1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

8 \includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} 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 \(\operatorname { arcs } P S\) 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 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

12 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} 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 \(A\).

1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
2. Find the length of the rope.
3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
4. Find the area of the complete region enclosed by the rope.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 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 \(10 \mathrm {~cm} ^ { 2 }\).

1. Find angle \(B A D\) in radians.
2. Find the perimeter of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751} The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.

1. Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
2. Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]

7 \includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-10_593_841_260_651} The diagram shows a sector \(O B A C\) of a circle with centre \(O\) and radius 10 cm . The point \(P\) lies on \(O C\) and \(B P\) is perpendicular to \(O C\). Angle \(A O C = \frac { 1 } { 6 } \pi\) and the length of the \(\operatorname { arc } A B\) is 2 cm .

1. Find the angle \(B O C\).
2. Hence find the area of the shaded region \(B P C\) giving your answer correct to 3 significant figures. [4]

9 The diagram shows triangle \(A B C\) with \(A B = B C = 6 \mathrm {~cm}\) and angle \(A B C = 1.8\) radians. The arc \(C D\) is part of a circle with centre \(A\) and \(A B D\) is a straight line.

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

4 The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius 8 cm . The area of the sector is \(\frac { 16 } { 3 } \pi \mathrm {~cm} ^ { 2 }\). The point \(D\) lies on the \(\operatorname { arc } B C\). Find the perimeter of the segment \(B C D\).

6 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-07_389_552_267_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B = \theta\) radians. It is given that the length of the \(\operatorname { arc } A B\) is 9.6 cm and that the area of the sector \(O A B\) is \(76.8 \mathrm {~cm} ^ { 2 }\).

1. Find the area of the shaded region.
2. Find the perimeter of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-05_483_561_287_753} The diagram shows a sector of a circle with centre \(C\). The radii \(C A\) and \(C B\) each have length \(r \mathrm {~cm}\) and the size of the reflex angle \(A C B\) is \(\theta\) radians. The sector, shaded in the diagram, has a perimeter of 65 cm and an area of \(225 \mathrm {~cm} ^ { 2 }\).

1. Find the values of \(r\) and \(\theta\).
2. Find the area of triangle \(A C B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-08_574_689_260_726} The diagram shows a sector \(A O B\) which is part of a circle with centre \(O\) and radius 6 cm and with angle \(A O B = 0.8\) radians. The point \(C\) on \(O B\) is such that \(A C\) is perpendicular to \(O B\). The arc \(C D\) is part of a circle with centre \(O\), where \(D\) lies on \(O A\). Find the area of the shaded region.

9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
   1. Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
   2. Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
2. For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively. Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-16_547_421_264_863} The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.

8 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.

9 \includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744} In the diagram, arc \(A B\) is part of a circle with centre \(O\) and radius 8 cm . Arc \(B C\) is part of a circle with centre \(A\) and radius 12 cm , where \(A O C\) is a straight line.

1. Find angle \(B A O\) in radians.
2. Find the area of the shaded region.
3. Find the perimeter of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-08_556_751_255_696} In the diagram the lengths of \(A B\) and \(A C\) are both 15 cm . The point \(P\) is the foot of the perpendicular from \(C\) to \(A B\). The length \(C P = 9 \mathrm {~cm}\). An arc of a circle with centre \(B\) passes through \(C\) and meets \(A B\) at \(Q\).

1. Show that angle \(A B C = 1.25\) radians, correct to 3 significant figures.
2. Calculate the area of the shaded region which is bounded by the \(\operatorname { arc } C Q\) and the lines \(C P\) and \(P Q\).

5 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .

1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
2. Find the perimeter of \(A B C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-06_494_542_260_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The length of the \(\operatorname { arc } A B\) is 8 cm . It is given that the perimeter of the sector is 20 cm .

1. Find the perimeter of the shaded segment.
2. Find the area of the shaded segment.

8 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).

1. Find the perimeter of the shaded region in terms of \(r\).
2. Find the area of the shaded region in terms of \(r\).

6 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-08_534_506_255_815} The diagram shows a motif formed by the major arc \(A B\) of a circle with radius \(r\) and centre \(O\), and the minor arc \(A O B\) of a circle, also with radius \(r\) but with centre \(C\). The point \(C\) lies on the circle with centre \(O\).

1. Given that angle \(A C B = k \pi\) radians, state the value of the fraction \(k\).
2. State the perimeter of the shaded motif in terms of \(\pi\) and \(r\).
3. Find the area of the shaded motif, giving your answer in terms of \(\pi , r\) and \(\sqrt { 3 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-12_552_582_255_778} The diagram shows points \(A , B\) and \(C\) lying on a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the circle with centre \(O\) and radius \(r\). The lower arc is part of a circle with centre \(C\) and radius \(R\).

1. State the size of angle \(A C O\) in radians.
2. Find \(R\) in terms of \(r\).
3. Find the area of the shaded region in terms of \(r\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).

1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
3. Find the area of the new cross-section (shown shaded in the diagram).

9 \includegraphics[max width=\textwidth, alt={}, center]{8214ccb9-0894-4c3c-a8d9-d8f8749fdbe1-3_321_636_267_758} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 8 cm . Angle \(A O B = \theta\) radians.

1. In the case where \(\theta = 1\), calculate the area of the sector BOC.
2. Find the value of \(\theta\) for which the perimeter of sector \(A O B\) is one half of the perimeter of sector BOC.
3. In the case where \(\theta = \frac { 1 } { 3 } \pi\), show that the exact length of the perimeter of triangle \(A B C\) is \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-2_501_682_1302_735} In the diagram, \(O C D\) is an isosceles triangle with \(O C = O D = 10 \mathrm {~cm}\) and angle \(C O D = 0.8\) radians. The points \(A\) and \(B\), on \(O C\) and \(O D\) respectively, are joined by an arc of a circle with centre \(O\) and radius 6 cm . Find

1. the area of the shaded region,
2. the perimeter of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.

1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).

1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.