Past paper questions

A box contains $n$ white balls, $5$ black balls and $8$ red balls. If a ball is randomly drawn from the box, then the probability of drawing a red ball is $\frac{2}{5}$. Find the value of $n$. (3 marks)

Factorize

The marked price of a vase is $30\%$ above its cost. A loss of \88$is made by selling the vase at a discount of$40\%$ on its marked price. Find the marked price of the vase. (5 marks)

In Figure 1, $ABCDE$ is a circle. It is given that $AB//ED$. $AD$ and $BE$ intersect at the point $F$.

A car travels from city $P$ to city $Q$ at an average speed of $72\text{ km/h}$ and then the car travels from city $Q$ to city $R$ at an average speed of $90\text{ km/h}$. It is given that the car travels $210\text{ km}$ in $161$ minutes for the whole journey. How long does the car take to travel from city $P$ to city $Q$? (5 marks)

The box-and-whisker diagram below shows the distribution of the ages of the clerks in team X of a company. It is given that the range and the inter-quartile range of this distribution are $43$ and $21$ respectively.

The following table shows the distribution of the numbers of children of some families:

[Table]

It is given that $k$ is a positive integer.

Let $f(x)=4x(x+1)^{2}+ax+b$, where $a$ and $b$ are constants. It is given that $x-3$ is a factor of $f(x)$. When $f(x)$ is divided by $x+2$, the remainder is $2b+165$.

In Figure 2, $ABCD$ is a trapezium with $\angle ABC = 90^\circ$ and $AB \parallel DC$. $E$ is a point lying on $BC$ such that $\angle AED = 90^\circ$.

A right circular cylindrical container of base radius $8\text{ cm}$ and height $64\text{ cm}$ and an inverted right circular conical vessel of base radius $20\text{ cm}$ and height $60\text{ cm}$ are held vertically. The container is fully filled with water. The water in the container is now poured into the vessel.

The coordinates of the points $A$, $B$ and $C$ are $(4, -8)$, $(-1, -2)$ and $(-2, 2)$ respectively. $D$ is a point lying on the line segment $BC$ such that $AD$ is perpendicular to $BC$. Let $E$ be the intersection point of $AD$ and the $y$-axis. Suppose that $F$ is a point lying on the line segment $AB$ such that the area of $\Delta AEF$ is $\frac{1}{3}$ of the area of $\Delta ABD$.

An eight-digit phone number is formed by a permutation of 2, 3, 4, 5, 6, 7, 8 and 9.

The 3rd term and the 4th term of a geometric sequence are 720 and 864 respectively.

It is given that $f(x)$ partly varies as $x^{2}$ and partly varies as $x$. Suppose that $f(2)=60$ and $f(3)=99$.

If $\frac{\alpha}{1-x} = \frac{\beta}{x}$, then $x =$

Which of the following statements about the graph of $y=16-(x-6)^{2}$ is true?

In the figure, the equations of the straight lines $L_{1}$ and $L_{2}$ are $3x + ay = b$ and $cx + y = d$ respectively. Which of the following is/are true?

If $f(x)=3x^{2}-2x+1$, then $f(2m-1)=$

Let $g(x)=x^{8}+ax^{7}+b$, where $a$ and $b$ are constants. If $g(x)$ is divisible by $x-1$, find the remainder when $g(x)$ is divided by $x+1$.

A sum of \100\,000$is deposited at an interest rate of$2\%$per annum for$3$ years, compounded monthly. Find the interest correct to the nearest dollar.

Let $a$, $b$ and $c$ be non-zero numbers. If $3a = 4b$ and $a:c = 2:5$, then $\frac{a+3b}{b+3c} =$

If $w$ varies directly as the square root of $u$ and inversely as the square of $v$, which of the following must be constant?

Let $a_n$ be the $n$th term of a sequence. If $a_3 = 21$, $a_6 = 89$ and $a_{n+2} = a_n + a_{n+1}$ for any positive integer $n$, then $a_1 =$

The solution of $\frac{1-2x}{3} \ge q x-3$ or $4x+9 < 1$ is

In the figure, ABCDEFGH is an octagon, where all the measurements are correct to the nearest cm. Let $x \, cm^2$ be the actual area of the octagon. Find the range of values of x.

In the figure, the volume of the solid right triangular prism is

In the figure, $ABCD$ is a parallelogram. $E$ is a point lying on $BC$ such that $BE:EC = 5:3$. $AE$ and $BD$ intersect at the point $F$. If the area of $\triangle ABF$ is $120\ cm^{2}$, then the area of the quadrilateral $CDFE$ is

In the figure, $O$ is the centre of the sector $OABCD$. $AD$ and $OC$ are perpendicular to each other and intersect at the point $E$. $F$ is a point lying on $AD$ such that $BF$ is perpendicular to $AD$. If $AF = 9$ cm, $DF = 39$ cm and $OE = 18$ cm, then the area of the sector $OBC$ is

In the figure, $ABCD$ is a rhombus. $E$ and $F$ are points lying on $AB$ and $AD$ respectively such that $AE = AF$ and $\angle ECF = 42^{\circ}$. If $\angle BAD = 110^{\circ}$, then $\angle BEC =$

In the figure, $ABCDE$ is a regular pentagon. $AD$ and $CE$ intersect at the point $F$. Which of the following are true?
I. $CD = CF$
II. $\Delta ABF \cong \Delta CBF$
III. $\angle AFB + \angle EAF = 90^{\circ}$

In the figure, $ABCD$ is a square. $E$ is a point lying on $AB$ produced such that $BE = 4$ cm. $BC$ and $DE$ intersect at the point $F$. If $EF = 5$ cm, then $DF =$

In the figure, $ABCD$ is a trapezium with $\angle ABC = \angle BAD = 90^{\circ}$. $E$ and $F$ are points lying on $AB$ such that $E$ and $F$ divide $AB$ into three equal parts. Which of the following must be true?

I. $AF\sin\alpha=BE\sin\beta$

II. $CE\cos\alpha=DF\cos\beta$

III. $AD\tan\alpha=BC\tan\beta$

In the figure, $ABCD$ is a circle. $AD$ produced and $BC$ produced meet at the point $E$. It is given that $BD=DE$, $\angle BAC=66^{\circ}$ and $\angle ABD=30^{\circ}$. Find $\angle CED$.

The figure below consists of eight identical squares. The number of folds of rotational symmetry of the figure is

The polar coordinates of the points $C$, $D$ and $E$ are $(16, 127^{\circ})$, $(12, 217^{\circ})$ and $(5, 307^{\circ})$ respectively. Find the perimeter of $\Delta CDE$.

The equations of the straight lines $L_{1}$ and $L_{2}$ are $3x - y + 7 = 0$ and $12x - 4y - 11 = 0$ respectively. Let $P$ be a moving point in the rectangular coordinate plane such that the perpendicular distance from $P$ to $L_{1}$ is equal to the perpendicular distance from $P$ to $L_{2}$. Find the equation of the locus of $P$.

The equation of the straight line $L_{1}$ is $4x + 3y - 36 = 0$. The straight line $L_{2}$ is perpendicular to $L_{1}$ and intersects $L_{1}$ at a point lying on the $y$-axis. Find the area of the region bounded by $L_{1}$, $L_{2}$ and the $x$-axis.

The equation of the circle $C$ is $5x^{2}+5y^{2}-30x+10y+6=0$. Which of the following is true?

Two numbers are randomly drawn at the same time from seven cards numbered 1, 1, 1, 2, 2, 3 and 4 respectively. Find the probability that the sum of the numbers drawn is 5.

The mean of the numbers of pages of 10 magazines is 132. If the mean of the numbers of pages of 6 of these 10 magazines is 108, then the mean of the numbers of pages of the remaining 4 magazines is

The stem-and-leaf diagram below shows the distribution of the numbers of books read by 20 students in a year.

$$\begin{array}{c|ccccc}{{\underline{S t e m}\textsc{(t e n s)}}}&{{\underline{L e a f}\textsc{(units)}}}&{{{8}}} \\{{{3}}}&{{{a}}}&{{{a}}} \\{{{4}}}&{{{0}}}&{{{2}}}&{{{4}}}&{{{5}}}&{{{5}}}&{{{7}}}&{{{8}}} \\{{{5}}}&{{{3}}} \\{{{6}}}&{{{b}}}&{{{b}}}&{{{9}}}&{{{9}}} \\{{{7}}}&{{{0}}}&{{{8}}} \\\end{array}$$

If the inter-quartile range of the above distribution is at most $25$, which of the following must be true

I. $5 \le q a \le q 9$
II. $0 \le q b \le q 4$
III. $1 \le q a - b \le q 6$