$\require{\cancel}$ $\require{\stix[upint]}$
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 |
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 $0^{\circ}< \theta <360^{\circ}$. $[4]$