$\require{\cancel}$ $\require{\stix[upint]}$
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 | 6 | 8 | 7 | 7 | 8 | 43 |
Score |
Question 1 Code: 9709/31/M/J/13/5, Topic: Differentiation
For each of the following curves, find the gradient at the point where the curve crosses the $y$-axis:
$\text{(i)}$ $\displaystyle y=\frac{1+x^{2}}{1+\mathrm{e}^{2 x}}$; $[3]$
$\text{(ii)}$ $2 x^{3}+5 x y+y^{3}=8$. $[4]$
Question 2 Code: 9709/32/M/J/13/5, Topic: Differentiation
The diagram shows the curve with equation
$$ x^{3}+x y^{2}+a y^{2}-3 a x^{2}=0, $$where $a$ is a positive constant. The maximum point on the curve is $M$. Find the $x$-coordinate of $M$ in terms of $a$. $[6]$
Question 3 Code: 9709/33/M/J/13/5, Topic: Algebra
The polynomial $8 x^{3}+a x^{2}+b x+3$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(2 x+1)$ is a factor of $\mathrm{p}(x)$ and that when $\mathrm{p}(x)$ is divided by $(2 x-1)$ the remainder is 1.
$\text{(i)}$ Find the values of $a$ and $b$. $[5]$
$\text{(ii)}$ When $a$ and $b$ have these values, find the remainder when $\mathrm{p}(x)$ is divided by $2 x^{2}-1$. $[3]$
Question 4 Code: 9709/31/O/N/13/5, Topic: Trigonometry, Integration
$\text{(i)}$ Prove that $\cot \theta+\tan \theta \equiv 2 \operatorname{cosec} 2 \theta$. $[3]$
$\text{(ii)}$ Hence show that $\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \operatorname{cosec} 2 \theta \mathrm{d} \theta=\frac{1}{2} \ln 3.$ $[4]$
Question 5 Code: 9709/32/O/N/13/5, Topic: Trigonometry, Integration
Question 6 Code: 9709/33/O/N/13/5, Topic: Integration, Numerical solutions of equations
It is given that $\displaystyle\int_{0}^{p} 4 x \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=9$, where $p$ is a positive constant.
$\text{(i)}$ Show that $\displaystyle p=2 \ln \left(\frac{8 p+16}{7}\right)$. $[5]$
$\text{(ii)}$ Use an iterative process based on the equation in part $\text{(i)}$ to find the value of $p$ correct to 3 significant figures. Use a starting value of $3.5$ and give the result of each iteration to 5 significant figures. $[3]$