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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) P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 3 3 3 4 8 7 8 8 8 10 9 9 80
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/14/1, Topic: Series Find the coefficient of$x$in the expansion of$\displaystyle\left(x^{2}-\frac{2}{x}\right)^{5}$.$$Question 2 Code: 9709/13/M/J/16/1, Topic: Series Find the coefficient of$x$in the expansion of$\displaystyle\left(\frac{1}{x}+3 x^{2}\right)^{5}$.$$Question 3 Code: 9709/13/M/J/21/1, Topic: Integration A curve with equation$y=\mathrm{f}(x)$is such that$\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$It is given that the curve passes through the point$(2,7)$. Find$\mathrm{f}(x)$.$$Question 4 Code: 9709/13/M/J/15/2, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=(2 x+1)^{\frac{1}{2}}$and the point$(4,7)$lies on the curve. Find the equation of the curve.$$Question 5 Code: 9709/13/M/J/14/7, Topic: Vectors The position vectors of points$A, B$and$C$relative to an origin$O$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -1 \\ 7 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 2 \\ 4 \\ 7 \end{array}\right)$$$\text{(i)}$Show that angle$B A C=\cos ^{-1}\left(\frac{1}{3}\right)$.$\text{(ii)}$Use the result in part$\text{(i)}$to find the exact value of the area of triangle$A B C$.$$Question 6 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry The point$A$has coordinates$(p, 1)$and the point$B$has coordinates$(9,3 p+1)$, where$p$is a constant.$\text{(i)}$For the case where the distance$A B$is 13 units, find the possible values of$p$.$\text{(ii)}$For the case in which the line with equation$2 x+3 y=9$is perpendicular to$A B$, find the value of$p$.$$Question 7 Code: 9709/13/M/J/20/7, Topic: Trigonometry$\text{(a)}$Show that$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}$.$\text{(b)}$Hence solve the equation$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta}=\frac{6}{\tan \theta}$for$0^{\circ}< \theta <180^{\circ}$.$$Question 8 Code: 9709/13/M/J/15/8, Topic: Differentiation The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x>-1$.$\text{(i)}$Find$\mathrm{f}^{\prime}(x)$.$\text{(ii)}$State, with a reason, whether$\mathrm{f}$is an increasing function, a decreasing function or neither.$$The function$\mathrm{g}$is defined by$\displaystyle\mathrm{g}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x<-1\text{(iii)}$Find the coordinates of the stationary point on the curve$y=\mathrm{g}(x)$.$$Question 9 Code: 9709/13/M/J/16/8, Topic: Trigonometry$\text{(i)}$Show that$3 \sin x \tan x-\cos x+1=0$can be written as a quadratic equation in$\cos x$and hence solve the equation$3 \sin x \tan x-\cos x+1=0$for$0 \leqslant x \leqslant \pi$.$\text{(ii)}$Find the solutions to the equation$3 \sin 2 x \tan 2 x-\cos 2 x+1=0$for$0 \leqslant x \leqslant \pi$.$$Question 10 Code: 9709/13/M/J/19/9, Topic: Trigonometry The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q\quad\text{(b)}\mathrm{f}(x)=q\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$$Question 11 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration Points$A(2,9)$and$B(3,0)$lie on the curve$y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at$A$intersects the$x$-axis at$C$. Showing all necessary working,$\text{(i)}$find the equation of the tangent$A C$and hence find the$x$-coordinate of$C$,$\text{(ii)}$find the area of the shaded region$A B C$.$$Question 12 Code: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{\sqrt{x}}-x$and points$A(1,7)$and$B(4,0)$which lie on the curve. The tangent to the curve at$B$intersects the line$x=1$at the point$C$.$\text{(i)}$Find the coordinates of$C$.$\text{(ii)}$Find the area of the shaded region.$\$

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