$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	6	Total
Marks	9	8	8	9	9	8	51
Score

Question 1 Code: 9709/31/M/J/10/7, Topic: Complex numbers

The complex number $2+2 \mathrm{i}$ is denoted by $u$.

$\text{(i)}$ Find the modulus and argument of $u$. $[3]$

$\text{(ii)}$ Sketch an Argand diagram showing the points representing the complex numbers 1 , i and $u$. Shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $|z-1| \leqslant|z-\mathrm{i}|$ and $|z-u| \leqslant 1$ $[4]$

$\text{(iii)}$ Using your diagram, calculate the value of $|z|$ for the point in this region for which $\arg z$ is least. $[3]$

Question 2 Code: 9709/32/M/J/10/7, Topic: Differentiation

The variables $x$ and $t$ are related by the differential equation

$$ \mathrm{e}^{2 t} \displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=\cos ^{2} x $$

where $t \geqslant 0$. When $t=0, x=0$.

$\text{(i)}$ Solve the differential equation, obtaining an expression for $x$ in terms of $t$. $[6]$

$\text{(ii)}$ State what happens to the value of $x$ when $t$ becomes very large. $[1]$

$\text{(iii)}$ Explain why $x$ increases as $t$ increases. $[1]$

Question 3 Code: 9709/33/M/J/10/7, Topic: Trigonometry, Integration

$\text{(i)}$ Prove the identity $\cos 3 \theta \equiv 4 \cos ^{3} \theta-3 \cos \theta$. $[4]$

$\text{(ii)}$ Using this result, find the exact value of $[4]$

$$ \displaystyle\int_{\frac{1}{3} \pi}^{\frac{1}{2} \pi} \cos ^{3} \theta \mathrm{d} \theta $$

Question 4 Code: 9709/31/O/N/10/7, Topic: Vectors

With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}$ and $\overrightarrow{O B}=3 \mathbf{i}+4 \mathbf{j}.$ The point $P$ lies on the line $A B$ and $O P$ is perpendicular to $A B$.

$\text{(i)}$ Find a vector equation for the line $A B$. $[1]$

$\text{(ii)}$ Find the position vector of $P$. $[4]$

$\text{(iii)}$ Find the equation of the plane which contains $A B$ and which is perpendicular to the plane $O A B$, giving your answer in the form $a x+b y+c z=d$. $[4]$

Question 5 Code: 9709/32/O/N/10/7, Topic: Vectors

Question 6 Code: 9709/33/O/N/10/7, Topic: Integration, Numerical solutions of equations

$\text{(i)}$ Given that $\displaystyle\int_{1}^{a} \frac{\ln x}{x^{2}} \mathrm{~d} x=\frac{2}{5}$, show that $a=\frac{5}{3}(1+\ln a)$. $[5]$

$\text{(ii)}$ Use an iteration formula based on the equation $a=\frac{5}{3}(1+\ln a)$ to find the value of $a$ correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places. $[3]$