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P1 January 2001

Fig. 3

Triangle ABC has AB = 9 cm, BC 10 cm and CA = 5 cm.

A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.

(a) Show that, to 3 decimal places, BAC = 1.504 radians. (3)

Calculate,

(b) the area, in cm2, of the sector APQ, (2)

(c) the area, in cm2, of the shaded region BPQC, (3)

(d) the perimeter, in cm, of the shaded region BPQC. (4)

P41487A 1 Turn over

10 cm

5 cm

9 cmP

Q

A

BC

P1 January 2002Figure 1

B C

A D

5m 7m

O

Figure 1 shows a gardener’s design for the shape of a flower bed with perimeter ABCD.

AD is an arc of a circle with centre O and radius 5 m.

BC is an arc of a circle with centre O and radius 7 m.

OAB and ODC are straight lines and the size of AOD is radians.

(a) Find, in terms of , an expression for the area of the flower bed.(3)

Given that the area of the flower bed is 15 m2,

(b) show that = 1.25,(2)

(c) calculate, in m, the perimeter of the flower bed.(3)

The gardener now decides to replace arc AD with the straight line AD.

(d) Find, to the nearest cm, the reduction in the perimeter of the flower bed.(2)

P41487A 2 Turn over

P1 June 2002Figure 1

The shape of a badge is a sector ABC of a circle with centre A and radius AB, as shown in Fig 1. The triangle ABC is equilateral and has a perpendicular height 3 cm.

(a) Find, in surd form, the length AB. (2)

(b) Find, in terms of , the area of the badge. (2)

(c) Prove that the perimeter of the badge is

2√33

(π+6) cm.

(3)

P1 January 2003

(ii) In the triangle ABC, AC = 18 cm, ABC = 60 and sin A = 13 .

(a) Use the sine rule to show that BC = 43. (4)

(b) Find the exact value of cos A.(2)

P41487A 3 Turn over

3 cm

CB

A


y

B C

A 14 M 14 D x

Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M.

Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.

(a) Show that the length of BM is 25 mm. (1)

(b) Show that, to 3 significant figures, BMC = 0.568 radians. (3)

(c) Hence calculate, in mm2, the area of the cross-section of the chocolate bar.(5)

Given that this chocolate bar has length 85 mm,

(d) calculate, to the nearest cm3, the volume of the bar. (2)

P41487A 4 Turn over

P1 June 2003Figure 1

Figure 1 shows the sector OAB of a circle of radius r cm. The area of the sector is 15 cm2

and AOB = 1.5 radians.

(a) Prove that r = 25.(3)

(b) Find, in cm, the perimeter of the sector OAB.(2)

The segment R, shaded in Fig 1, is enclosed by the arc AB and the straight line AB.

(c) Calculate, to 3 decimal places, the area of R.(3)

P41487A 5 Turn over

rr

BA

O

R


O

6.5 cm

A B

Figure 1 shows the sector AOB of a circle, with centre O and radius 6.5 cm, and AOB = 0.8 radians.

(a) Calculate, in cm2, the area of the sector AOB. (2)

(b) Show that the length of the chord AB is 5.06 cm, to 3 significant figures.(3)

The segment R, shaded in Fig. 1, is enclosed by the arc AB and the straight line AB.

(c) Calculate, in cm, the perimeter of R. (2)

P41487A 6 Turn over


C

27 cmD

A 2 cm

B

In ΔABC, AB = 2 cm, AC = 6 cm and BC = 27 cm.

(a) Use the cosine rule to show that BAC =

π3 radians.

(3)

The circle with centre A and radius 2 cm intersects AC at the point D, as shown in Figure 2.

Calculate

(b) the length, in cm, of the arc BD,(2)

(c) the area, in cm2, of the shaded region BCD.(4)

P41487A 7 Turn over

A2 cm

D

6 cm

P1 January 2005 Figure 2

O

r cm

L M

A major sector LOM of a circle, with centre O and radius r cm, has LOM = radians, as shown in Figure 2. The perimeter of the sector is P cm and the area of the sector is A cm2.

(a) Write down, in terms of r and , expressions for P and A. (2)

Given that r = 22 and that P = A,

(b) show that =

2√ 2−1 .

(3)

(c) Express in the form a + b2, where a and b are integers to be found.(3)

P41487A 8 Turn over

P1 June 2005 Figure 1

A fence from a point A to a point B is in the shape of an arc AB of a circle with centre O and radius 45 m, as shown in Figure 1. The length of the fence is 63 m.

(a) Show that the size of AOB is exactly 1.4 radians. (2)

The points C and D are on the lines OB and OA respectively, with OC = OD = 30 m.

A plot of land ABCD, shown shaded in Figure 1, is enclosed by the arc AB and the straight lines BC, CD and DA.

(b) Calculate, to the nearest m2, the area of this plot of land. (5)

P41487A 9 Turn over

O B C

D

30 m

45 m

A


Figure 2 shows the cross-section ABC of a metal cutter used for making biscuits. The straight sides AB and AC are both of length 6 cm and BAC is 1.2 radians. The curved portion BC is an arc of a circle with centre A.

(a) Find the perimeter of the cross-section of the cutter.(2)

(b) Find an equation the area of the cross-section ABC.(2)

Figure 3

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 3.

(c) Find the value of a and the value of b, giving your answers to 3 significant figures.(4)

P41487A 10 Turn over

6 cm

A

B C

6 cm 1.2

1.2

1.2

a cm

b cm 6 cm

C2 January 2005 Figure 1

Figure 1 shows the triangle ABC, with AB = 8 cm, AC = 11 cm and BAC = 0.7 radians. The arc BD, where D lies on AC, is an arc of a circle with centre A and radius 8 cm. The region R, shown shaded in Figure 1, is bounded by the straight lines BC and CD and the arc BD.

Find

(a) the length of the arc BD,(2)

(b) the perimeter of R, giving your answer to 3 significant figures, (4)

(c) the area of R, giving your answer to 3 significant figures.(5)

C2 June 2005In the triangle ABC, AB = 8 cm, AC = 7 cm, ABC = 0.5 radians and ACB = x radians.

(a) Use the sine rule to find the value of sin x, giving your answer to 3 decimal places.(3)

Given that there are two possible values of x,

(b) find these values of x, giving your answers to 2 decimal places.(3)


A

D

11 cm0.7 rad

8 cm R C

B

C2 January 2006

Figure 2

In Figure 2 OAB is a sector of a circle, radius 5 m. The chord AB is 6 m long.

(a) Show that cosA O B =

725 .

(2)(b) Hence find the angleA O B in radians, giving your answer to 3 decimal places.

(1)(c) Calculate the area of the sector OAB.

(2)(d) Hence calculate the shaded area.

(3)


5 m

BA

5 m

6 m

O

C2 June 2006Figure 2

Figure 2 shows the cross-section ABCD of a small shed.

The straight line AB is vertical and has length 2.12 m.

The straight line AD is horizontal and has length 1.86 m.

The curve BC is an arc of a circle with centre A, and CD is a straight line.

Given that the size of BAC is 0.65 radians, find

(a) the length of the arc BC, in m, to 2 decimal places,(2)

(b) the area of the sector BAC, in m2, to 2 decimal places, (2)

(c) the size of CAD, in radians, to 2 decimal places,(2)

(d) the area of the cross-section ABCD of the shed, in m2, to 2 decimal places.(3)


B

C

A D

2.12 m

1.86 m

C2 January 2007Figure 2

Figure 2 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 63 m,

(a) find the exact size of angle PQR in radians.(3)

(b) Show that the area of the patio PQRS is 12 m2.(2)

(c) Find the exact area of the triangle PQR. (2)

(d) Find, in m2 to 1 decimal place, the area of the segment PRS.(2)

(e) Find, in m to 1 decimal place, the perimeter of the patio PQRS.(2)


R P

Q

S

63 m

6 m 6 m

C2 June 2007

Figure 1

Figure 1 shows the triangle ABC, with AB = 6 cm, BC = 4 cm and CA = 5 cm.

(a) Show that cos A = 34 .

(3)(b) Hence, or otherwise, find the exact value of sin A.

(2)


5 cm

6 cm

4 cm A

C

B

C2 January 2008Figure 1

Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane. Yacht B is 500 m due north of yacht A and yacht C is 700 m from A. The bearing of C from A is 015.

(a) Calculate the distance between yacht B and yacht C, in metres to 3 significant figures.(3)

The bearing of yacht C from yacht B is , as shown in Figure 1.

(b) Calculate the value of .(4)


C

700 m500 m

B

15

A

N

C2 June 2008

Figure 1

Figure 1 shows ABC, a sector of a circle with centre A and radius 7 cm.

Given that the size of BAC is exactly 0.8 radians, find

(a) the length of the arc BC,(2)

(b) the area of the sector ABC.(2)

The point D is the mid-point of AC. The region R, shown shaded in Figure 1, is bounded by CD, DB and the arc BC.

Find

(c) the perimeter of R, giving your answer to 3 significant figures,(4)

(d) the area of R, giving your answer to 3 significant figures.(4)


C2 January 2009

Figure 3

The shape BCD shown in Figure 3 is a design for a logo.

The straight lines DB and DC are equal in length. The curve BC is an arc of a circle with centre A and radius 6 cm. The size of BAC is 2.2 radians and AD = 4 cm.

Find

(a) the area of the sector BAC, in cm2,(2)

(b) the size of DAC, in radians to 3 significant figures,(2)

(c) the complete area of the logo design, to the nearest cm2.(4)


C2 January 2010

Figure 1

An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight line with AB = 5 cm and BD = 4 cm. Angle BAC = 0.6 radians and AC is the longest side of the triangle ABC.

(a) Show that angle ABC = 1.76 radians, correct to three significant figures.

(4)(b) Find the area of the emblem.

(3)


C

A DB

0.6 rad

4 cm

4 cm5 cm

C2 June 2010

Figure 1

Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians.

(a) Find the length of the arc AB.(2)

(b) Find the area of the sector OAB.(2)

The line AC shown in Figure 1 is perpendicular to OA, and OBC is a straight line.

(c) Find the length of AC, giving your answer to 2 decimal places.(2)

The region H is bounded by the arc AB and the lines AC and CB.

(d) Find the area of H, giving your answer to 2 decimal places.(3)

C2 January 2011In the triangle ABC, AB = 11 cm, BC = 7 cm and CA = 8 cm.

(a) Find the size of angle C, giving your answer in radians to 3 significant figures.(3)

(b) Find the area of triangle ABC, giving your answer in cm2 to 3 significant figures.


C2 June 2011

Figure 1

The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle

centre O, of radius 6 cm, and angle AOB =

π3 . The circle C, inside the sector, touches the two

straight edges, OA and OB, and the arc AB as shown.

Find

(a) the area of the sector OAB,(2)

(b) the radius of the circle C.(3)

The region outside the circle C and inside the sector OAB is shown shaded in Figure 1.

(c) Find the area of the shaded region.(2)


C2 January 2012

Figure 2

Figure 2 shows ABC, a sector of a circle of radius 6 cm with centre A. Given that the size of angle BAC is 0.95 radians, find

(a) the length of the arc BC,(2)

(b) the area of the sector ABC.(2)

The point D lies on the line AC and is such that AD = BD. The region R, shown shaded in Figure 2, is bounded by the lines CD, DB and the arc BC.

(c) Show that the length of AD is 5.16 cm to 3 significant figures.(2)

Find

(d) the perimeter of R,(2)

(e) the area of R, giving your answer to 2 significant figures.(4)


C2 January 2013

Figure 2

The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = . The point W lies on the line XY.

The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm.

(a) Show that, to 3 significant figures, = 2.22 radians.(2)

(b) Find the area, in cm2, of the major sector XZWX.(3)

The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shaded in Figure 1.

Calculate

(c) the area of this shaded region,(3)

(d) the perimeter ZWYZ of this shaded region.(4)


ANSWERS:

C2 January 2005(a) 5.6 cm (b) 15.7 cm (c) 5.95 cm2

C2 June 2005(a) 0.548 (b) 0.58, 2.56

C2 January 2006(b) 1.287 (c) 16.087 (d) 4.1

C2 June 2006(a) 1.38 m (b) 1.46 m2 (c) 0.92 radians (d) 3.03

C2 January 2007

(a)

2π3 (c) 93 m2 (d) 22.1 m2 (e) 24.6 m

C2 June 2007

(b)

14 √7

C2 January 2008(a) 253 m (b) = 45.8

C2 June 2008(a) 5.6 (b) 19.6 cm2 (c) 14.3 cm (d) = 10.8 cm2

C2 January 2009(a) 39.6 (b) 2.04 (c) 61

C2 January 2010(b) 20.9 cm2

C2 June 2010(a) 6.3 (b) 28.35 (c) 7.58 (d) 5.76

C2 January 2011(a) 1.64 (b) 27.9 cm2

C2 June 2011(a) 18.8 cm2 (b) r = 2 (c) 2 cm2


C2 January 2012(a) 5.7 cm (b) 17.1 cm2 (d) 11.7 cm (e) 4.5 cm2


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