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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 6 7 8 9 10 11 12 Total
Marks 4 4 5 4 6 6 7 7 8 8 11 12 82
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/10/1, Topic: Trigonometry The acute angle$x$radians is such that$\tan x=k$, where$k$is a positive constant. Express, in terms of$k$,$\text{(i)}\tan (\pi-x)$,$[1]\text{(ii)}\tan \left(\frac{1}{2} \pi-x\right)$,$[1]\text{(iii)}\sin x$.$[2]$Question 2 Code: 9709/13/M/J/13/1, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$and$(2,5)$is a point on the curve. Find the equation of the curve.$[4]$Question 3 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry Find the coordinates of the point at which the perpendicular bisector of the line joining$(2,7)$to$(10,3)$meets the$x$-axis.$[5]$Question 4 Code: 9709/12/M/J/15/2, Topic: Circular measure In the diagram,$A Y B$is a semicircle with$A B$as diameter and$O A X B$is a sector of a circle with centre$O$and radius$r$. Angle$A O B=2 \theta$radians. Find an expression, in terms of$r$and$\theta$, for the area of the shaded region.$[4]$Question 5 Code: 9709/13/M/J/19/3, Topic: Circular measure The diagram shows triangle$A B C$which is right-angled at$A$. Angle$A B C=\frac{1}{5} \pi$radians and$A C=8 \mathrm{~cm}$. The points$D$and$E$lie on$B C$and$B A$respectively. The sector$A D E$is part of a circle with centre$A$and is such that$B D C$is the tangent to the$\operatorname{arc} D E$at$D$.$\text{(i)}$Find the length of$A D$.$[3]\text{(ii)}$Find the area of the shaded region.$[3]$Question 6 Code: 9709/12/M/J/12/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\tan x+\frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}$.$[2]\text{(ii)}$Solve the equation$\displaystyle\frac{2}{\sin x \cos x}=1+3 \tan x$, for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[4]$Question 7 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry The point$C$lies on the perpendicular bisector of the line joining the points$A(4,6)$and$B(10,2)$.$C$also lies on the line parallel to$A B$through$(3,11)$.$\text{(i)}$Find the equation of the perpendicular bisector of$A B$.$[4]\text{(ii)}$Calculate the coordinates of$C$.$[3]$Question 8 Code: 9709/12/M/J/19/7, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned}\text{(i)}$Obtain expressions for$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$, stating the value of$x$for which$\mathrm{g}^{-1}(x)$is not defined.$[4]\text{(ii)}$Solve the equation$\mathrm{fg}(x)=\frac{7}{3}$.$[3]$Question 9 Code: 9709/12/M/J/13/8, Topic: Differentiation The volume of a solid circular cylinder of radius$r \mathrm{~cm}$is$250 \pi \mathrm{cm}^{3}$.$\text{(i)}$Show that the total surface area,$S \mathrm{~cm}^{2}$, of the cylinder is given by$[2]$$$S=2 \pi r^{2}+\frac{500 \pi}{r}$$$\text{(ii)}$Given that$r$can vary, find the stationary value of$S$.$[4]\text{(iii)}$Determine the nature of this stationary value.$[2]$Question 10 Code: 9709/13/M/J/15/8, Topic: Differentiation The function$\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)$.$[3]\text{(ii)}$State, with a reason, whether$\mathrm{f}$is an increasing function, a decreasing function or neither.$[1]$The function$\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)$.$[4]$Question 11 Code: 9709/11/M/J/18/9, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-2 \\ &\mathrm{~g}: x \mapsto 4+x-\frac{1}{2} x^{2} \end{aligned}\text{(i)}$Find the points of intersection of the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{g}(x)$.$[3]\text{(ii)}$Find the set of values of$x$for which$\mathrm{f}(x)>\mathrm{g}(x)$.$[2]\text{(iii)}$Find an expression for$\mathrm{fg}(x)$and deduce the range of$\mathrm{fg}$.$[4]$The function h is defined by h :$x \mapsto 4+x-\frac{1}{2} x^{2}$for$x \geqslant k$.$\text{(iv)}$Find the smallest value of$k$for which$\mathrm{h}$has an inverse.$[2]$Question 12 Code: 9709/12/M/J/15/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x^{2}-6 x+5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$p$for which the equation$\mathrm{f}(x)=p$has no real roots.$[3]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$for$0 \leqslant x \leqslant 4$.$\text{(ii)}$Express$\mathrm{g}(x)$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]\text{(iii)}$Find the range of$\mathrm{g}$.$[2]$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(iv)}$State the smallest value of$k$for which$\mathrm{h}$has an inverse.$[1]\text{(v)}$For this value of$k$, find an expression for$\mathrm{h}^{-1}(x)$.$[3]\$

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