Past paper questions

In a recreation club, there are 180 members and the number of male members is $40\%$ more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)

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.

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 $444\pi$ cm^{3} of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm }$.

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.

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

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.

Simplify $\frac{(xy)^2}{x^{-5}y^6}$ and express your answer with positive indices.

Make $b$ the subject of the formula $a(b+7)=a+b$

Factorize

The marked price of a handbag is \560$. It is given that the marked price of the handbag is$40\%$ higher than the cost.

In a football league, each team gains 3 points for a win, 1 point for a draw and 0 point for a loss. The champion of the league plays 36 games and gains a total of 84 points. Given that the champion does not lose any games, find the number of games that the champion wins.

In Figure 2, $O$ is the centre of the semicircle $ABCD$. If $AB \parallel OC$ and $\angle BAD = 38^{\circ}$, find $\angle BDC$.

In Figure 3, the coordinates of the point $A$ are $(-2,5)$. $A$ is rotated clockwise about the origin $O$ through $90^{\circ}$ to $A'$. $A''$ is the reflection image of $A$ with respect to the $y$-axis.

In a factory, the production cost of a carpet of perimeter $s$ metres is $C$. It is given that $C$ is a sum of two parts, one part varies as $s$ and the other part varies as the square of $s$. When $s=2$, $C=356$; when $s=5$, $C=1250$.

In Figure 6, the straight line $L_{1}$: $4x-3y+12=0$ and the straight line $L_{2}$ are perpendicular to each other and intersect at $A$. It is given that $L_{1}$ cuts the $y$-axis at $B$ and $L_{2}$ passes through the point $(4,9)$.

The data below show the percentages of customers who bought newspaper A from a magazine stall in city H for five days randomly selected in a certain week:

$62\%$ $63\%$ $55\%$ $62\%$ $58\%$

(a) Find the median and the mean of the above data. (2 marks)

(b) Let $a\%$ and $b\%$ be the percentages of customers who bought newspaper A from the stall for the other two days in that week. The two percentages are combined with the above data to form a set of seven data.

(i) Write down the least possible value of the median of the combined set of seven data.

(ii) It is known that the median and the mean of the combined set of seven data are the same as that found in (a). Write down one pair of possible values of $a$ and $b$.

(c) The stall-keeper claims that since the median and the mean found in (a) exceed $50\%$, newspaper A has the largest market share among the newspapers in city H. Do you agree? Explain your answer. (2 marks)

A committee consists of 5 teachers from school A and 4 teachers from school B. Four teachers are randomly selected from the committee.

A researcher defined Scale A and Scale B to represent the magnitude of an explosion as shown in the following table:

In Figure 8(a), $ABC$ is a triangular paper card. $D$ is a point lying on $AB$ such that $CD$ is perpendicular to $AB$. It is given that $AC=20$ cm, $\angle CAD=45^{\circ}$ and $\angle CBD=30^{\circ}$.

In Figure 9, the circle passes through four points A, B, C and D. PQ is the tangent to the circle at C and is parallel to BD. AC and BD intersect at E. It is given that AB = AD.