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

Qtn No.	1	2	3	4	5	6	7	8	9	10	Total
Marks	4	4	4	5	6	9	8	10	10	10	70
Score

Question 1 Code: 9709/32/O/N/12/1, Topic: Algebra

31

Question 2 Code: 9709/32/O/N/15/1, Topic: Algebra

31

Question 3 Code: 9709/32/O/N/14/2, Topic: Numerical solutions of equations

Question 4 Code: 9709/32/O/N/12/3, Topic: Trigonometry

Question 5 Code: 9709/33/O/N/20/4, Topic: Logarithmic and exponential functions

Solve the equation $$ \log _{10}(2 x+1)=2 \log _{10}(x+1)-1 $$

Give your answers correct to 3 decimal places. $[6]$

Question 6 Code: 9709/32/O/N/16/6, Topic: Numerical solutions of equations

$\text{(i)}$ By sketching a suitable pair of graphs, show that the equation

$$ \operatorname{cosec} \frac{1}{2} x=\frac{1}{3} x+1 $$

has one root in the interval $0$[2]$

$\text{(ii)}$ Show by calculation that this root lies between $1.4$ and $1.6$. $[2]$

$\text{(iii)}$ Show that, if a sequence of values in the interval $0 < x \leqslant \pi$ given by the iterative formula $[2]$

$$ \displaystyle x_{n+1}=2 \sin ^{-1}\left(\frac{3}{x_{n}+3}\right) $$

converges, then it converges to the root of the equation in part $\text{(i)}$. $[2]$

$\text{(iv)}$ Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

Question 7 Code: 9709/32/O/N/20/6, Topic: Complex numbers

The complex number $u$ is defined by

$$ u=\frac{7+\mathrm{i}}{1-\mathrm{i}} $$

$\text{(a)}$ Express $u$ in the form $x+$ i $y$, where $x$ and $y$ are real. $[3]$

$\text{(b)}$ Show on a sketch of an Argand diagram the points $A, B$ and $C$ representing $u, 7+\mathrm{i}$ and $1-\mathrm{i}$ respectively. $[2]$

$\text{(c)}$ By considering the arguments of $7+\mathrm{i}$ and $1-\mathrm{i}$, show that $[3]$

$$ \tan ^{-1}\left(\frac{4}{3}\right)=\tan ^{-1}\left(\frac{1}{7}\right)+\frac{1}{4} \pi $$

Question 8 Code: 9709/32/O/N/12/9, Topic: Algebra

Question 9 Code: 9709/32/O/N/18/9, Topic: Complex numbers

$\text{(a)}\,\,\text{(i)}$ Without using a calculator, express the complex number $\displaystyle\frac{2+6 i}{1-2 i}$ in the form $x+\mathrm{i} y$, where $x$ and $y$ are real. $[2]$

$\text{(ii)}$ Hence, without using a calculator, express $\displaystyle\frac{2+6 i}{1-2 i}$ in the form $r(\cos \theta+i \sin \theta)$, where $r>0$ and $-\pi < \theta \leqslant \pi$, giving the exact values of $r$ and $\theta$. $[3]$

$\text{(b)}$ On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying both the inequalities $|z-3 \mathrm{i}| \leqslant 1$ and $\operatorname{Re} z \leqslant 0$, where $\operatorname{Re} z$ denotes the real part of $z$. Find the greatest value of $\arg z$ for points in this region, giving your answer in radians correct to 2 decimal places. $[5]$

Question 10 Code: 9709/32/O/N/19/9, Topic: Integration, Numerical solutions of equations

It is given that $\displaystyle\int_{0}^{a} x \cos \frac{1}{3} x \mathrm{~d} x=3$, where the constant $a$ is such that $0 < a < \frac{3}{2} \pi$.

$\text{(i)}$ Show that $a$ satisfies the equation

$$ a=\frac{4-3 \cos \frac{1}{3} a}{\sin \frac{1}{3} a} $$

$\text{(ii)}$ Verify by calculation that $a$ lies between $2.5$ and 3.

$\text{(iii)}$ Use an iterative formula based on the equation in part $\text{(i)}$ to calculate $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$