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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 5 5 6 4 6 3 6 6 6 7 12 9 75
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/21/2, Topic: Coordinate geometry$\text{(a)}$The graph of$y=\mathrm{f}(x)$is transformed to the graph of$y=2 \mathrm{f}(x-1)$. Describe fully the two single transformations which have been combined to give the resulting transformation.$[3]\text{(b)}$The curve$y=\sin 2 x-5 x$is reflected in the$y$-axis and then stretched by scale factor$\frac{1}{3}$in the$x$-direction. Write down the equation of the transformed curve.$[2]$Question 2 Code: 9709/12/M/J/14/3, Topic: Trigonometry The reflex angle$\theta$is such that$\cos \theta=k$, where$0 < k < 1$.$\text{(i)}$Find an expression, in terms of$k$, for$\text{(a)}\sin \theta$,$[2]\text{(b)}\tan \theta$.$[1]\text{(ii)}$Explain why$\sin 2 \theta$is negative for$0 < k < 1$.$[2]$Question 3 Code: 9709/12/M/J/17/3, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2} \equiv \frac{1-\sin \theta}{1+\sin \theta}$.$[3]\text{(ii)}$Hence solve the equation$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2}=\frac{1}{2}$, for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$[3]$Question 4 Code: 9709/13/M/J/20/3, Topic: Coordinate geometry In each of parts$\text{(a)}, \text{(b)}$and$\text{(c)}$, the graph shown with solid lines has equation$y=\mathrm{f}(x)$. The graph shown with broken lines is a transformation of$y=\mathrm{f}(x)$.$\text{(a)}$State, in terms of$\mathrm{f}$, the equation of the graph shown with broken lines.$[1]\text{(b)}$State, in terms of f, the equation of the graph shown with broken lines.$[1]\text{(c)}$State, in terms of$\mathrm{f}$, the equation of the graph shown with broken lines.$[2]$Question 5 Code: 9709/11/M/J/10/4, Topic: Integration The diagram shows the curve$y=6 x-x^{2}$and the line$y=5$. Find the area of the shaded region.$[6]$Question 6 Code: 9709/11/M/J/21/4, Topic: Trigonometry The diagram shows part of the graph of$y=a \tan (x-b)+c$Given that$0 < b < \pi$, state the values of the constants$a, b$and$c$.$[3]$Question 7 Code: 9709/12/M/J/11/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\cos \theta}{\tan \theta(1-\sin \theta)} \equiv 1+\frac{1}{\sin \theta}$.$[3]\text{(ii)}$Hence solve the equation$\displaystyle\frac{\cos \theta}{\tan \theta(1-\sin \theta)}=4$, for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$[3]$Question 8 Code: 9709/12/M/J/21/5, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2 x^{2}+3$for$x \geqslant 0$.$\text{(a)}$Find and simplify an expression for$\mathrm{ff}(x)$.$[2]\text{(b)}$Solve the equation$\mathrm{ff}(x)=34 x^{2}+19$.$[4]$Question 9 Code: 9709/11/M/J/20/8, Topic: Circular measure In the diagram,$A B C$is a semicircle with diameter$A C$, centre$O$and radius$6 \mathrm{~cm}$. The length of the$\operatorname{arc} A B$is$15 \mathrm{~cm}$. The point$X$lies on$A C$and$B X$is perpendicular to$A X$. Find the perimeter of the shaded region$B X C$.$[6]$Question 10 Code: 9709/13/M/J/21/8, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined as follows: \begin{aligned}&\mathrm{f}: x \mapsto x^{2}-1 \text { for } x<0, \\&\mathrm{~g}: x \mapsto \frac{1}{2 x+1} \text { for } x<-\frac{1}{2}.\end{aligned}\text{(a)}$Solve the equation$\mathrm{fg}(x)=3$.$[4]\text{(b)}$Find an expression for$(\mathrm{fg})^{-1}(x)$.$[3]$Question 11 Code: 9709/12/M/J/16/10, Topic: Differentiation, Integration, Coordinate geometry The diagram shows the part of the curve$\displaystyle y=\frac{8}{x}+2 x$for$x>0$, and the minimum point$M\text{(i)}$Find expressions for$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}, \displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$and$\displaystyle\int y^{2} \mathrm{~d} x$.$[5]\text{(ii)}$Find the coordinates of$M$and determine the coordinates and nature of the stationary point on the part of the curve for which$x<0$.$[5]\text{(iii)}$Find the volume obtained when the region bounded by the curve, the$x$-axis and the lines$x=1$and$x=2$is rotated through$360^{\circ}$about the$x$-axis.$[2]$Question 12 Code: 9709/13/M/J/16/10, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=2 x+3$for$x \geqslant 0$. The function$\mathrm{g}$is such that$\mathrm{g}(x)=a x^{2}+b$for$x \leqslant q$, where$a, b$and$q$are constants. The function fg is such that fg$(x)=6 x^{2}-21$for$x \leqslant q\text{(i)}$Find the values of$a$and$b$.$[3]\text{(ii)}$Find the greatest possible value of$q$.$[2]$It is now given that$q=-3$.$\text{(iii)}$Find the range of$\mathrm{fg}$.$[1]\text{(iv)}$Find an expression for$\mathrm{(f g)^{-1}}(x)$and state the domain of$(f g)^{-1}$.$[3]\$

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