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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 6 5 7 6 6 8 7 6 10 11 82
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/13/2, Topic: Circular measure The diagram shows a circle$C$with centre$O$and radius$3 \mathrm{~cm}$. The radii$O P$and$O Q$are extended to$S$and$R$respectively so that$O R S$is a sector of a circle with centre$O$. Given that$P S=6 \mathrm{~cm}$and that the area of the shaded region is equal to the area of circle$C$,$\text{(i)}$show that angle$P O Q=\frac{1}{4} \pi$radians,$\text{(ii)}$find the perimeter of the shaded region.$$Question 2 Code: 9709/11/M/J/13/3, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$8 \mathrm{~cm}$. Angle$B O A$is$\alpha$radians.$O A C$is a semicircle with diameter$O A$. The area of the semicircle$O A C$is twice the area of the sector$O A B$.$\text{(i)}$Find$\alpha$in terms of$\pi$.$\text{(ii)}$Find the perimeter of the complete figure in terms of$\pi$.$$Question 3 Code: 9709/13/M/J/19/3, Topic: Circular measure The diagram shows triangle$A B C$which is right-angled at$A$. Angle$A B C=\frac{1}{5} \pi$radians and$A C=8 \mathrm{~cm}$. The points$D$and$E$lie on$B C$and$B A$respectively. The sector$A D E$is part of a circle with centre$A$and is such that$B D C$is the tangent to the$\operatorname{arc} D E$at$D$.$\text{(i)}$Find the length of$A D$.$\text{(ii)}$Find the area of the shaded region.$$Question 4 Code: 9709/12/M/J/14/4, Topic: Circular measure The diagram shows a sector of a circle with radius$r \mathrm{~cm}$and centre$O$. The chord$A B$divides the sector into a triangle$A O B$and a segment$A X B$. Angle$A O B$is$\theta$radians.$\text{(i)}$In the case where the areas of the triangle$A O B$and the segment$A X B$are equal, find the value of the constant$p$for which$\theta=p \sin \theta$.$\text{(ii)}$In the case where$r=8$and$\theta=2.4$, find the perimeter of the segment$A X B$.$$Question 5 Code: 9709/12/M/J/17/4, Topic: Circular measure The diagram shows a circle with radius$r \mathrm{~cm}$and centre$O$. Points$A$and$B$lie on the circle and$A B C D$is a rectangle. Angle$A O B=2 \theta$radians and$A D=r \mathrm{~cm}$.$\text{(i)}$Express the perimeter of the shaded region in terms of$r$and$\theta$.$\text{(ii)}$In the case where$r=5$and$\theta=\frac{1}{6} \pi$, find the area of the shaded region.$$Question 6 Code: 9709/13/M/J/20/5, Topic: Circular measure The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre$O$and radius$5 \mathrm{~cm}$. The thickness of the cord and the size of the pin$P$can be neglected. The pin is situated 13 cm vertically below$O$. Points$A$and$B$are on the circumference of the circle such that$A P$and$B P$are tangents to the circle. The cord passes over the major arc$A B$of the circle and under the pin such that the cord is taut. Calculate the length of the cord.$$Question 7 Code: 9709/12/M/J/16/6, Topic: Circular measure The diagram shows a circle with radius$r \mathrm{~cm}$and centre$O$. The line$P T$is the tangent to the circle at$P$and angle$P O T=\alpha$radians. The line$O T$meets the circle at$Q.\text{(i)}$Express the perimeter of the shaded region$P Q T$in terms of$r$and$\alpha$.$\text{(ii)}$In the case where$\alpha=\frac{1}{3} \pi$and$r=10$, find the area of the shaded region correct to 2 significant figures.$$Question 8 Code: 9709/13/M/J/10/7, Topic: Circular measure The diagram shows a metal plate$A B C D E F$which has been made by removing the two shaded regions from a circle of radius 10 cm and centre$O.$The parallel edges$A B$and$E D$are both of length 12 cm.$\text{(i)}$Show that angle$D O E$is 1.287 radians, correct to 4 significant figures.$\text{(ii)}$Find the perimeter of the metal plate.$\text{(iii)}$Find the area of the metal plate.$$Question 9 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.$$Question 10 Code: 9709/11/M/J/20/8, Topic: Circular measure In the diagram,$A B C$is a semicircle with diameter$A C$, centre$O$and radius$6 \mathrm{~cm}$. The length of the$\operatorname{arc} A B$is$15 \mathrm{~cm}$. The point$X$lies on$A C$and$B X$is perpendicular to$A X$. Find the perimeter of the shaded region$B X C$.$$Question 11 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.$\text{(b)}$Find the exact perimeter of the plate.$\text{(c)}$Show that the area of the plate is$\left(\frac{20}{3}\pi + 8\sqrt{3}\right)$cm$^{2}$.$$Question 12 Code: 9709/12/M/J/21/12, Topic: Circular measure The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are$A, B, C, D$,$E$and$F$. Points$P$and$Q$are situated where straight sections of the rope meet the pipe with centre$A.\text{(a)}$Show that angle$P A Q=\frac{1}{3} \pi$radians.$\text{(b)}$Find the length of the rope.$\text{(c)}$Find the area of the hexagon$A B C D E F$, giving your answer in terms of$\sqrt{3}$.$\text{(d)}$Find the area of the complete region enclosed by the rope.$\$

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