[Maximum number: 4]

In the diagram, A Y B is a semicircle with A B as diameter and O A X B is a sector of a circle with centre O and radius r. Angle $A O B=2 \theta$ radians. Find an expression, in terms of r and $\theta$ , for the area of the shaded region.

In the diagram, A B C is a triangle in which angle A B C is a right angle and B C=a. A circular arc, with centre C and radius a, joins B and the point M on A C. The angle A C B is $\theta$ radians. The area of the sector C M B is equal to one third of the area of the triangle A B C.

(a)

Show that $\theta$ satisfies the equation

\tan \theta=3 \theta .

[Maximum number: 6]

In the diagram, O A D C is a sector of a circle with centre O and radius $3 \mathrm{~cm} . A B$ and C B are tangents to the circle and angle $A B C=\frac{1}{3} \pi$ radians. Find, giving your answer in terms of $\sqrt{ } 3$ and $\pi$ ,

(a)

the perimeter of the shaded region,

[ 3 ]

(b)

the area of the shaded region.

[ 3 ]

[Maximum number: 6]

The diagram shows a triangle A O B in which O A is $12 \mathrm{~cm}, O B$ is 5 cm and angle A O B is a right angle. Point P lies on A B and O P is an arc of a circle with centre A. Point Q lies on A B and O Q is an arc of a circle with centre B.

(a)

Show that angle B A O is 0.3948 radians, correct to 4 decimal places.

[ 1 ]

(b)

Calculate the area of the shaded region.

[ 5 ]

[Maximum number: 6]

In the diagram, C X D is a semicircle of radius 7 cm with centre A and diameter C D. The straight line Y A B X is perpendicular to C D, and the $\operatorname{arc} C Y D$ is part of a circle with centre B and radius 8 cm . Find the total area of the region enclosed by the two arcs.

[Maximum number: 6]

The diagram shows an arc B C of a circle with centre A and radius 5 cm . The length of the arc B C is 4 cm . The point D is such that the line B D is perpendicular to B A and D C is parallel to B A.

(a)

Find angle B A C in radians.

[ 1 ]

(b)

Find the area of the shaded region B D C.

[ 5 ]

[Maximum number: 4]

In the diagram, D lies on the side A B of triangle A B C and C D is an arc of a circle with centre A and radius 2 cm . The line B C is of length $2 \sqrt{ } 3 \mathrm{~cm}$ and is perpendicular to A C. Find the area of the shaded region B D C, giving your answer in terms of $\pi$ and $\sqrt{ } 3$ .

[Maximum number: 7]

The diagram shows a circle with centre O and radius $r \mathrm{~cm}$ . Points A and B lie on the circle and angle $A O B=2 \theta$ radians. The tangents to the circle at A and B meet at T.

(a)

Express the perimeter of the shaded region in terms of r and $\theta$ .

[ 3 ]

(b)

In the case where r=5 and $\theta=1.2$ , find the area of the shaded region.

[ 4 ]

[Maximum number: 6]

Fig. 1

Fig. 2

Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm . The paper is cut from A to O and opened out to form the sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The angle of the sector is $\theta$ radians. Calculate

(a)

the value of $\theta$ ,

[ 4 ]

(b)

the area of paper needed to make the cone.

[ 2 ]

[Maximum number: 5]

In the diagram, O A B is a sector of a circle with centre O and radius 8 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.

(a)

Find $\alpha$ in terms of $\pi$ .

[ 3 ]

(b)

Find the perimeter of the complete figure in terms of $\pi$ .

[ 2 ]