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Qtn No.	1	2	3	4	5	6	Total
Marks	7	5	5	6	6	7	36
Score

Question 1 Code: 9709/31/M/J/11/3, Topic: Vectors

Points $A$ and $B$ have coordinates $(-1,2,5)$ and $(2,-2,11)$ respectively. The plane $p$ passes through $B$ and is perpendicular to $A B$.

$\text{(i)}$ Find an equation of $p$, giving your answer in the form $a x+b y+c z=d$. $[3]$

$\text{(ii)}$ Find the acute angle between $p$ and the $y$-axis. $[4]$

Question 2 Code: 9709/32/M/J/11/3, Topic: Trigonometry

Solve the equation $$ \cos \theta+4 \cos 2 \theta=3 $$

giving all solutions in the interval $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[5]$

Question 3 Code: 9709/33/M/J/11/3, Topic: Integration

Show that $\displaystyle\int_{0}^{1}(1-x) \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=4 \mathrm{e}^{-\frac{1}{2}}-2$. $[5]$

Question 4 Code: 9709/31/O/N/11/3, Topic: Algebra

The polynomial $x^{4}+3 x^{3}+a x+3$ is denoted by $\mathrm{p}(x)$. It is given that $\mathrm{p}(x)$ is divisible by $x^{2}-x+1$.

$\text{(i)}$ Find the value of $a$. $[4]$

$\text{(ii)}$ When $a$ has this value, find the real roots of the equation $\mathrm{p}(x)=0$. $[2]$

Question 5 Code: 9709/32/O/N/11/3, Topic: Algebra

Question 6 Code: 9709/33/O/N/11/3, Topic: Trigonometry

$\text{(i)}$ Express $8 \cos \theta+15 \sin \theta$ in the form $R \cos (\theta-\alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$. Give the value of $\alpha$ correct to 2 decimal places. $[3]$

$\text{(ii)}$ Hence solve the equation $8 \cos \theta+15 \sin \theta=12$, giving all solutions in the interval $0^{\circ}< \theta <360^{\circ}$. $[4]$