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### HENRYTAIGO

#### Cambridge International AS and A Level

 Name of student HENRYTAIGO Date Adm. number Year/grade HenryTaigo Stream HenryTaigo Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

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

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/19/4, Topic: Trigonometry By first expressing the equation$\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]\$

Worked solutions: P1, P3 & P6 (S1)

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