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

Qtn No.	1	2	3	Total
Marks	10	10	10	30
Score

Question 1 Code: 9709/33/M/J/12/9, Topic: Vectors

The lines $l$ and $m$ have equations $\mathbf{r}=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}+\lambda(-\mathbf{i}+2 \mathbf{j}+\mathbf{k})$ and $\mathbf{r}=4 \mathbf{i}+4 \mathbf{j}+2 \mathbf{k}+\mu(a \mathbf{i}+b \mathbf{j}-\mathbf{k})$ respectively, where $a$ and $b$ are constants.

$\text{(i)}$ Given that $l$ and $m$ intersect, show that $[4]$

$$ 2 a-b=4. $$

$\text{(ii)}$ Given also that $l$ and $m$ are perpendicular, find the values of $a$ and $b$. $[4]$

$\text{(iii)}$ When $a$ and $b$ have these values, find the position vector of the point of intersection of $l$ and $m$. $[2]$

Question 2 Code: 9709/31/O/N/12/9, Topic: Algebra

The complex number $1+(\sqrt{2}) \mathrm{i}$ is denoted by $u$. The polynomial $x^{4}+x^{2}+2 x+6$ is denoted by $\mathrm{p}(x)$.

$\text{(i)}$ Showing your working, verify that $u$ is a root of the equation $\mathrm{p}(x)=0$, and write down a second complex root of the equation. $[4]$

$\text{(ii)}$ Find the other two roots of the equation $\mathrm{p}(x)=0$. $[6]$

Question 3 Code: 9709/32/O/N/12/9, Topic: Algebra