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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 5 4 6 7 9 7 9 9 12 9 87
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/20/2, Topic: Series The coefficient of$\displaystyle \frac{1}{x}$in the expansion of$\displaystyle \left(k x+\frac{1}{x}\right)^{5}+\left(1-\frac{2}{x}\right)^{8}$is 74. Find the value of the positive constant$k$.$[5]$Question 2 Code: 9709/13/M/J/11/3, Topic: Coordinate geometry The line$\displaystyle\frac{x}{a}+\frac{y}{b}=1$, where$a$and$b$are positive constants, meets the$x$-axis at$P$and the$y$-axis at$Q$. Given that$P Q=\sqrt{(} 45)$and that the gradient of the line$P Q$is$-\frac{1}{2}$, find the values of$a$and$b$.$[5]$Question 3 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$The point$P(2,9)$lies on the curve.$\text{(i)}$A point moves on the curve in such a way that the$x$-coordinate is decreasing at a constant rate of$0.05$units per second. Find the rate of change of the$y$-coordinate when the point is at$P$.$[2]\text{(ii)}$Find the equation of the curve.$[3]$Question 4 Code: 9709/11/M/J/21/5, Topic: Series The fifth, sixth and seventh terms of a geometric progression are$8 k,-12$and$2 k$respectively. Given that$k$is negative, find the sum to infinity of the progression.$[4]$Question 5 Code: 9709/13/M/J/21/5, Topic: Circular measure The diagram shows a triangle$A B C$, in which angle$A B C=90^{\circ}$and$A B=4 \mathrm{~cm}.$The sector$A B D$is part of a circle with centre$A$. The area of the sector is$10 \mathrm{~cm}^{2}$.$\text{(a)}$Find angle$B A D$in radians.$[2]\text{(b)}$Find the perimeter of the shaded region.$[4]$Question 6 Code: 9709/13/M/J/13/6, Topic: Differentiation The non-zero variables$x, y$and$u$are such that$u=x^{2} y$. Given that$y+3 x=9$, find the stationary value of$u$and determine whether this is a maximum or a minimum value.$[7]$Question 7 Code: 9709/13/M/J/18/7, Topic: Trigonometry$\text{(a)}\quad\text{(i)}$Express$\displaystyle \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}$in the form$a \sin ^{2} \theta+b$, where$a$and$b$are constants to be found.$[3]\qquad \text{(ii)}$Hence, or otherwise, and showing all necessary working, solve the equation.$[2]$$$\frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}=\frac{1}{4}$$$\qquad$for$-90^{\circ} \leqslant \theta \leqslant 0^{\circ}\text{(b)}$The diagram shows the graphs of$y=\sin x$and$y=2 \cos x$for$-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points$A$and$B$.$\text{(i)}$Find the$x$-coordinate of$A$.$[2]\text{(ii)}$Find the$y$-coordinate of$B$.$[2]$Question 8 Code: 9709/13/M/J/12/8, Topic: Circular measure In the diagram,$A B$is an arc of a circle with centre$O$and radius$r$. The line$X B$is a tangent to the circle at$B$and$A$is the mid-point of$O X$.$\text{(i)}$Show that angle$A O B=\frac{1}{3} \pi$radians.$[2]$Express each of the following in terms of$r, \pi$and$\sqrt{3}$:$\text{(ii)}$the perimeter of the shaded region,$[3]\text{(iii)}$the area of the shaded region.$[2]$Question 9 Code: 9709/11/M/J/15/8, Topic: Functions The function$\mathrm{f}: x \mapsto 5+3 \cos \left(\frac{1}{2} x\right)$is defined for$0 \leqslant x \leqslant 2 \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)=7$, giving your answer correct to 2 decimal places.$[3]\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$.$[2]\text{(iii)}$Explain why$\mathrm{f}$has an inverse.$[1]\text{(iv)}$Obtain an expression for$\mathrm{f}^{-1}(x)$.$[3]$Question 10 Code: 9709/12/M/J/10/10, Topic: Differentiation The equation of a curve is$y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.$\text{(i)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$.$[3]\text{(ii)}$Find the equation of the tangent to the curve at the point where the curve intersects the$y$-axis.$[3]\text{(iii)}$Find the set of values of$x$for which$\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$is an increasing function of$x$.$[3]$Question 11 Code: 9709/13/M/J/11/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned}\text{(i)}$Evaluate$\mathrm{fg(2)}$.$[2]\text{(ii)}$Sketch in a single diagram the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs.$[3]\text{(iii)}$Obtain an expression for$\mathrm{g}^{\prime}(x)$and use your answer to explain why$\mathrm{g}$has an inverse.$[3]\text{(iv)}$Express each of$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$in terms of$x$.$[4]$Question 12 Code: 9709/13/M/J/18/10, Topic: Functions The one-one function$\mathrm{f}$is defined by$\mathrm{f}(x)=(x-2)^{2}+2$for$x \geqslant c$, where$c$is a constant.$\text{(i)}$State the smallest possible value of$c$.$[1]$In parts$\text{(ii)}$and$\text{(iii)}$the value of$c$is 4.$\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$.$[3]\text{(iii)}$Solve the equation$\mathrm{ff}(x)=51$, giving your answer in the form$a+\sqrt{b}$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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