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### MATHEMATICS 9709

#### Cambridge International AS and A Level

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

Qtn No. 1 2 3 4 5 6 7 Total
Marks 4 6 7 8 8 11 10 54
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 7 questions Question 1 Code: 9709/31/O/N/18/2, Topic: Logarithmic and exponential functions Showing all necessary working, solve the equation$\displaystyle\frac{2 \mathrm{e}^{x}+\mathrm{e}^{-x}}{\mathrm{e}^{x}-\mathrm{e}^{-x}}=4$, giving your answer correct to 2 decimal places.$[4]$Question 2 Code: 9709/31/O/N/14/4, Topic: Differentiation The parametric equations of a curve are $$x=\frac{1}{\cos ^{3} t}, \quad y=\tan ^{3} t,$$ where$0 \leqslant t < \frac{1}{2} \pi$.$\text{(i)}$Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin t$.$[4]\text{(ii)}$Hence show that the equation of the tangent to the curve at the point with parameter$t$is$y=x \sin t-\tan t$.$[3]$Question 3 Code: 9709/31/O/N/13/5, Topic: Trigonometry, Integration$\text{(i)}$Prove that$\cot \theta+\tan \theta \equiv 2 \operatorname{cosec} 2 \theta$.$[3]\text{(ii)}$Hence show that$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \operatorname{cosec} 2 \theta \mathrm{d} \theta=\frac{1}{2} \ln 3.[4]$Question 4 Code: 9709/31/O/N/11/6, Topic: Trigonometry$\text{(i)}$Express$\cos x+3 \sin x$in the form$R \cos (x-\alpha)$, where$R>0$and$0^{\circ}< \alpha <90^{\circ}$, giving the exact value of$R$and the value of$\alpha$correct to 2 decimal places.$[3]\text{(ii)}$Hence solve the equation$\cos 2 \theta+3 \sin 2 \theta=2$, for$0^{\circ}< \theta <90^{\circ}$.$[5]$Question 5 Code: 9709/31/O/N/19/6, Topic: Differentiation, Integration$\text{(i)}$By differentiating$\displaystyle\frac{\cos x}{\sin x}$, show that if$\displaystyle y=\cot x$then$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=-\operatorname{cosec}^{2} x$.$[2]\text{(ii)}$Show that$\displaystyle\int_{\frac{1}{4} \pi}^{\frac{1}{2} \pi} x \operatorname{cosec}^{2} x \mathrm{~d} x=\frac{1}{4}(\pi+\ln 4)$.$[6]$Question 6 Code: 9709/31/O/N/13/9, Topic: Vectors The diagram shows three points$A, B$and$C$whose position vectors with respect to the origin$O$are given by$\overrightarrow{O A}=\left(\begin{array}{r}2 \\ -1 \\ 2\end{array}\right), \overrightarrow{O B}=\left(\begin{array}{l}0 \\ 3 \\ 1\end{array}\right)$and$\overrightarrow{O C}=\left(\begin{array}{l}3 \\ 0 \\ 4\end{array}\right).$The point$D$lies on$B C$, between$B$and$C$, and is such that$C D=2 D B$.$\text{(i)}$Find the equation of the plane$A B C$, giving your answer in the form$a x+b y+c z=d$.$[6]\text{(ii)}$Find the position vector of$D$.$[1]\text{(iii)}$Show that the length of the perpendicular from$A$to$O D$is$\frac{1}{3} \sqrt{(65)}$.$[4]$Question 7 Code: 9709/31/O/N/18/9, Topic: Algebra, Integration Let$\displaystyle\mathrm{f}(x)=\frac{6 x^{2}+8 x+9}{(2-x)(3+2 x)^{2}}\text{(i)}$Express$\mathrm{f}(x)$in partial fractions.$[5]\text{(ii)}$Hence, showing all necessary working, show that$\displaystyle\int_{-1}^{0} \mathrm{f}(x) \, \mathrm{d} x=1+\frac{1}{2} \ln \left(\frac{3}{4}\right)$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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