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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 1 (P1) Variant(s) P11, P12, P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 Total
Marks 4 6 6 5 8 29
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 5 questions Question 1 Code: 9709/13/M/J/16/2, Topic: Integration The diagram shows part of the curve$\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$and the point$P(2,3)$lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$$Question 2 Code: 9709/12/M/J/17/3, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2} \equiv \frac{1-\sin \theta}{1+\sin \theta}$.$\text{(ii)}$Hence solve the equation$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2}=\frac{1}{2}$, for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$$Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$.$\text{(ii)}$Hence solve the equation$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$for$0 \leqslant x \leqslant 2 \pi$.$$Question 4 Code: 9709/12/M/J/19/4, Topic: Trigonometry Angle$x$is such that$\sin x=a+b$and$\cos x=a-b$, where$a$and$b$are constants.$\text{(i)}$Show that$a^{2}+b^{2}$has a constant value for all values of$x$.$\text{(ii)}$In the case where$\tan x=2$, express$a$in terms of$b$.$$Question 5 Code: 9709/12/M/J/19/8, Topic: Vectors The position vectors of points$A$and$B$, relative to an origin$O$, are given by $$\overrightarrow{O A}=\left(\begin{array}{r} 6 \\ -2 \\ -6 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 3 \\ k \\ -3 \end{array}\right)$$ where$k$is a constant.$\text{(i)}$Find the value of$k$for which angle$A O B$is$90^{\circ}$.$\text{(ii)}$Find the values of$k$for which the lengths of$O A$and$O B$are equal. The point$C$is such that$\overrightarrow{A C}=2 \overrightarrow{C B}$.$\text{(iii)}$In the case where$k=4$, find the unit vector in the direction of$\overrightarrow{O C}$.$\$

Worked solutions: P1, P3 & P6 (S1)

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