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

Qtn No.	1	2	3	4	Total
Marks	11	10	9	10	40
Score

Question 1 Code: 9709/31/M/J/16/9, Topic: Vectors

With respect to the origin $O$, the points $A, B, C, D$ have position vectors given by

$$ \overrightarrow{O A}=\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}, \quad \overrightarrow{O B}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}, \quad \overrightarrow{O C}=2 \mathbf{i}+4 \mathbf{j}+\mathbf{k}, \quad \overrightarrow{O D}=-3 \mathbf{i}+\mathbf{j}+2 \mathbf{k} $$

$\text{(i)}$ Find the equation of the plane containing $A, B$ and $C$, giving your answer in the form $a x+b y+c z=d$. $[6]$

$\text{(ii)}$ The line through $D$ parallel to $O A$ meets the plane with equation $x+2 y-z=7$ at the point $P$. Find the position vector of $P$ and show that the length of $D P$ is $2 \sqrt{(} 14)$. $[5]$

Question 2 Code: 9709/32/M/J/16/9, Topic: Vectors

The points $A, B$ and $C$ have position vectors, relative to the origin $O$, given by $\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$ $\overrightarrow{O B}=4 \mathbf{j}+\mathbf{k}$ and $\overrightarrow{O C}=2 \mathbf{i}+5 \mathbf{j}-\mathbf{k}.$ A fourth point $D$ is such that the quadrilateral $A B C D$ is a parallelogram.

$\text{(i)}$ Find the position vector of $D$ and verify that the parallelogram is a rhombus. $[5]$

$\text{(ii)}$ The plane $p$ is parallel to $O A$ and the line $B C$ lies in $p$. Find the equation of $p$, giving your answer in the form $a x+b y+c z=d$. $[5]$

Question 3 Code: 9709/33/M/J/16/9, Topic: Complex numbers

Throughout this question the use of a calculator is not permitted.

The complex numbers $-1+3 \mathrm{i}$ and $2-\mathrm{i}$ are denoted by $u$ and $v$ respectively. In an Argand diagram with origin $O$, the points $A, B$ and $C$ represent the numbers $u, v$ and $u+v$ respectively.

$\text{(i)}$ Sketch this diagram and state fully the geometrical relationship between $O B$ and $A C$. $[4]$

$\text{(ii)}$ Find, in the form $x+\mathrm{i} y$, where $x$ and $y$ are real, the complex number $\displaystyle\frac{u}{v}$. $[3]$

$\text{(iii)}$ Prove that angle $A O B=\frac{3}{4} \pi$. $[2]$

Question 4 Code: 9709/31/O/N/16/9, Topic: Complex numbers

Throughout this question the use of a calculator is not permitted.

$\text{(a)}$ Solve the equation $(1+2 \mathrm{i}) w^{2}+4 w-(1-2 \mathrm{i})=0$, giving your answers in the form $x+\mathrm{i} y$, where $x$ and $y$ are real. $[5]$

$\text{(b)}$ On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $|z-1-\mathrm{i}| \leqslant 2$ and $-\frac{1}{4} \pi \leqslant \arg z \leqslant \frac{1}{4} \pi$. $[5]$