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

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

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

\begin{matrix} δ x & = (0.5 + p) - 0.5 = p δ y & = ? \end{matrix}

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

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

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.

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$\begin{aligned} x &=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \\ &=\frac{4+2 \sqrt{2}+\sqrt{[-(4+2 \sqrt{2})]^{2}-4(5-2 \sqrt{2})(-2)}}{2(5-2 \sqrt{2})} \\ &=\frac{4+2 \sqrt{2}+\sqrt{16+16 \sqrt{2}+8+40-16 \sqrt{2}}}{2(5-2 \sqrt{2})} \\ &=\frac{4+2 \sqrt{2}+\sqrt{64}}{10-4 \sqrt{2}} \\ &=\frac{4+2 \sqrt{2}+8}{10-4 \sqrt{2}} \cdot \frac{10+4 \sqrt{2}}{10+4 \sqrt{2}} \\ &=\frac{120+48 \sqrt{2}+20 \sqrt{2}+16}{68} \\ &=\frac{136+68 \sqrt{2}}{68} \\ &=2+\sqrt{2} \end{aligned}$

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.

$\begin{aligned}{ }^{6} P_{2}=30\end{aligned}$

Question 1

Question 2

Question 4

Question 5

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