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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 6 4 9 10 11 9 13 9 12 98
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/19/2, Topic: Series$\text{(i)}$In the binomial expansion of$\displaystyle \left(2 x-\frac{1}{2 x}\right)^{5}$, the first three terms are$32 x^{5}-40 x^{3}+20 x$. Find the remaining three terms of the expansion.$[3]\text{(ii)}$Hence find the coefficient of$x$in the expansion of$\displaystyle \left(1+4 x^{2}\right)\left(2 x-\frac{1}{2 x}\right)^{5}$.$[2]$Question 2 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 3 Code: 9709/12/M/J/21/4, Topic: Series The coefficient of$x$in the expansion of$\displaystyle \left(4 x+\frac{10}{x}\right)^{3}$is$p$. The coefficient of$\displaystyle \frac{1}{x}$in the expansion of$\displaystyle \left(2 x+\frac{k}{x^{2}}\right)^{5}$is$q$. Given that$p=6 q$, find the possible values of$k$.$[5]$Question 4 Code: 9709/13/M/J/21/4, Topic: Trigonometry$\text{(a)}$Show that the equation$[2]$$$\frac{\tan x+\sin x}{\tan x-\sin x}=k$$ where$k$is a constant, may be expressed as $$\frac{1+\cos x}{1-\cos x}=k$$$\text{(b)}$Hence express$\cos x$in terms of$k$.$[2]\text{(c)}$Hence solve the equation$\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$for$-\pi < x < \pi$.$[2]$Question 5 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry Points$A$and$B$have coordinates$(8,3)$and$(p, q)$respectively. The equation of the perpendicular bisector of$A B$is$y=-2 x+4$. Find the values of$p$and$q$.$[4]$Question 6 Code: 9709/13/M/J/20/8, Topic: Series The first term of a progression is$\sin ^{2} \theta$, where$0 < \theta < \frac{1}{2} \pi$. The second term of the progression is$\sin ^{2} \theta \cos ^{2} \theta$.$\text{(a)}$Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic.$[3]\text{(b)} \quad \text{(i)}$Find the common difference of the progression in terms of$\sin \theta$.$[3] \quad \quad \text{(ii)}$Find the sum of the first 16 terms when$\theta=\frac{1}{3} \pi$.$[3]$Question 7 Code: 9709/11/M/J/21/8, Topic: Circular measure The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre$C$. The boundary of the plate consists of two arcs$P S$and$Q R$of the original circle and two semicircles with$P Q$and$R S$as diameters. The radius of the circle with centre$C$is$4 \mathrm{~cm}$, and$P Q=R S=4 \mathrm{~cm}$also.$\text{(a)}$Show that angle$P C S=\frac{2}{3} \pi$radians.$[2]\text{(b)}$Find the exact perimeter of the plate.$[3]\text{(c)}$Show that the area of the plate is$\left(\frac{20}{3}\pi + 8\sqrt{3}\right)$cm$^{2}$.$[5]$Question 8 Code: 9709/11/M/J/21/9, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined as follows: \begin{aligned} &\mathrm{f}(x)=(x-2)^{2}-4 \text { for } x \geqslant 2, \\ &\mathrm{~g}(x)=a x+2 \text { for } x \in \mathbb{R}, \end{aligned} wherea$is a constant.$\text{(a)}$State the range of$\mathrm{f}$.$[1]\text{(b)}$Find$\mathrm{f}^{-1}(x)$.$[2]\text{(c)}$Given that$a=-\frac{5}{3}$, solve the equation$\mathrm{f}(x)=\mathrm{g}(x)$.$[3]\text{(d)}$Given instead that$\operatorname{ggf}^{-1}(12)=62$, find the possible values of$a$.$[5]$Question 9 Code: 9709/11/M/J/19/10, Topic: Integration, Differentiation A curve for which$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 x-5$has a stationary point at$(3,6)$.$\text{(i)}$Find the equation of the curve.$[6]\text{(ii)}$Find the$x$-coordinate of the other stationary point on the curve.$[1]\text{(iii)}$Determine the nature of each of the stationary points.$[2]$Question 10 Code: 9709/13/M/J/19/10, Topic: Coordinate geometry, Integration The diagram shows part of the curve with equation$y=(3 x+4)^{\frac{1}{2}}$and the tangent to the curve at the point$A$. The$x$-coordinate of$A$is 4.$\text{(i)}$Find the equation of the tangent to the curve at$A$.$[5]\text{(ii)}$Find, showing all necessary working, the area of the shaded region.$[5]\text{(iii)}$A point is moving along the curve. At the point$P$the$y$-coordinate is increasing at half the rate at which the$x$-coordinate is increasing. Find the$x$-coordinate of$P$.$[3]$Question 11 Code: 9709/12/M/J/20/10, Topic: Differentiation The equation of a curve is$y=54 x-(2 x-7)^{3}$.$\text{(a)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$[4]\text{(b)}$Find the coordinates of each of the stationary points on the curve.$[3]\text{(c)}$Determine the nature of each of the stationary points.$[2]$Question 12 Code: 9709/12/M/J/18/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{x}{2}+\frac{6}{x}$. The line$y=4$intersects the curve at the points$P$and$Q$.$\text{(i)}$Show that the tangents to the curve at$P$and$Q$meet at a point on the line$y=x$.$[6]\text{(ii)}$Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis. Give your answer in terms of$\pi$.$[6]\$

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