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

Qtn No.	1	2	3	4	5	6	7	8	9	10	Total
Marks	5	4	6	5	7	8	8	8	9	9	69
Score

Question 1 Code: 9709/31/M/J/12/2, Topic: Algebra

$\text{(i)}$ Expand $\displaystyle\frac{1}{\sqrt{(} 1-4 x)}$ in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients. $[3]$

$\text{(ii)}$ Hence find the coefficient of $x^{2}$ in the expansion of $\displaystyle\frac{1+2 x}{\sqrt{(} 4-16 x)}$. $[2]$

Question 2 Code: 9709/32/M/J/15/2, Topic: Logarithmic and exponential functions

Using the substitution $u=4^{x}$, solve the equation $4^{x}+4^{2}=4^{x+2}$, giving your answer correct to 3 significant figures. $[4]$

Question 3 Code: 9709/31/M/J/18/2, Topic: Trigonometry

$\text{(i)}$ Given that $\sin \left(x-60^{\circ}\right)=3 \cos \left(x-45^{\circ}\right)$, find the exact value of $\tan x$. $[4]$

$\text{(ii)}$ Hence solve the equation $\sin \left(x-60^{\circ}\right)=3 \cos \left(x-45^{\circ}\right)$, for $0^{\circ} < x < 360^{\circ}$. $[2]$

Question 4 Code: 9709/32/M/J/16/3, Topic: Integration

Find the exact value of $\displaystyle\int_{0}^{\frac{1}{2} \pi} x^{2} \sin 2 x \mathrm{~d} x$. $[5]$

Question 5 Code: 9709/31/O/N/14/5, Topic: Complex numbers

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

The complex numbers $w$ and $z$ satisfy the relation

$$ w=\frac{z+\mathrm{i}}{\mathrm{i} z+2}. $$

$\text{(i)}$ Given that $z=1+\mathrm{i}$, find $w$, giving your answer in the form $x+\mathrm{i} y$, where $x$ and $y$ are real. $[4]$

$\text{(ii)}$ Given instead that $w=z$ and the real part of $z$ is negative, find $z$, giving your answer in the form $x+\mathrm{i} y$, where $x$ and $y$ are real. $[4]$

Question 6 Code: 9709/31/M/J/17/5, Topic: Numerical solutions of equations

 

The diagram shows a semicircle with centre $O$, radius $r$ and diameter $A B$. The point $P$ on its circumference is such that the area of the minor segment on $A P$ is equal to half the area of the minor segment on $B P$. The angle $A O P$ is $x$ radians.

$\text{(i)}$ Show that $x$ satisfies the equation $x=\frac{1}{3}(\pi+\sin x)$. $[3]$

$\text{(ii)}$ Verify by calculation that $x$ lies between 1 and $1.5.$ $[2]$

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

Question 7 Code: 9709/31/M/J/12/7, Topic: Differential equations

The variables $x$ and $y$ are related by the differential equation

$$ \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 x \mathrm{e}^{3 x}}{y^{2}} $$

It is given that $y=2$ when $x=0$. Solve the differential equation and hence find the value of $y$ when $x=0.5$, giving your answer correct to 2 decimal places. $[7]$

Question 8 Code: 9709/31/O/N/12/7, Topic: Differentiation

The equation of a curve is $\ln (x y)-y^{3}=1$.

$\text{(i)}$ Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y}{x\left(3 y^{3}-1\right)}$. $[4]$

$\text{(ii)}$ Find the coordinates of the point where the tangent to the curve is parallel to the $y$-axis, giving each coordinate correct to 3 significant figures. $[4]$

Question 9 Code: 9709/31/O/N/16/7, Topic: Differentiation, Integration

 

The diagram shows part of the curve $y=\left(2 x-x^{2}\right) \mathrm{e}^{\frac{1}{2} x}$ and its maximum point $M$.

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

$\text{(ii)}$ Find the exact value of the area of the shaded region bounded by the curve and the positive $x$-axis. $[5]$

Question 10 Code: 9709/32/M/J/18/9, Topic: Algebra

Let $\displaystyle\mathrm{f}(x)=\frac{x-4 x^{2}}{(3-x)\left(2+x^{2}\right)}$.

$\text{(i)}$ Express $\displaystyle\mathrm{f}(x)$ in the form $\displaystyle\frac{A}{3-x}+\frac{B x+C}{2+x^{2}}$. $[4]$

$\text{(ii)}$ Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^{3}$. $[5]$