Cambridge Additional Mathematics 2011 Past Paper Oct Nov Paper 23

Question 1

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

[3]

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Question 2

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

\begin{aligned} y &=(12-4 x)^{5} \\ \frac{d y}{d x} &=5(12-4 x)^{4}(-4) \\ &=-20(12-4 x)^{4} \end{aligned}

(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} \delta x &=(0.5+p)-0.5 \\ &=p \\ \delta y &=? \end{aligned}

\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,

\begin{array}{l}
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.

\begin{aligned}
& 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}

(iii) Calculate the number of different ways in which a chairperson and a secretary can be selected.
\begin{aligned}{ }^{6} P_{2}=30\end{aligned}