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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 5 4 6 6 6 4 8 9 7 10 9 78
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/16/2, Topic: Trigonometry Solve the equation$3 \sin ^{2} \theta=4 \cos \theta-1$for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$[3]$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: 9709/13/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 4 Code: 9709/11/M/J/13/4, Topic: Series The third term of a geometric progression is$-108$and the sixth term is 32. Find$\text{(i)}$the common ratio,$[3]\text{(ii)}$the first term,$[1]\text{(iii)}$the sum to infinity.$[2]$Question 5 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry The diagram shows the curve$y=7 \sqrt{x}$and the line$y=6 x+k$, where$k$is a constant. The curve and the line intersect at the points$A$and$B$.$\text{(i)}$For the case where$k=2$, find the$x$-coordinates of$A$and$B$.$[4]\text{(ii)}$Find the value of$k$for which$y=6 x+k$is a tangent to the curve$y=7 \sqrt{x}$.$[2]$Question 6 Code: 9709/12/M/J/18/6, Topic: Circular measure The diagram shows points$A$and$B$on a circle with centre$O$and radius$r$. The tangents to the circle at$A$and$B$meet at$T$. The shaded region is bounded by the minor$\operatorname{arc} A B$and the lines$A T$and$B T$. Angle$A O B$is$2 \theta$radians.$\text{(i)}$In the case where the area of the sector$A O B$is the same as the area of the shaded region, show that$\tan \theta=2 \theta$.$[3]\text{(ii)}$In the case where$r=8 \mathrm{~cm}$and the length of the minor$\operatorname{arc} A B$is$19.2 \mathrm{~cm}$, find the area of the shaded region.$[3]$Question 7 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 8 Code: 9709/13/M/J/14/9, Topic: Differentiation The base of a cuboid has sides of length$x \mathrm{~cm}$and$3 x \mathrm{~cm}$. The volume of the cuboid is$288 \mathrm{~cm}^{3}$.$\text{(i)}$Show that the total surface area of the cuboid,$A \mathrm{~cm}^{2}$, is given by$[3]$$$A=6 x^{2}+\frac{768}{x}.$$$\text{(ii)}$Given that$x$can vary, find the stationary value of$A$and determine its nature.$[5]$Question 9 Code: 9709/11/M/J/16/9, Topic: Series$\text{(a)}$The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression.$[3]\text{(b)}$The first three terms of an arithmetic progression are$2 \sin x, 3 \cos x$and$(\sin x+2 \cos x)$respectively, where$x$is an acute angle.$\text{(i)}$Show that$\tan x=\frac{4}{3}$.$[3]\text{(ii)}$Find the sum of the first twenty terms of the progression.$[3]$Question 10 Code: 9709/11/M/J/19/9, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2-3 \cos x$for$0 \leqslant x \leqslant 2 \pi$.$\text{(i)}$State the range of$\mathrm{f}$.$[2]\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2-3 \cos x$for$0 \leqslant x \leqslant p$, where$p$is a constant.$[2]\text{(iii)}$State the largest value of$p$for which$\mathrm{g}$has an inverse.$[1]\text{(iv)}$For this value of$p$, find an expression for$\mathrm{g}^{-1}(x)$.$[3]$Question 11 Code: 9709/13/M/J/19/9, Topic: Trigonometry The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$[2]\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q[1]\quad\text{(b)}\mathrm{f}(x)=q[1]\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q[1]\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$[5]$Question 12 Code: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{\sqrt{x}}-x$and points$A(1,7)$and$B(4,0)$which lie on the curve. The tangent to the curve at$B$intersects the line$x=1$at the point$C$.$\text{(i)}$Find the coordinates of$C$.$[4]\text{(ii)}$Find the area of the shaded region.$[5]\$

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