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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 4 4 6 7 8 7 8 8 9 9 11 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/11/1, Topic: Trigonometry The coefficient of$x^{3}$in the expansion of$(a+x)^{5}+(1-2 x)^{6}$, where$a$is positive, is 90. Find the value of$a$.$$Question 2 Code: 9709/13/M/J/13/1, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$and$(2,5)$is a point on the curve. Find the equation of the curve.$$Question 3 Code: 9709/11/M/J/12/2, Topic: Series Find the coefficient of$x^{6}$in the expansion of$\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$.$$Question 4 Code: 9709/12/M/J/12/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\tan x+\frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}$.$\text{(ii)}$Solve the equation$\displaystyle\frac{2}{\sin x \cos x}=1+3 \tan x$, for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$$Question 5 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.$$Question 6 Code: 9709/12/M/J/12/6, Topic: Circular measure The diagram shows a metal plate made by removing a segment from a circle with centre$O$and radius$8 \mathrm{~cm}.$The line$A B$is a chord of the circle and angle$A O B=2.4$radians. Find$\text{(i)}$the length of$A B$,$\text{(ii)}$the perimeter of the plate,$\text{(iii)}$the area of the plate.$$Question 7 Code: 9709/11/M/J/12/7, Topic: Series$\text{(a)}$The first two terms of an arithmetic progression are 1 and$\cos ^{2} x$respectively. Show that the sum of the first ten terms can be expressed in the form$a-b \sin ^{2} x$, where$a$and$b$are constants to be found.$\text{(b)}$The first two terms of a geometric progression are 1 and$\frac{1}{3} \tan ^{2} \theta$respectively, where$0< \theta <\frac{1}{2} \pi$.$\text{(i)}$Find the set of values of$\theta$for which the progression is convergent.$\text{(ii)}$Find the exact value of the sum to infinity when$\theta=\frac{1}{6} \pi$.$$Question 8 Code: 9709/13/M/J/20/7, Topic: Trigonometry$\text{(a)}$Show that$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}$.$\text{(b)}$Hence solve the equation$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta}=\frac{6}{\tan \theta}$for$0^{\circ}< \theta <180^{\circ}$.$$Question 9 Code: 9709/13/M/J/14/8, Topic: Quadratics$\text{(i)}$Express$2 x^{2}-10 x+8$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants, and use your answer to state the minimum value of$2 x^{2}-10 x+8$.$\text{(ii)}$Find the set of values of$k$for which the equation$2 x^{2}-10 x+8=k x$has no real roots.$$Question 10 Code: 9709/11/M/J/11/9, Topic: Circular measure In the diagram,$O A B$is an isosceles triangle with$O A=O B$and angle$A O B=2 \theta$radians. Arc$P S T$has centre$O$and radius$r$, and the line$A S B$is a tangent to the$\operatorname{arc} P S T$at$S$.$\text{(i)}$Find the total area of the shaded regions in terms of$r$and$\theta$.$\text{(ii)}$In the case where$\theta=\frac{1}{3} \pi$and$r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of$\sqrt{3}$and$\pi$.$$Question 11 Code: 9709/13/M/J/17/9, Topic: Functions$\text{(i)}$Express$9 x^{2}-6 x+6$in the form$(a x+b)^{2}+c$, where$a, b$and$c$are constants.$$The function$\mathrm{f}$is defined by$\mathrm{f}(x)=9 x^{2}-6 x+6$for$x \geqslant p$, where$p$is a constant.$\text{(ii)}$State the smallest value of$p$for which$\mathrm{f}$is a one-one function.$\text{(iii)}$For this value of$p$, obtain an expression for$\mathrm{f}^{-1}(x)$, and state the domain of$\mathrm{f}^{-1}$.$\text{(iv)}$State the set of values of$q$for which the equation$\mathrm{f}(x)=q$has no solution.$$Question 12 Code: 9709/12/M/J/11/10, Topic: Series$\text{(a)}$A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is$5 \mathrm{~cm}$, find the perimeter of the smallest sector.$\text{(b)}$The first, second and third terms of a geometric progression are$2 k+3, k+6$and$k$, respectively. Given that all the terms of the geometric progression are positive, calculate$\text{(i)}$the value of the constant$k$,$\text{(ii)}$the sum to infinity of the progression.$\$

Worked solutions: P1, P3 & P6 (S1)

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