## Question 1

Solve the inequality \(x(2 x-1)>15\)

[3]

## Question 2

(i) Given that \(y=(12-4 x)^{5},\) find \(\frac{ d y}{ d x}\).

(ii) Hence find the approximate change in \(y\) as \(x\) increases from 0.5 to \(0.5+p,\) where \(p\) is small. [2]

\(\begin{aligned} \frac{\delta y}{\delta x} & \approx \frac{d y}{d x} \\ \delta y &=\delta x \times \frac{d y}{d x} \\ & \approx \delta x \times\left[-20(12-4 x)^{4}\right] \\ & \approx p \times\left[-20(12-4(0.5)]^{4}\right.\\ & \approx p \times[-200000] \\ &=-200000 p \end{aligned}\)

## Question 4

Without using a calculator, find the positive root of the equation

\[

(5-2 \sqrt{2}) x^{2}-(4+2 \sqrt{2}) x-2=0

\]

giving your answer in the form \(a+b \sqrt{2},\) where \(a\) and \(b\) are integers.

[6]

\begin{aligned}

x &=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \\

&=\frac{4+2 \sqrt{2}+\sqrt{[-(4+2 \sqrt{2})]^{2}-4(5-2 \sqrt{2})(-2)}}{2(5-2 \sqrt{2})} \\

&=\frac{4+2 \sqrt{2}+\sqrt{16+16 \sqrt{2}+8+40-16 \sqrt{2}}}{2(5-2 \sqrt{2})} \\

&=\frac{4+2 \sqrt{2}+\sqrt{64}}{10-4 \sqrt{2}} \\

&=\frac{4+2 \sqrt{2}+8}{10-4 \sqrt{2}} \cdot \frac{10+4 \sqrt{2}}{10+4 \sqrt{2}} \\

&=\frac{120+48 \sqrt{2}+20 \sqrt{2}+16}{68} \\

&=\frac{136+68 \sqrt{2}}{68} \\

&=2+\sqrt{2}

\end{aligned}

## Question 5

A school council of 6 people is to be chosen from a group of 8 students and 6 teachers. Calculate the number of different ways that the council can be selected if

(i) there are no restrictions,

14 \text { choose } 6 \\

=14 C _{6} \\

=3003

\end{array}

(ii) there must be at least 1 teacher on the council and more students than teachers. After the council is chosen, a chairperson and a secretary have to be selected from the 6 council members.

& 1 T 5 S+2 T 4 S \\

=&{ }^{6} C_{1} \times{ }^{8} C_{5}+{ }^{6} C_{2} \times{ }^{2} C_{4} \\

=& 336+1050 \\

=& 1386

\end{aligned}