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Cambridge International AS and A Level

Name of student HENRYTAIGO Date
Adm. number Year/grade HenryTaigo Stream HenryTaigo
Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33
Start time Duration Stop time

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

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
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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 $0^{\circ}< \theta <360^{\circ}$. $[4]$

Worked solutions: P1, P3 & P6 (S1)

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