of tokens got in the lucky draw.

Consider the following data:

$32$ $68$ $79$ $86$ $88$ $98$ $98$ $a$ $b$ $c$

If the mean and the mode of the above data are $77$ and $68$ respectively, then the median of the above data is

The L.C.M. of $9a^{2}b$, $12a^{4}b^{3}$ and $15a^{6}$ is

The graph in the figure shows the linear relation between $x$ and $\log_{9}y$. If $y=ab^{x}$, then $b=$

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 $\frac{1}{\nu}$.

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?

[Figure 1]

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