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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 5 5 5 5 5 5 30

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject.
Attempt all the 6 questions

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 $\left(x^{2}+1\right)$. $[3]$

Worked solutions: P1, P3 & P6 (S1)

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