[Maximum number: 6]

The following diagram shows a triangle ABC .

\mathrm{AB}=5 \mathrm{~cm}, \hat{\mathrm{~A}} \mathrm{~A}=50^{\circ} \text { and } \mathrm{A} \hat{\mathrm{C}} \mathrm{~B}=112^{\circ}

(a)

Find BC .

[ 3 ]

(b)

Find the area of triangle ABC .

[ 3 ]

[Maximum number: 6]

The following diagram shows a circle with centre O and radius 3 cm .

diagram not to scale

Points A, B, and C lie on the circle, and $\mathrm{A} \hat{\mathrm{O}} \mathrm{C}=1.3$ radians.

(a)

Find the length of arc ABC .

[ 2 ]

(b)

Find the area of the shaded region.

[ 4 ]

[Maximum number: 10]

The following diagram shows a circle with centre O and radius 5 cm .

The points A, B and C lie on the circumference of the circle, and $\mathrm{A} \hat{\mathrm{O}} \mathrm{C}=0.7$ radians.

(a)

(i)

Find the length of the arc ABC .

[ 4 ]

(ii)

Find the perimeter of the shaded sector.

[ 4 ]

(b)

Find the area of the shaded sector.

[ 2 ]

[Maximum number: 2]

Point P has coordinates (-3,2), and point Q has coordinates (15,-8). Point M is the midpoint of [PQ].

(a)

Point P has coordinates (-3,2), and point Q has coordinates (15,-8). Point M is the midpoint of [PQ].

Find the coordinates of M .

Line L is perpendicular to $[\mathrm{PQ}]$ and passes through M .

[ 2 ]

[Maximum number: 6]

The following diagram shows a regular pentagon inscribed in a circle with centre O and radius $r \mathrm{~cm}$ .
The angle $\mathrm{A} \hat{\mathrm{O}} \mathrm{B}$ is $\theta$ , where $\theta$ is measured in radians.
The $\operatorname{arc} A B$ is 12 cm .

(a)

Find

[ 3 ]

(i)

$\theta$ ;

(ii)

r.

[ 3 ]

(b)

Find the area of the shaded region.

[ 3 ]

[Maximum number: 6]

The following diagram shows triangle ABC .
$\mathrm{AB}=7 \mathrm{~cm}, \mathrm{BC}=9 \mathrm{~cm}$ and $\mathrm{A} \hat{\mathrm{B}} \mathrm{C}=120^{\circ}$ .

(a)

Find AC .

[ 3 ]

(b)

Find $\mathrm{B} \hat{\mathrm{A}} \mathrm{C}$ .

[ 3 ]

[Maximum number: 5]

The following diagram shows a right-angled triangle, ABC , where $\sin A=\frac{5}{13}$ .

(a)

Show that $\cos A=\frac{12}{13}$ .

[ 2 ]

(b)

Find $\cos 2 A$ .

[ 3 ]

[Maximum number: 6]

The following diagram shows triangle ABC .

\mathrm{AB}=6 \mathrm{~cm}, \mathrm{BC}=10 \mathrm{~cm}, \text { and } \mathrm{A} \hat{\mathrm{~B}} \mathrm{C}=100^{\circ}

(a)

Find AC.

[ 3 ]

(b)

Find $B \hat{C} A$ .

[ 3 ]

[Maximum number: 6]

The following diagram shows triangle ABC .

\mathrm{BC}=10 \mathrm{~cm}, \mathrm{~A} \hat{\mathrm{~B} \mathrm{C}}=80^{\circ} \text { and } \mathrm{B} \hat{\mathrm{~A}} \mathrm{C}=35^{\circ} .

(a)

Find AC.

[ 3 ]

(b)

Find the area of triangle ABC .

[ 3 ]

[Maximum number: 6]

The following diagram shows a circle with centre O and radius r.

Points A and B lie on the circumference of the circle, and $\mathrm{A} \hat{\mathrm{O}} \mathrm{B}=1$ radian .
The perimeter of the shaded region is 12 .

(a)

Find the value of r.

[ 3 ]

(b)

Hence, find the exact area of the non-shaded region.

[ 3 ]

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