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Qtn No.	1	2	3	4	5	6	Total
Marks	6	7	7	8	7	7	42
Score

Question 1 Code: 9709/31/M/J/19/4, Topic: Trigonometry

By first expressing the equation $\cot \theta-\cot \left(\theta+45^{\circ}\right)=3$ as a quadratic equation in $\tan \theta$, solve the equation for $0^{\circ} < \theta < 180^{\circ}$. $\qquad$ $[6]$

Question 2 Code: 9709/32/M/J/19/4, Topic: Differentiation

Find the exact coordinates of the point on the curve $\displaystyle y=\frac{x}{1+\ln x}$ at which the gradient of the tangent is equal to $\frac{1}{4}$. $[7]$

Question 3 Code: 9709/33/M/J/19/4, Topic: Differentiation

The equation of a curve is $\displaystyle y=\frac{1+\mathrm{e}^{-x}}{1-\mathrm{e}^{-x}}$, for $x>0.$

$\text{(i)}$ Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ is always negative. $[3]$

$\text{(ii)}$ The gradient of the curve is equal to $-1$ when $x=a$. Show that $a$ satisfies the equation $\mathrm{e}^{2 a}-4 \mathrm{e}^{a}+1=0$. Hence find the exact value of $a$. $[4]$

Question 4 Code: 9709/31/O/N/19/4, Topic: Differential equations

The number of insects in a population $t$ weeks after the start of observations is denoted by $N$. The population is decreasing at a rate proportional to $N \mathrm{e}^{-0.02 t}$. The variables $N$ and $t$ are treated as continuous, and it is given that when $t=0, N=1000$ and $\displaystyle\frac{\mathrm{d} N}{\mathrm{~d} t}=-10$.

$\text{(i)}$ Show that $N$ and $t$ satisfy the differential equation $[1]$

$$ \displaystyle\frac{\mathrm{d} N}{\mathrm{~d} t}=-0.01 \mathrm{e}^{-0.02 t} N $$

$\text{(ii)}$ Solve the differential equation and find the value of $t$ when $N=800$. $[6]$

$\text{(iii)}$ State what happens to the value of $N$ as $t$ becomes large. $[1]$

Question 5 Code: 9709/32/O/N/19/4, Topic: Trigonometry

$\text{(i)}$ Express $\big(\sqrt{6}\,\big) \sin x+\cos x$ in the form $R \sin (x+\alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$. State the exact value of $R$ and give $\alpha$ correct to 3 decimal places. $[3]$

$\text{(ii)}$ Hence solve the equation $\big(\sqrt{6}\,\big) \sin 2 \theta+\cos 2 \theta=2$, for $0^{\circ} < \theta < 180^{\circ}$. $[4]$

Question 6 Code: 9709/33/O/N/19/4, Topic: Trigonometry

$\text{(i)}$ By first expanding $\tan (2 x+x)$, show that the equation $\tan 3 x=3 \cot x$ can be written in the form $\tan ^{4} x-12 \tan ^{2} x+3=0$.$\quad$ $[4]$

$\text{(ii)}$ Hence solve the equation $\tan 3 x=3 \cot x$ for $0^{\circ} < x < 90^{\circ}$. $[4]$