Past paper questions

Factorize

Let $a$, $b$ and $c$ be non-zero numbers such that $5a = 6c$ and $\frac{2b + 7c}{b + c} = 4$. Find $\frac{5a + 8b}{2b + 3c}$.

The marked price of a calculator is $40\%$ higher than its cost. The calculator is sold at a discount of $25\%$ on its marked price and the profit is \13$. Find the marked price of the calculator.

In Figure 1, $E$ is the point of intersection of $AC$ and $BD$. It is given that $\angle ACB = \angle ADB = 90^{\circ}$ and $AD = BC$.

The table below shows the distribution of the numbers of keys owned by a group of housewives.

[Table]

If a housewife is randomly selected from the group, then the probability that she owns more than 6 keys is $\frac{5}{18}$.

The stem-and-leaf diagram below shows the distribution of the numbers of hours spent on reading journals in a month by a group of researchers.

\begin{array}{c|cccccccccc}{{{\underline{S t e m}(t e n s)}}}&{{{\underline{L e a f}(u n i t s)}}} \\{{{\hline2}}}&{{{0}}}&{{{0}}}&{{{1}}}&{{{a}}}&{{{a}}}&{{{a}}}&{{{8}}}&{{{8}}}&{{{9}}}&{{{9}}} \\{{{3}}}&{{{0}}}&{{{0}}}&{{{2}}}&{{{3}}}&{{{4}}}&{{{4}}}&{{{7}}}&{{{9}}} \\{{{4}}}&{{{0}}}&{{{b}}} \\\end{array}}

The mean of the distribution is $30$.

Let $\mathrm{F}(x) = (6x^{2} + x + p)(qx^{2} + rx - 10)$, where $p$, $q$ and $r$ are constants. The constant term of $\mathrm{F}(x)$ is $40$.

It is given that $\log_9 y$ is a linear function of $\log_3 x$. Denote the graph of the linear function by $L$. The slope of $L$ is $4$ and $L$ passes through the point $(5, 22)$.

In a bag, there are 16 red cups and 4 white cups. If 5 cups are randomly drawn from the bag at the same time, find

The coordinates of the points $Q$ and $R$ are $(10,-1)$ and $(-4,-9)$ respectively.

If $k = \frac{5}{2m} + n$, then $m =$

The price of $2$ apples and $3$ lemons is \38$. If the price of$3$apples and$2$lemons is$\$47$, then the price of $4$ apples and $7$ lemons is

If $a$, $b$ and $c$ are non-zero constants such that $4x^{2}+2ax+3a \equiv x(4x+b)+2c$, then $a:b:c=$

Let $m$ be a constant. Solve the equation $x^{2}-3x=(m-1)^{2}-3(m-1)$.

Let $g(x)=(x+1)(x+a)$, where $a$ is a constant. If $g(1)=g(2)$, then $g(a)=$

Let $f(x)=x^{3}+kx^{2}+5x+10$, where $k$ is a constant. If $f(x)$ is divisible by $x+k$, find the remainder when $f(x)$ is divided by $x+1$.

The solution of $\frac{1-x}{2} \geq 4$ or $7+5x \leq -3$ is

In a school, $40\%$ of the students are girls and $\beta\%$ of the girls are foreign students. It is given that $30\%$ of the boys in the school are foreign students. In the school, the number of foreign students and the number of girls are equal. Find $\beta$.

A car travels at an average speed of $60$ km/h for $18$ minutes and then the car travels at an average speed of $40$ km/h for $27$ minutes. The average speed of the car for the whole journey is

It is given that $z$ varies directly as the square of $x$ and inversely as $y$. If $x$ is increased by $20\%$ and $y$ is decreased by $20\%$, then $z$

Which of the following statements about the graph of $y=2(6-x)^{2}-7$ is true?

If the arc length and the area of a sector are $8\pi$ cm and $80\pi$ cm$^2$ respectively, then the angle of the sector is

The ratio of the height of a right circular cylinder to the height of a right circular cone is $32:15$ while the ratio of the volume of the circular cylinder to the volume of the circular cone is $10:9$. If the base radius of the circular cylinder is $25\text{ cm}$, then the base radius of the circular cone is

In the figure, $ABCD$ is a square. Let $M$ be the mid-point of $BC$. $E$ is a point lying on $AD$ such that $AE:ED=3:1$. $F$ is a point lying on $BC$ produced such that $EF$ // $AM$. $CD$ and $EF$ intersect at the point $G$ while $AM$ and $BG$ intersect at the point $H$. If the area of $\triangle BMH$ is $4\text{ cm}^{2}$, then the area of the trapezium $AEGH$ is

In the figure, $ABC$ is a straight line. It is given that $AD=37\text{ cm}$, $BC=5\text{ cm}$, $BD=12\text{ cm}$, $CD=13\text{ cm}$ and $CE=9\text{ cm}$. If $\angle ACE=90^{\circ}$, find the perimeter of the quadrilateral $ADCE$.

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

If the sum of the interior angles of a regular polygon is $900^{\circ}$, which of the following is/are true?

I. The number of diagonals of the polygon is 7.

II. The number of folds of rotational symmetry of the polygon is 7.

III. The number of axes of reflectional symmetry of the polygon is 7.

In the figure, $ABCD$ is a rhombus. Denote the point of intersection of $AC$ and $BD$ by $E$. Let $F$ be a point such that $BH//EF$ and $CFHG$ is a rectangle, where $G$ and $H$ are points lying on $AC$ produced and $BC$ produced respectively. Denote the point of intersection of $CD$ and $EF$ by $I$. Which of the following must be true?

I. $CI = FI$

II. $\angle ABE = \angle GCH$

III. $\triangle ADE \cong \triangle HCF$

In the figure, $ABCDE$ is a circle. $AC$ and $BE$ are diameters of the circle. Let $P$ be the point of intersection of $AC$ and $BD$. If $\angle ABE = 46^{\circ}$ and $\angle DBE = 16^{\circ}$, then $\angle APD =$

In the figure, $ABC$ is a straight line. $\frac{BC}{AD}=$

The coordinates of the point $U$ are $(-3, -8)$. $U$ is rotated anticlockwise about the origin through $90^{\circ}$ to the point $V$. $V$ is then reflected with respect to the straight line $x=2$ to the point $W$. Find the $x$-coordinate of $W$.

The coordinates of the points $A$ and $B$ are $(-3,1)$ and $(-7,-5)$ respectively. If $P$ is a point lying on the straight line $x-y+13=0$ such that $AP=PB$, then the $y$-coordinate of $P$ is

Find the constant $k$ such that the straight lines $6x-8y=7k$ and $kx+12y=5$ do not intersect with each other.

Denote the circle $3x^{2}+3y^{2}-6x+12y-4=0$ by $C$. Which of the following are true?

I. The origin lies inside $C$.

II. The circumference of $C$ is less than $16$.

III. The perpendicular distance from the centre of $C$ to the $x$-axis is $2$.

Two numbers are randomly drawn at the same time from six cards numbered $1$, $2$, $3$, $4$, $5$ and $6$ respectively. Find the probability that the product of the numbers drawn is not less than $12$.

The box-and-whisker diagram below shows the distribution of the numbers of tokens got by a group of children in a game. If the range of the distribution is the triple of its inter-quartile range, find $m$.

Consider the following positive integers:

Let $p$, $q$ and $r$ be the standard deviation, the mode and the median of the above positive integers respectively. If the mean of the above positive integers is $7$, which of the following must be true?

I. $p>7$

II. $q=5$

III. $r<7$

The H.C.F. of $u^{2}v^{3}w$, $u^{3}vw^{2}$ and $u^{2}v^{3}w^{4}$ is