Past Exam Questions

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows a sector of a circle with centre \(O\) and radius 6 . (a) It is given that the area of triangle \(A O B\) is \(9 \mathrm{~cm}^{2}\).

Find the value of \(\sin \theta\).

(b) It is also given that the exact area of the shaded segment is \((15 \pi-9) \mathrm{cm}^{2}\).

Find the exact length of the arc \(A B\).

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows a sector \(A O B\) of a circle, centre \(O\), radius 15 . Angle \(A O B=\frac{6 \pi}{5}\). The sector is made into a cone with points \(A\) and \(B\) touching, as shown. (a) Find the curved surface area of the cone.

The top of the cone is a horizontal circle. (b) Find the circumference of the circular top.

(c) Hence find the radius of the circular top and the perpendicular height of the cone.

(d) Water is poured into the cone.

When the depth of the water in the cone is \(h\), the radius of the circular top of the water is \(r\). (i) Find an expression for \(r\) in terms of \(h\).

(ii) The water is poured into the cone at a constant rate of \(27 \mathrm{~cm}^{3}\) per second.

Find the rate at which the depth of the water is rising when the depth of the water is 4 .

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows a circle with centre \(O\) and radius 12, and a circle with centre \(C\) and radius 5 . The circles intersect at the points \(A\) and \(B\), such that \(O A\) and \(O B\) are tangents to the circle with centre \(C\). (a) Show that the obtuse angle \(A C B\) is 2.35 radians, correct to 2 decimal places.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

DO NOT USE A CALCULATOR IN THIS QUESTION. In this question, all lengths are in centimetres.

The diagram shows the trapezium \(A B C D\). The lengths of \(A B, B C\) and \(C D\) are \(8 \sqrt{7}-7, \sqrt{7}+2\) and \(9 \sqrt{7}-9\) respectively. The line \(B C\) is perpendicular to the lines \(A B\) and \(C D\). (a) Find the perimeter of the trapezium, giving your answer in its simplest form.

(b) Find the area of the trapezium, giving your answer in the form \(p \sqrt{7}+q\), where \(p\) and \(q\) are rational numbers.

(c) Find \(\cot D B C\), giving your answer in the form \(r \sqrt{7}+s\), where \(r\) and \(s\) are simplified rational numbers.

A circle, centre \(O\) and radius \(r\) cm, has a sector \(OAB\) of fixed area \(10\text{ cm}^2\). Angle \(AOB\) is \(\theta\) radians and the perimeter is \(P\) cm.

(a) Find \(P\) in terms of \(r\).

(b) Find \(r\) for which \(P\) has a stationary value.

(c) Determine the nature of this stationary value.

(d) Find \(\theta\) at this stationary value.

In this question the units are metres.

The diagram shows a circle, centre \(O\) and radius 2 .
The chord \(A B\) has length \(2 \sqrt{3}\).
The point \(Q\) lies on the circle such that \(A Q=B Q\).
The \(\operatorname{arc} A P B\) is part of a circle, centre \(Q\).
(a) Find the exact value of angle \(A Q B\) in radians.

(b) Hence find the area of the shaded region. Give your answer in terms of \(\pi\).

The diagram shows the shaded region \(ABCD\). The lines \(AC\) and \(BD\) each have length \(12\text{ cm}\) and bisect each other at \(O\). The lines \(AD\) and \(BC\) are parallel and each has length \(4\text{ cm}\). The arcs \(AB\) and \(DC\) are part of a circle with centre \(O\).

(a) Find the obtuse angle \(AOB\), giving your answer in radians.

(b) Use your answer to part (a) to find

(i) the perimeter of the shaded region,

(ii) the area of the shaded region.

(a) The diagram shows an isosceles triangle. Find the value of \(\theta\) in radians.

(b) The diagram shows a shape made of two arcs. Each arc is part of a circle with radius \(5\text{ cm}\). The height of the shape is \(8\text{ cm}\).

Use your answer to part (a) to find (i) the perimeter of the shape and (ii) the area of the shape.

The diagram shows a circle with centre \(O\) and radius \(5\text{ cm}\).

The point \(A\) lies on the circle. The point \(B\) is such that the line \(AB\) is a tangent to the circle. \(OB\) has length \(13\text{ cm}\).

(a) Find angle \(AOB\), giving your answer in radians.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

In this question all lengths are in metres.

The diagram shows a shape \(A B C D E F\). \(A B, B D\) and \(D E\) are three sides of a rectangle. \(O\) is the mid-point of \(B D\). \(A F E\) is an arc of a circle whose centre is \(O\). \(A B=\sqrt{3}, B C=C D=5\) and \(B D=6\). (a) Find the exact value of the perimeter of the shape, giving your answer in terms of \(\pi\).

(b) Find the exact value of the area of the shape, giving your answer in terms of \(\pi\).

The diagram shows a design for a logo. The logo is a sector of a circle, radius \(r \mathrm{~cm}\), with angle \(\alpha\) radians.

The area of the logo is \(9 \mathrm{~cm}^{2}\). (a) Show that the perimeter, \(P \mathrm{~cm}\), of the logo is given by \(P=2 r+\frac{18}{r} .\) (b) Given that \(r\) can vary, find the stationary value of \(P\) and determine its nature.

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a company logo. Each part of the logo is a sector of a circle with centre \(O\). Sector \(A O B\) has radius \(x\). Sector \(C O D\) has radius \(x+2\). Sector \(E O F\) has radius \(y\). The shaded region has area \(A \mathrm{~cm}^{2}\) and perimeter 24 . It is given that \(x\) and \(y\) can vary. (a) Show that \(A=\frac{91}{8} x^{2}-68 x+132\).

(b) Use differentiation to find the minimum possible area of the logo.

In this question, all lengths are in metres and all angles are in radians.

The diagram shows a circle with centre \(O\) and radius 5. The points \(A, B, C\) and \(D\) lie on the circumference of the circle. Angle \(D O C=\theta\). Angle \(A O D=\) angle \(C O B=0.5\). The length of the minor \(\operatorname{arc} D C\) is 3.75 . (a) Show that \(\theta=0.75\).

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle centre \(O\) with radius \(6\). The line \(AB\) is a tangent to the circle at the point \(B\). The point \(C\) lies on the circle such that \(AOC\) is a straight line. \(AB=8\).

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

In the diagram, \(A D\) and \(B C\) are arcs of circles with common centre \(O\). \(O D C\) and \(O A B\) are straight lines with \(O A=5 \mathrm{~cm}\) and \(A B=4 \mathrm{~cm}\). Angle \(B O C=\theta\) radians. The area of the shaded region \(A B C D\) is \(4 \pi \mathrm{~cm}^{2}\). (a) Find \(\theta\).

(b)

The straight line \(A C\) is added to the diagram and the region \(A C D\) is now shaded. Find the perimeter of the shaded region \(A C D\).

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a circle with centre \(O\) and radius \(r\). The points \(A\) and \(B\) lie on the circumference of the circle such that the angle \(AOB\) is \(\theta\) and the length of the minor arc \(AB\) is 12. The area of the minor sector \(AOB\) is \(57.6\text{ cm}^2\). The point \(C\) lies on the tangent to the circle at \(A\) such that \(OBC\) is a straight line.

(a) Find the values of \(r\) and \(\theta\).

(b) Find the area of the shaded region. Give your answer correct to 1 decimal place.

In this question lengths are in centimetres and angles are in radians.

The diagram shows a circle with centre \(O\) and radius \(r\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the minor sector \(OAB\) is \(25\text{ cm}^2\). The angle \(AOB\) is \(\theta\).

(a) Find an expression for the perimeter, \(P\), of the minor sector \(OAB\), in terms of \(r\).

(b) Given that \(r\) can vary, show that \(P\) has a minimum value and find this minimum value.

In this question all lengths are in centimetres and all angles are in radians.

(a) The area of a sector of a circle of radius \(24\) is \(432\text{ cm}^2\). Find the length of the arc of the sector.

(b) The diagram shows an isosceles triangle \(OAB\), with \(AO=AB=y\) and height \(AD\). \(OCD\) is a sector of the circle with centre \(O\). Angle \(AOB\) is \(\alpha\).

(i) Find an expression for \(OB\) in terms of \(y\) and \(\alpha\).

(ii) Hence show that the area of the shaded region can be written as

\(\frac{y^2}{2}\cos\alpha(2\sin\alpha-\alpha\cos\alpha).\)

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a badge which consists of a minor sector \(OAB\), of the circle with centre \(O\) and radius 12, and a kite \(OBCD\), where \(OB=OD\) and \(CD=CB\). The arc \(AB\) has length 27. The line \(OB\) is perpendicular to the line \(CB\), and \(COA\) is a straight line.

(a) Find the perimeter of the badge.

(b) Find the area of the badge.

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows the sector \(OAB\) of a circle with centre \(O\) and radius \(20\). The perimeter of this sector is \(65\). The lines \(CA\) and \(CB\) are both tangents to the circle at the points \(A\) and \(B\), so that the triangle \(ABC\) is isosceles, with \(AC=CB\). The angle \(AOB\) is equal to \(\theta\).

Find the area of the shaded region.