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

Qtn No.	1	2	3	Total
Marks	10	9	9	28
Score

Question 1 Code: 9709/31/M/J/18/8, Topic: Integration, Numerical solutions of equations

The positive constant $a$ is such that $\displaystyle\int_{0}^{a} x \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=2$.

$\text{(i)}$ Show that $a$ satisfies the equation $a=2 \ln (a+2)$. $[5]$

$\text{(ii)}$ Verify by calculation that $a$ lies between 3 and $3.5$. $[2]$

$\text{(iii)}$ Use an iteration based on the equation in part $\text{(i)}$ to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

Question 2 Code: 9709/32/M/J/18/8, Topic: Differentiation, Integration

The diagram shows the curve $y=(x+1) \mathrm{e}^{-\frac{1}{3} x}$ and its maximum point $M$.

$\text{(i)}$ Find the $x$-coordinate of $M$. $[4]$

$\text{(ii)}$ Find the area of the shaded region enclosed by the curve and the axes, giving your answer in terms of e. $[5]$

Question 3 Code: 9709/33/O/N/18/8, Topic: Complex numbers

$\text{(a)}$ Showing all necessary working, express the complex number $\displaystyle\frac{2+3 \mathrm{i}}{1-2 \mathrm{i}}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$, where $r>0$ and $-\pi<\theta \leqslant \pi$. Give the values of $r$ and $\theta$ correct to 3 significant figures. $[5]$

$\text{(b)}$ On an Argand diagram sketch the locus of points representing complex numbers $z$ satisfying the equation $|z-3+2 i|=1$. Find the least value of $|z|$ for points on this locus, giving your answer in an exact form. $[4]$