$\require{\cancel}$ $\require{\stix[upint]}$
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 | 5 | 5 | 5 | 5 | 5 | 5 | 30 |
Score |
Question 1 Code: 9709/31/M/J/14/3, Topic: Differentiation
The parametric equations of a curve are
$$ x=\ln (2 t+3), \quad y=\frac{3 t+2}{2 t+3} . $$Find the gradient of the curve at the point where it crosses the $y$-axis. $[6]$
Question 2 Code: 9709/32/M/J/14/3, Topic: Trigonometry
Solve the equation $$ \cos \left(x+30^{\circ}\right)=2 \cos x $$giving all solutions in the interval $-180^{\circ} < x < 180^{\circ}$. $[5]$
Question 3 Code: 9709/33/M/J/14/3, Topic: Trigonometry
$\text{(i)}$ Show that the equation
$$ \tan \left(x-60^{\circ}\right)+\cot x=\sqrt{3} $$can be written in the form $[3]$
$$ 2 \tan ^{2} x+(\sqrt{3}) \tan x-1=0 $$$\text{(ii)}$ Hence solve the equation
$$ \tan \left(x-60^{\circ}\right)+\cot x=\sqrt{3}, $$for $0^{\circ} < x < 180^{\circ}$. $[3]$
Question 4 Code: 9709/31/O/N/14/3, Topic: Algebra
The polynomial $a x^{3}+b x^{2}+x+3$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(3 x+1)$ is a factor of $\mathrm{p}(x)$, and that when $\mathrm{p}(x)$ is divided by $(x-2)$ the remainder is 21. Find the values of $a$ and $b$. $[5]$
Question 5 Code: 9709/32/O/N/14/3, Topic: Algebra
Question 6 Code: 9709/33/O/N/14/3, Topic: Algebra
The polynomial $4 x^{3}+a x^{2}+b x-2$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(x+1)$ and $(x+2)$ are factors of $\mathrm{p}(x)$.
$\text{(i)}$ Find the values of $a$ and $b$. $[4]$
$\text{(ii)}$ When $a$ and $b$ have these values, find the remainder when $\mathrm{p}(x)$ is divided by $\left(x^{2}+1\right)$. $[3]$