## A-level H2 Mathematics (9740) Suggested Solutions

After teaching H2 Mathematics for so many years and sharing my suggested solutions post A-levels, I thought I collate them nicely and type them all neatly here in Latex. 🙂 Please be kind to be should I make any careless mistakes as I typed them all out and spell check does not work with math equations. Feel free to comment.

Challenging A’levels Questions

MF15

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

## Random Questions from 2016 Prelims #8

ACJC P2 Q3

The function g is defined by

$g: x \mapsto \begin{cases} 2x, & \text{for }0 \le x \le \frac{1}{2} \\ 2-2x, & \text{for } \frac{1}{2} \le x \le 1 \end{cases}$

(ii) Explain why the composite function $gg$ exist.

(iii) Sketch the graph of $gg(x)$.

## Random Questions from 2016 Prelims #7

VJC P1 Q9

(i) Sketch the graph with equation $x^2 +(y-r)^2 = r^2$, where $r >0$ and $y le r$

A hemispherical bowl of fixed radius $r$ cm is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate. When the depth of water if $h$cm (where $h \le r$).

(ii) Use your graph in (i) to show that the volume of water in the bowl is given by $V = \frac{\pi h^2}{3} (3r-h)$.

(iii) Find the rate of decrease of the depth of water in the bowl, given that a full bow of water would become empty in 24 s,

(iv) without any differentiation, determine the slowest rate at which the depth of water is decreasing.

## Random Questions from 2016 Prelims #6

HCI P1 Q5

Sketch on a single Argand diagram, the loci defined by $-\frac{\pi}{4} \textless \text{arg}(z+1+2i) \le \frac{\pi}{4}$ and $|(2+i)w+5| \le \sqrt{5}$

(i) Find the minimum value of $\text{arg}(w)$

(ii) Find the minimum value of $|z-w|$

(iii) Given that $\text{arg}(z-w) \textless \theta, - \pi \textless \theta \le \pi$, state the minimum value of $\theta$

## ‘Religion divides rather than unites.’ Discuss. (RI 2016 Prelims)

I realized that I have not done a post about religion, so here is a basic outline for this essay. This question requires students to compare whether religion is more divisive or more unifying in nature.

The world’s major religions preach love, peace, tolerance, kindness, forgiveness and reconciliation. However in reality, religion often divides rather than unites when violent or harmful acts are committed in the name of God against humanity. In the history of the world thus far, we have examples of the Crusaders, as well as today’s religious extremists who have waged war and suicide bomb attacks on innocents, all in the name of religion and their twisted mission to ‘avenge’ or ‘bring glory to God’s name’. Recent examples include the Bali and London suicide bomb attacks by religious extremists. The division comes about in the aftermath of such attacks when people begin to place blame or be overly suspicious or hostile towards a certain religion or race. Again, differences are emphasised and the society is divided into an ‘us’ versus ‘them’ mentality, with unfounded prejudices, and immense anger and bitterness.

Religion has been shown to be able to unite rather than divide people during times of crisis, by providing a common denominator through which strangers can reach out and relate to one another. In the aftermath of 9/11, religious places of worship throughout America provided a place of solace, comfort and strength for believers seeking answers and support. Believers gathered together in mosques, temples, churches and synagogues for mass prayer and vigil sessions, which united communities and pulled the nation together as one as all shared in the immense grief and shock following the tragedy.

Religion can also unite rather than divide by playing an important role in society as moral compass as well as peacemaker. In times of societal conflict and tension, religious leaders and institutions are often called upon to help out as they are deemed to be highly respected and influential in the community. In the face of potentially volatile racial and religious tension and discrimination in Singapore following a string of extremist bombings in the region, an inter-faith dialogue was set up to promote better communication, understanding and respect amongst the nation’s major religions. This sought to ensure that the local community would not be vulnerable to false teachings/ prejudices, or liable to instigate violent retaliations against any group. The dialogue also sought to find common ground between the religions, thus uniting rather than dividing the nation’s citizens during an otherwise volatile time.

Ideally, religion should unite people by emphasising the similarities between each other, regardless of race or nationality – that we are all equally fallible and imperfect, and in need of a higher guidance. However in reality, religion can often create division in societies as it creates and emphasises differences in beliefs and opinions between people. This can result in conflict and tension especially when people begin to think and view things using an ‘us’ versus ‘them’ mentality, and start taking sides or criticising others’ way of doing things. For example, the recent AWARE saga generated nation-wide debate and controversy as people were outraged and concerned about Christian values and opinions being promoted in a secular organisation. The debate culminated in a dramatic and at times, bitter showdown between the two sides, where the new ex-co was given a vote of no confidence. Thus, religion can divide rather than unite when it is deemed to be threatening the harmony within a society.

## Random Questions from 2016 Prelims #5

NYJC P1 Q4

Referred to the origin , the points A and B have position vectors a and b respectively. A point C is such that OACB forms a parallelogram. Given that M is the mid-point of AC, find the position vector of point N if M lies on ON produced such that OM:ON is in ratio 3:2. Hence show that A, B, and N are collinear.

Point P is on AB such that MP is perpendicular to AB. Given that angle AOB is $60^{\circ}, |a|=2 \text{~and~} |b|=3$, find the position vector of P in terms of a and b.

## Random Questions from 2016 Prelims #4

VJC P1 Q6

A curve has equation $y^2=4x$ and a line $l$ has equation $2x-y+1=0$ as shown above.

$B(b, 2\sqrt{b})$ is a fixed point on C and A is an arbitrary point on $l$. State the geometrical relationship between the line segment AB and $l$ is the distance from B to A is the least.

Taking the coordinates of A as $(a, 2a+1)$, find an equation relating $a$ and $b$ for which AB is the least.

Deduce that when AB is the least, $(AB)^2 = m (2b - 2\sqrt{b} +1)^2$ where $m$ is a constant to be found. Hence or otherwise, find the coordinates of the point on C that is nearest on $l$.

## Random Questions from 2016 Prelims #3

AJC P1 Q7

The function f is defined by $f:x \mapsto e^{|4-x|}, x \in \mathbb{R}, x \le a$.

(i) State the greatest value of $a$ for which $f^{-1}$ exists.

(ii) Define $f^{-1}$ in similar form.