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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 10 7 10 10 10 10 57
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/14/8, Topic: Numerical solutions of equations$\text{(i)}$By sketching each of the graphs$y=\operatorname{cosec} x$and$y=x(\pi-x)$for$0 < x < \pi$, show that the equation $$\operatorname{cosec} x=x(\pi-x)$$ has exactly two real roots in the interval$0 < x < \pi$.$[2]\text{(ii)}$Show that the equation$\operatorname{cosec} x=x(\pi-x)$can be written in the form$\displaystyle x=\frac{1+x^{2} \sin x}{\pi \sin x}$.$[2]\text{(iii)}$The two real roots of the equation$\operatorname{cosec} x=x(\pi-x)$in the interval$0 < x < \pi$are denoted by$\alpha$and$\beta$, where$\alpha < \beta$.$\text{(a)}$Use the iterative formula $$\displaystyle x_{n+1}=\frac{1+x_{n}^{2} \sin x_{n}}{\pi \sin x_{n}}$$ to find$\alpha$correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]\text{(b)}$Deduce the value of$\beta$correct to 2 decimal places.$[1]$Question 2 Code: 9709/32/M/J/14/8, Topic: Differentiation The diagram shows the curve$y=x \cos \frac{1}{2} x$for$0 \leqslant x \leqslant \pi$.$\text{(i)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and show that$4 \displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+y+4 \sin \frac{1}{2} x=0$.$[5]\text{(ii)}$Find the exact value of the area of the region enclosed by this part of the curve and the$x$-axis.$[5]$Question 3 Code: 9709/33/M/J/14/8, Topic: Algebra, Integration Let$\displaystyle\mathrm{f}(x)=\frac{6+6 x}{(2-x)\left(2+x^{2}\right)}$.$\text{(i)}$Express$\mathrm{f}(x)$in the form$\displaystyle\frac{A}{2-x}+\frac{B x+C}{2+x^{2}}$.$[4]\text{(ii)}$Show that$\displaystyle\int_{-1}^{1} \mathrm{f}(x) \mathrm{d} x=3 \ln 3$.$[5]$Question 4 Code: 9709/31/O/N/14/8, Topic: Trigonometry$\text{(i)}$By first expanding$\sin (2 \theta+\theta)$, show that$[4]$$$\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta.$$$\text{(ii)}$Show that, after making the substitution$\displaystyle x=\frac{2 \sin \theta}{\sqrt{3}}$, the equation$x^{3}-x+\frac{1}{6} \sqrt{3}=0$can be written in the form$\sin 3 \theta=\frac{3}{4}$.$[1]\text{(iii)}$Hence solve the equation $$x^{3}-x+\frac{1}{6} \sqrt{3}=0,$$ giving your answers correct to 3 significant figures.$[4]$Question 5 Code: 9709/32/O/N/14/8, Topic: Trigonometry Question 6 Code: 9709/33/O/N/14/8, Topic: Differentiation The variables$x$and$y$are related by the differential equation $$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{5} x y^{\frac{1}{2}} \sin \left(\frac{1}{3} x\right) \text {. }$$$\text{(i)}$Find the general solution, giving$y$in terms of$x$.$[6]\text{(ii)}$Given that$y=100$when$x=0$, find the value of$y$when$x=25$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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