$\require{\cancel}$ $\require{\stix[upint]}$
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 | Total |
---|---|---|---|---|---|
Marks | 5 | 7 | 5 | 5 | 22 |
Score |
Question 1 Code: 9709/32/M/J/19/3, Topic: Trigonometry
Showing all necessary working, solve the equation $\cot 2 \theta=2 \tan \theta$ for $0^{\circ} < \theta < 180^{\circ}$. $[5]$
Question 2 Code: 9709/33/M/J/19/3, Topic: Trigonometry, Integration
Let $\displaystyle\mathrm{f}(\theta)=\frac{1-\cos 2 \theta+\sin 2 \theta}{1+\cos 2 \theta+\sin 2 \theta}$
$\text{(i)}$ Show that $\mathrm{f}(\theta)=\tan \theta$. $[3]$
$\text{(ii)}$ Hence show that $\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{4} \pi} \mathrm{f}(\theta) \, \mathrm{d} \theta=\frac{1}{2} \ln \frac{3}{2}$. $[4]$
Question 3 Code: 9709/31/O/N/19/3, Topic: Differentiation
The parametric equations of a curve are
$$ x=2 t+\sin 2 t, \quad y=\ln (1-\cos 2 t). $$Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\operatorname{cosec} 2 t$. $[5]$
Question 4 Code: 9709/32/O/N/19/3, Topic: Algebra
The polynomial $x^{4}+3 x^{3}+a x+b$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x).$ When $\mathrm{p}(x)$ is divided by $x^{2}+x-1$ the remainder is $2 x+3$. Find the values of $a$ and $b$. $[5]$