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

Qtn No.	1	2	3	4	5	6	Total
Marks	12	11	11	11	11	10	66
Score

Question 1 Code: 9709/31/M/J/14/10, Topic: Differentiation

 

The diagram shows the curve $y=10 \mathrm{e}^{-\frac{1}{2} x} \sin 4 x$ for $x \geqslant 0$. The stationary points are labelled $T_{1}, T_{2}$, $T_{3}, \ldots$ as shown.

$\text{(i)}$ Find the $x$-coordinates of $T_{1}$ and $T_{2}$, giving each $x$-coordinate correct to 3 decimal places. $[6]$

$\text{(ii)}$ It is given that the $x$-coordinate of $T_{n}$ is greater than 25. Find the least possible value of $n$. $[4]$

Question 2 Code: 9709/32/M/J/14/10, Topic: Vectors

Referred to the origin $O$, the points $A, B$ and $C$ have position vectors given by

$$ \overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}, \quad \overrightarrow{O B}=2 \mathbf{i}+4 \mathbf{j}+\mathbf{k} \quad \text { and } \quad \overrightarrow{O C}=3 \mathbf{i}+5 \mathbf{j}-3 \mathbf{k} $$

$\text{(i)}$ Find the exact value of the cosine of angle $B A C$. $[4]$

$\text{(ii)}$ Hence find the exact value of the area of triangle $A B C$. $[3]$

$\text{(iii)}$ Find the equation of the plane which is parallel to the $y$-axis and contains the line through $B$ and $C$. Give your answer in the form $a x+b y+c z=d$. $[5]$

Question 3 Code: 9709/33/M/J/14/10, Topic: Vectors

The line $l$ has equation $\mathbf{r}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}+\lambda(3 \mathbf{i}-2 \mathbf{j}+2 \mathbf{k})$ and the plane $p$ has equation $2 x+3 y-5 z=18$

$\text{(i)}$ Find the position vector of the point of intersection of $l$ and $p$. $[3]$

$\text{(ii)}$ Find the acute angle between $l$ and $p$. $[4]$

$\text{(iii)}$ A second plane $q$ is perpendicular to the plane $p$ and contains the line $l$. Find the equation of $q$, giving your answer in the form $a x+b y+c z=d$. $[5]$

Question 4 Code: 9709/31/O/N/14/10, Topic: Vectors

The line $l$ has equation $\mathbf{r}=4 \mathbf{i}-9 \mathbf{j}+9 \mathbf{k}+\lambda(-2 \mathbf{i}+\mathbf{j}-2 \mathbf{k})$. The point $A$ has position vector $3 \mathbf{i}+8 \mathbf{j}+5 \mathbf{k}$.

$\text{(i)}$ Show that the length of the perpendicular from $A$ to $l$ is $15.$ $[5]$

$\text{(ii)}$ The line $l$ lies in the plane with equation $a x+b y-3 z+1=0$, where $a$ and $b$ are constants. Find the values of $a$ and $b$. $[5]$

Question 5 Code: 9709/32/O/N/14/10, Topic: Vectors

Question 6 Code: 9709/33/O/N/14/10, Topic: Integration

By first using the substitution $u=\mathrm{e}^{x}$, show that $[10]$

$$ \displaystyle\int_{0}^{\ln 4} \frac{\mathrm{e}^{2 x}}{\mathrm{e}^{2 x}+3 \mathrm{e}^{x}+2} \mathrm{~d} x=\ln \left(\frac{8}{5}\right) $$