Past paper questions

The figure shows the graph of $y = b^{x}$ and the graph of $y = c^{x}$ on the same rectangular coordinate system, where $b$ and $c$ are positive constants. If a horizontal line $L$ cuts the y-axis, the graph of $y = b^{x}$ and the graph of $y = c^{x}$ at $A$, $B$ and $C$ respectively, which of the following are true?

I. $b<c$

II. $bc>1$

III. $\frac{AB}{AC}=\log_{b}c$

Which of the following is the greatest?

Let $f(x)=3x^{2}-6x+k$, where $k$ is a constant. If the $y$-coordinate of the vertex of the graph of $y=f(x)$ is $7$, then $k=$

If $\beta$ is a real number, then $\frac{\beta^{2}+4}{\beta+2i}=$

If $m>1$, which of the following are geometric sequences?

Which of the following may represent the graph of $y = f(x)$ and the graph of $y = 1 - f(x)$ on the same rectangular coordinate system?

For $0^{\circ} \le q x \le q 360^{\circ}$, how many roots does the equation $7\sin^{2}x=\sin x$ have?

In the figure, $AB$ is a vertical pole standing on the horizontal ground $BCD$, where $\angle CBD = 90^{\circ}$. If the angle between the plane $ACD$ and the horizontal ground is $\theta$, then $\tan \theta =$

In the figure, $PQS$ is a circle. $PQ$ is produced to $R$ such that $RS$ is the tangent to the circle at $S$. $I$ is the in-centre of $\Delta QRS$. If $\angle IRQ = 12^{\circ}$ and $\angle PSQ = 70^{\circ}$, then $\angle QPS =$

If the straight line $x - y = k$ and the circle $x^{2} + y^{2} + 2x - 4y - 1 = 0$ intersect at $A$ and $B$, then the x-coordinate of the mid-point of $AB$ is

There are 13 boys and 17 girls in a class. If a team of 2 boys and 3 girls is selected from the class to participate in a voluntary service, how many different teams can be formed?

In an examination, Peter gets 55 marks and his standard score is $-3$ while Mary gets 95 marks and her standard score is $2$. Find the mean of the examination scores.

If the variance of the four numbers $a$, $b$, $c$ and $d$ is 9, then the variance of the four numbers $14-a$, $14-b$, $14-c$ and $14-d$ is

If $p$ and $q$ are constants such that $px(x-1)+x^{2}=qx(x-2)+4x$, then $p=$

Let $a$ be a constant. If the quadratic equation $x^{2}+ax+a=1$ has equal roots, then $a=$

The figure shows the graph of $y = mx^{2} + x + n$, where $m$ and $n$ are constants. Which of the following is true?

If $a > b$ and $k < 0$, which of the following must be true?

The solution of $-3x<6<2x$ is

The price of 2 bowls and 3 cups is \506$. If the price of 5 bowls and the price of 4 cups are the same, then the price of a bowl is

There are 792 workers in a factory. If the number of male workers is $20\%$ less than that of female workers, then the number of male workers is

If the angle and the radius of a sector are decreased by $x\%$ and $50\%$ respectively so that its area is decreased by $90\%$, then $x=$

The width and the length of a thin rectangular metal sheet are measured as $8\text{ cm}$ and $10\text{ cm}$ correct to the nearest cm respectively. Let $x\text{ cm}^{2}$ be the actual area of the metal sheet. Find the range of values of $x$.

It is given that $\frac{4}{5a} = \frac{5}{7b} = \frac{7}{9c}$, where $a$, $b$ and $c$ are positive numbers. Which of the following is true?

If $z$ varies inversely as $x$ and directly as the cube of $y$, which of the following must be constant?

Let $a_n$ be the $n$th term of a sequence. If $a_2 = 7$, $a_4 = 63$ and $a_{n+2} = a_{n+1} + a_n$ for any positive integer $n$, then $a_5 =$

In the figure, $AB = AE$ and $\angle BAE = \angle BCD = \angle CDE = 90^{\circ}$. If $BC = CD = DE = 16\text{ cm}$, then the area of the pentagon $ABCDE$ is

In the figure, $ABCD$ is a square. $BC$ is produced to $G$ such that $\angle CDG = 25^\circ$. $E$ is a point lying on $AB$ such that $AE = CG$. If $F$ is a point lying on $BC$ such that $\angle CDF = 20^\circ$, then $\angle DFE =$

In the figure, $B$ is a point lying on $AC$ such that $AB:BC = 3:2$. It is given that $AE//BD$. If the area of $\triangle BCD$ and the area of $\triangle CDE$ are $4\text{ cm}^{2}$ and $8\text{ cm}^{2}$ respectively, then the area of the trapezium $ABDE$ is

In the figure, $\angle ABD = \angle ADC = \angle BCD = 90^\circ$. If $AB = \ell$, then $CD =$

In the figure, $AC$ is a diameter of the circle $ABCDE$. If $\angle ADE = 28^{\circ}$, then $\angle CBE =$

In the figure, $O$ is the centre of the circle $ABCDEF$. $\Delta PQR$ intersects the circle at $A$, $B$, $C$, $D$, $E$ and $F$. If $\angle QPR = 38^\circ$ and $AB = CD = EF$, then $\angle QOR =$

If an interior angle of a regular $n$-sided polygon is greater than an exterior angle by $100^{\circ}$, which of the following are true?

I. The value of $n$ is 10.

II. Each exterior angle of the polygon is $40^{\circ}$.

III. The number of axes of reflectional symmetry of the polygon is 9.

The rectangular coordinates of the point $P$ are $(-1, \sqrt{3})$. If $P$ is reflected with respect to the $x$-axis, then the polar coordinates of its image are

The equations of the straight lines $L_{1}$ and $L_{2}$ are $2x + 3y = 5$ and $4x + 6y = 7$ respectively. If $P$ is 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 $L_{2}$, then the locus of $P$ is a

In the figure, the two straight lines intersect at a point on the positive y-axis. Which of the following are true?

I. $a < 0$

II. $c > 0$

III. $b = d$

If a diameter of the circle $x^{2}+y^{2}-8x+ky-214=0$ passes through the point $(6,-5)$ and the slope of the diameter is $-4$, then $k=$

A box contains $m$ yellow balls and $20$ black balls. If a ball is randomly drawn from the box, then the probability of drawing a yellow ball is $\frac{1}{m}$. Find the value of $m$.

The mean height of 25 teachers and 140 students is $150\text{ cm }$. If the mean height of the students is $145\text{ cm }$, then the mean height of the teachers is

The pie chart below shows the expenditure of John in a certain week. John spends $240 on clothing that week. Find his expenditure on transportation that week.

The stem-and-leaf diagram below shows the distribution of the ages of the passengers in a bus.

$$\begin{array}{c|ccccc}{{{\underline{S t e m}\mathrm{(t e n s)}}}}&{{{\underline{L e a f}\mathrm{(u n i t s)}}}} \\{{{1}}}&{{{h}}}&{{{4}}}&{{{6}}} \\{{{2}}}&{{{3}}}&{{{3}}}&{{{3}}}&{{{4}}}&{{{6}}}&{{{7}}}&{{{7}}} \\{{{3}}}&{{{1}}}&{{{2}}}&{{{2}}}&{{{2}}}&{{{6}}}&{{{8}}} \\{{{4}}}&{{{0}}}&{{{k}}} \\ \end{array}$$

If the range of the above distribution is at least $33$, which of the following must be true?

I. $0 \le q h \le q 3$
II. $3 \le q k \le q 9$
III. $3 \le q k - h \le q 5$

The H.C.F. of $3x^{4}y^{2}z$, $4xy^{5}z$ and $6x^{2}y^{3}$ is

The figure shows the graph of $y = b^{x}$ and the graph of $y = c^{x}$ on the same rectangular coordinate system, where $b$ and $c$ are positive constants. If a horizontal line $L$ cuts the y-axis, the graph of $y = b^{x}$ and the graph of $y = c^{x}$ at $A$, $B$ and $C$ respectively, which of the following are true?

Which of the following is the greatest?

Let $f(x)=3x^{2}-6x+k$, where $k$ is a constant. If the y-coordinate of the vertex of the graph of $y=f(x)$ is 7, then $k=$

If $\beta$ is a real number, then $\frac{\beta^{2}+4}{\beta+2i}=$