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1 1. The diagram above shows the sector OA of a circle with centre O, radius 9 cm and angle 0.7 radians. Find the length of the arc A. Find the area of the sector OA. The line AC shown in the diagram above is perpendicular to OA, and OC is a straight line. Find the length of AC, giving your answer to 2 decimal places. The region H is bounded by the arc A and the lines AC and C. (d) Find the area of H, giving your answer to 2 decimal places. (Total 9 marks) Edexcel Internal Review 1

2 2. An emblem, as shown in the diagram above, consists of a triangle AC joined to a sector CD of a circle with radius 4 cm and centre. The points A, and D lie on a straight line with A = 5 cm and D = 4 cm. Angle AC = 0.6 radians and AC is the longest side of the triangle AC. Show that angle AC = 1.76 radians, correct to 3 significant figures. Find the area of the emblem. (Total 7 marks) 3. The diagram above shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height h cm. The cross section is a sector of a circle. The sector has radius r cm and angle 1 radian. The volume of the box is 300 cm 3. Edexcel Internal Review 2

3 Show that the surface area of the box, S cm 2, is given by S = r r (5) Use calculus to find the value of r for which S is stationary. Prove that this value of r gives a minimum value of S. (d) Find, to the nearest cm 2, this minimum value of S. (Total 13 marks) Edexcel Internal Review 3

4 4. The shape CD shown above is a design for a logo. The straight lines D and DC are equal in length. The curve C is an arc of a circle with centre A and radius 6 cm. The size of AC is 2.2 radians and AD = 4 cm. Find the area of the sector AC, in cm 2, the size of DAC, in radians to 3 significant figures, the complete area of the logo design, to the nearest cm 2. (Total 8 marks) Edexcel Internal Review 4

5 5. 7 cm R A 0.8 rad D C The diagram above shows AC, a sector of a circle with centre A and radius 7 cm. Given that the size of AC is exactly 0.8 radians, find the length of the arc C, the area of the sector AC. The point D is the mid-point of AC. The region R, shown shaded in the diagram above, is bounded by CD, D and the arc C. Find the perimeter of R, giving your answer to 3 significant figures, (d) the area of R, giving your answer to 3 significant figures. (Total 12 marks) Edexcel Internal Review 5

6 6. A circle C has centre M (6, 4) and radius 3. Write down the equation of the circle in the form (x a) 2 + (y b) 2 = r 2. y T 3 P (12, 6) Q C M (6, 4) O x The diagram above shows the circle C. The point T lies on the circle and the tangent at T passes through the point P (12, 6). The line MP cuts the circle at Q. Show that the angle TMQ is radians to 4 decimal places. The shaded region TPQ is bounded by the straight lines TP, QP and the arc TQ, as shown in the diagram above. Find the area of the shaded region TPQ. Give your answer to 3 decimal places. (5) (Total 11 marks) Edexcel Internal Review 6

7 7. S P 6 3m R 6m 6m Q The diagram above shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 6 3m, find the exact size of angle PQR in radians. Show that the area of the patio PQRS is 12π m 2. Find the exact area of the triangle PQR. (d) Find, in m 2 to 1 decimal place, the area of the segment PRS. (e) Find, in m to 1 decimal place, the perimeter of the patio PQRS. (Total 11 marks) Edexcel Internal Review 7

8 8. C 2.12 m A 1.86 m D The figure above shows the cross section ACD of a small shed. The straight line A is vertical and has length 2.12 m. The straight line AD is horizontal and has length 1.86 m. The curve C is an arc of a circle with centre A, and CD is a straight line. Given that the size of AC is 0.65 radians, find the length of the arc C, in m, to 2 decimal places, the area of the sector AC, in m 2, to 2 decimal places, (d) the size of CAD, in radians, to 2 decimal places, the area of the cross section ACD of the shed, in m 2, to 2 decimal places. (Total 9 marks) Edexcel Internal Review 8

9 9. Figure 1 A 6 cm cm C Figure 1 shows the cross-section AC of a metal cutter used for making biscuits. The straight sides A and AC are both of length 6 cm and AC is 1.2 radians. The curved portion C is an arc of a circle with centre A. Find the perimeter of the cross-section of the cutter. Find the area of the cross-section AC. Figure 2 a cm 1.2 b cm 6 cm 1.2 A pair of these cutters are kept together in a rectangular box of length a cm and width b cm. The cutters fit into the box as shown in Figure 2. Find the value of a and the value of b, giving your answers to 3 significant figures. (Total 8 marks) Edexcel Internal Review 9

10 10. A 6 m 5 m 5 m O In the figure above OA is a sector of a circle radius 5 m. The chord A is 6 m long. 7 Show that cos A O ˆ =. 25 Hence find the angle AO ˆ in radians, giving your answer to 3 decimal places. (1) Calculate the area of the sector OA. (d) Hence calculate the shaded area. (Total 8 marks) Edexcel Internal Review 10

11 11. A D 30m O C 45m A fence from a point A to a point is in the shape of an arc A of a circle with centre O and radius 45 m, as shown in the diagram. The length of the fence is 63 m. Show that the size of AO is exactly 1.4 radians. The points C and D are on the lines O and OA respectively, with OC = OD = 30 m. A plot of land ACD, shown shaded in the figure above, is enclosed by the arc A and the straight lines C, CD and DA. Calculate, to the nearest m 2, the area of this plot of land. (5) (Total 7 marks) Edexcel Internal Review 11

12 12. O r cm L M A major sector LOM of a circle, with centre O and radius r cm, has LOM = θ radians, as shown in the diagram. The perimeter of the sector is P cm and the area of the sector is A cm 2. Write down, in terms of r and θ, expressions for P and A. Given that r = 2 2 and that P = A, show that θ = Express θ in the form a + b 2, where a and b are integers to be found. (Total 8 marks) Edexcel Internal Review 12

13 13. 8 cm R C A 0.7 rad 11 cm D This diagram shows the triangle AC, with A = 8 cm, AC = 11 cm and AC = 0.7 radians. The arc D, where D lies on AC, is an arc of a circle with centre A and radius 8 cm. The region R, shown shaded in the diagram, is bounded by the straight lines C and CD and the arc D. Find the length of the arc D, the perimeter of R, giving your answer to 3 significant figures, the area of R, giving your answer to 3 significant figures. (5) (Total 11 marks) Edexcel Internal Review 13

14 14. C 6 cm D 2 7 cm A 2 cm In AC, A = 2 cm, AC = 6 cm and C = 2 7 cm. Use the cosine rule to show that AC = 3 π radians. The circle with centre A and radius 2 cm intersects AC at the point D, as shown in the diagram above. Calculate the length, in cm, of the arc D, the area, in cm 2, of the shaded region CD. (Total 9 marks) Edexcel Internal Review 14

15 15. O 6.5 cm A The diagram above shows the sector AO of a circle, with centre O and radius 6.5 cm, and AO = 0.8 radians. Calculate, in cm 2, the area of the sector AO. Show that the length of the chord A is 5.06 cm, to 3 significant figures. The segment R, shaded in the diagram above, is enclosed by the arc A and the straight line A. Calculate, in cm, the perimeter of R. (Total 7 marks) 16. A R r r O Edexcel Internal Review 15

16 The diagram above shows the sector OA of a circle of radius r cm. The area of the sector is 15 cm 2 and AO = 1.5 radians. Prove that r = 2 5. Find, in cm, the perimeter of the sector OA. The segment R, shaded in the diagram above, is enclosed by the arc A and the straight line A. Calculate, to 3 decimal places, the area of R. (Total 8 marks) Edexcel Internal Review 16

17 17. y C A 14 M 14 D x The diagram above shows the cross-section ACD of a chocolate bar, where A, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and C is an arc of a circle with centre M. Taking A as the origin,, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively. Show that the length of M is 25 mm. Show that, to 3 significant figures, MC = radians. Hence calculate, in mm 2, the area of the cross-section of the chocolate bar. (1) (5) Given that this chocolate bar has length 85 mm, (d) calculate, to the nearest cm 3, the volume of the bar. (Total 11 marks) Edexcel Internal Review 17

18 18. C A 2a A flat plate S, which is part of a child s toy, is shown in the diagram above. The points A, and C are the vertices of an equilateral triangle and the distance between A and is 2a. The circular arc A has centre C and radius 2a. The circular arcs C and CA have centres at A and respectively and radii 2a. Find, in terms of π and a, the perimeter of S. Prove that the area of the plate S is 2a 2 (π 3 ). (6) (Total 8 marks) Edexcel Internal Review 18

19 19. 6 cm A 6 cm D cm C The diagram above shows a logo AD. The logo is formed from triangle AC. The mid-point of AC is D and C = AD = DC = 6 cm. CA = 0.4 radians. The curve D is an arc of a circle with centre C and radius 6 cm. Write down the length of the arc D. Find the length of A. (1) Write down the perimeter of the logo AD, giving your answer to 3 significant figures. (1) (Total 5 marks) Edexcel Internal Review 19

20 C2 Trigonometry: Trigonometric Graphs 1. Sketch, for 0 x 2π, the graph of y = sin. 6 x + π Write down the exact coordinates of the points where the graph meets the coordinate axes. Solve, for 0 x 2π, the equation sin x + π = 0.65, 6 giving your answers in radians to 2 decimal places (5) (Total 10 marks) π 3. The curve C has equation y = cos x +, 0 x 2π. 4 Sketch C. Write down the exact coordinates of the points at which C meets the coordinate axes. Solve, for x in the interval 0 x 2π, cos π x + = 0.5, 4 giving your answers in terms of p. (Total 9 marks) Edexcel Internal Review 1

21 C2 Trigonometry: Trigonometric Equations 7. Find all the solutions, in the interval 0 x < 2π, of the equation 2 cos 2 x + 1 = 5 sin x, giving each solution in terms of π. (Total 6 marks) 13. Solve, for 0 θ < 2π, the equation sin 2 θ = 1 + cos θ, giving your answers in terms of π. (Total 5 marks) π 19. The curve C has equation y = cos x +, 0 x 2π. 4 Sketch C. Write down the exact coordinates of the points at which C meets the coordinate axes. Solve, for x in the interval 0 x 2π, cos π x + = 0.5, 4 giving your answers in terms of p. (Total 9 marks) Edexcel Internal Review 3

22 C2 Trigonometry: Trigonometric Identities 3. Solve, for 0 θ < 2π, the equation sin 2 θ = 1 + cos θ, giving your answers in terms of π. (Total 5 marks) (ii) Given that sin α = 13 5, 0 < α < 2 π, find the exact value of cos α, cos 2α. Given also that 13 cos (x + α) + 5 sin x = 6, and 0 < α < 2 π, find the value of x. (5) (Total 14 marks) Edexcel Internal Review 1

23 C2 Trigonometry: Sine & Cosine Rule 3. C 5 cm 4 cm A 6 cm The diagram above shows the triangle AC, with A = 6 cm, C = 4 cm and CA = 5 cm. 3 Show that cos A =. 4 Hence, or otherwise, find the exact value of sin A. (Total 5 marks) 4. In the triangle AC, A = 8 cm, AC = 7 cm, AC = 0.5 radians and AC = x radians. Use the sine rule to find the value of sin x, giving your answer to 3 decimal places. Given that there are two possible values of x, find these values of x, giving your answers to 2 decimal places. (Total 6 marks) Edexcel Internal Review 3