### 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 bowl 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 $\theta$

### 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.

### Challenging A’levels Questions

Below is a list of questions that students should take note of! The questions are not there due to copyright issues. But linked are the suggested solutions by KS. 🙂

2015 P1: 3, 9

2015 P2: 9, 11

2014 P1: 3, 7, 9

2014 P2: 1, 4, 6, 9

2013 P1: 5, 6

2013 P2: 1, 4, 8, 11

2012 P1: 5, 6, 7, 9, 11

2012 P2: 4, 6, 7, 10

2011 P1: 3, 4, 5, 6, 7

2011 P2: 11

2010 P1: None

2010 P2: 7, 8, 11

2009 P1: 2, 10 iii, 11 iv

2009 P2: 2 iii, 2 iv, 3, 7 ii, 8, 9

2008 P1: 1, 4 iii, 5 ii, 6a, 8, 9, 11

2008 P2: 1 iv, 2 ii, 3a, 3b ii, 7, 10, 11 last part

2007 P1: 7 iii, 9, 11

2007 P2: 8 iv, 9, 10

### Question of the Day #12

(a) $\frac{d}{dx} x^x$
(b) $\frac{d}{dx} \mathrm{cot}^{-1}x$

### Question of the Day #11

Given that $y = e^x \mathrm{ln}x$, find
(a) $\frac{dy}{dx}$
(b) Show that $\frac{d^2y}{dx^2}x + \frac{dy}{dx} -(1+x)y = 2e^x$

### Question of the Day #10

Given that $\mathrm{sin}^{-1}(x+y) = xy$, find $\frac{dy}{dx}$ in terms of x and y

### Question of the Day #9

Find the area bounded by the curve $x=y^3$ and the lines $x=0$, $y=2$ and $y=-1$

### Question of the Day #8

Find the area bounded by the curve $y^2 = 4a^2 (x-1)$ and the lines $x=1$ and $y=4a$