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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 6 5 7 7 8 8 9 12 9 84
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/12/1, Topic: Integration The diagram shows the region enclosed by the curve$\displaystyle y=\frac{6}{2 x-3}$, the$x$-axis and the lines$x=2$and$x=3$. Find, in terms of$\pi$, the volume obtained when this region is rotated through$360^{\circ}$about the$x$-axis.$$Question 2 Code: 9709/13/M/J/16/2, Topic: Integration The diagram shows part of the curve$\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$and the point$P(2,3)$lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$$Question 3 Code: 9709/11/M/J/16/3, Topic: Integration The diagram shows part of the curve$\displaystyle x=\frac{12}{y^{2}}-2$. The shaded region is bounded by the curve, the$y$-axis and the lines$y=1$and$y=2$. Showing all necessary working, find the volume, in terms of$\pi$, when this shaded region is rotated through$360^{\circ}$about the$y$-axis.$$Question 4 Code: 9709/13/M/J/21/3, Topic: Quadratics A line with equation$y=m x-6$is a tangent to the curve with equation$y=x^{2}-4 x+3$. Find the possible values of the constant$m$, and the corresponding coordinates of the points at which the line touches the curve.$$Question 5 Code: 9709/13/M/J/19/4, Topic: Functions The function$\mathrm{f}$is defined by$\displaystyle \mathrm{f}(x)=\frac{48}{x-1}$for$3 \leqslant x \leqslant 7$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2 x-4$for$a \leqslant x \leqslant b$, where$a$and$b$are constants.$\text{(i)}$Find the greatest value of$a$and the least value of$b$which will permit the formation of the composite function$\mathrm{gf}$.$$It is now given that the conditions for the formation of gf are satisfied.$\text{(ii)}$Find an expression for$\operatorname{gf}(x)$.$\text{(iii)}$Find an expression for$(\mathrm{gf})^{-1}(x)$.$$Question 6 Code: 9709/11/M/J/10/6, Topic: Differentiation A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-6$and the point$(9,2)$lies on the curve.$\text{(i)}$Find the equation of the curve.$\text{(ii)}$Find the$x$-coordinate of the stationary point on the curve and determine the nature of the stationary point.$$Question 7 Code: 9709/12/M/J/20/7, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$2 r$, and angle$A O B=\frac{1}{6} \pi$radians. The point$C$is the midpoint of$O A$.$\text{(a)}$Show that the exact length of$B C$is$r \sqrt{5-2 \sqrt{3}}$.$\text{(b)}$Find the exact perimeter of the shaded region.$\text{(c)}$Find the exact area of the shaded region.$$Question 8 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$0^{\circ}< \theta <180^{\circ}$.$$Question 9 Code: 9709/12/M/J/11/9, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=3-4 \cos ^{k} x$, for$0 \leqslant x \leqslant \pi$, where$k$is a constant.$\text{(i)}$In the case where$k=2$,$\text{(a)}$find the range of$\mathrm{f}$,$\text{(b)}$find the exact solutions of the equation$\mathrm{f}(x)=1$.$\text{(ii)}$In the case where$k=1$,$\text{(a)}$sketch the graph of$y=\mathrm{f}(x)$,$\text{(b)}$state, with a reason, whether$\mathrm{f}$has an inverse.$$Question 10 Code: 9709/12/M/J/17/9, Topic: Differentiation, Coordinate geometry The equation of a curve is$y=8 \sqrt{x}-2 x$.$\text{(i)}$Find the coordinates of the stationary point of the curve.$\text{(ii)}$Find an expression for$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$and hence, or otherwise, determine the nature of the stationary point.$\text{(iii)}$Find the values of$x$at which the line$y=6$meets the curve.$\text{(iv)}$State the set of values of$k$for which the line$y=k$does not meet the curve.$$Question 11 Code: 9709/12/M/J/12/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned}\text{(i)}$Obtain expressions, in terms of$x$, for$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$, stating the value of$x$for which$\mathrm{g}^{-1}(x)$is not defined.$\text{(ii)}$Sketch the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$on the same diagram, making clear the relationship between the two graphs.$\text{(iii)}$Given that the equation$f g(x)=5-k x$, where$k$is a constant, has no solutions, find the set of possible values of$k$.$$Question 12 Code: 9709/13/M/J/13/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where$k$is a constant.$\text{(i)}$In the case where$k=3$, solve the equation$\mathrm{ff}(x)=25$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.$\text{(ii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=\mathrm{g}(x)$has no real solutions.$$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.$\text{(iii)}$Find an expression for$\mathrm{h}^{-1}(x)$.$\$

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