Past Exam Questions

The diagram shows the sector \(AOB\) of a circle, centre \(O\) and radius \(15\) cm. Angle \(AOB\) is \(\frac{\pi}{6}\) radians. Point \(C\) lies on \(OB\) such that \(CB\) is \(a\) cm. \(AC\) is a straight line.

(a) Find the exact value of \(a\) such that the area of triangle \(AOC\) is equal to the area of the shaded region \(ACB\).

(b) For the value of \(a\) found in part (a), find the perimeter of the shaded region. Give your answer correct to \(1\) decimal place.

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

(a) The diagram shows two sectors, \(AOB\) and \(COD\), with common centre \(O\). The angle \(AOB\) is \(\frac{3\pi}{8}\), \(OC=6.5\), and \(OA:OC=4:5\). Find the perimeter of the shaded region.

(b) The diagram shows a circle with centre \(O\) and radius \(a\). The sector \(PQR\) is part of a circle with centre \(R\) and radius \(y\). The angle \(OPR\) is \(\phi\). Find the total area of the three shaded regions in terms of \(a\) and \(\phi\), giving your answer in its simplest form.

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

The diagram shows a circle, centre \(O\), radius \(8\). The points \(A\), \(B\) and \(C\) lie on the circumference of the circle. The chord \(AB\) has length \(10\).

(a) Show that angle \(BOA\) is \(1.35\) correct to \(2\) decimal places.

(b) Given that the minor arc \(BC\) has a length of \(18\), find angle \(BOC\).

(c) Find the area of the minor sector \(AOC\).

In this question all lengths are in centimetres.

The diagram shows a circle, centre \(O\), radius \(a\). The lines \(PT\) and \(QT\) are tangents to the circle at \(P\) and \(Q\) respectively. Angle \(POQ\) is \(2\phi\) radians.

(a) In the case when the area of the sector \(OPQ\) is equal to the area of the shaded region, show that \(\tan\phi=2\phi\).

(b) In the case when the perimeter of the sector \(OPQ\) is equal to half the perimeter of the shaded region, find an expression for \(\tan\phi\) in terms of \(\phi\).

The diagram shows a circle with centre \(O\) and radius \(r\). \(OAB\) and \(OCD\) are sectors of a circle with centre \(O\) and radius \(x\), where \(0\lt x\lt r\). Angle \(AOB=\) angle \(COD=\theta\) radians, where \(0\lt \theta\lt \pi\).

(a) Find, in terms of \(r\), \(x\) and \(\theta\), the perimeter of the shaded region.

(b) Find, in terms of \(r\), \(x\) and \(\theta\), the area of the shaded region.

It is given that \(x\) can vary and that \(r\) and \(\theta\) are constant.

(c) Write down the least possible area of the shaded region in terms of \(r\) and \(\theta\).

The diagram shows a circle, centre \(O\), radius \(10\) cm. The points \(A\) and \(B\) lie on the circumference of the circle. The tangent at \(A\) and the tangent at \(B\) meet at the point \(C\). The angle \(AOB\) is \(\theta\) radians. The length of the minor arc \(AB\) is \(28\) cm.

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

(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 circle, centre \(O\), radius \(7\). The points \(A\) and \(B\) lie on the circumference of the circle. The line \(BC\) is a tangent to the circle at the point \(B\) such that the length of \(BC\) is \(24\). The length of the minor arc \(AB\) is \(12.25\).

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

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

\(AOB\) is a sector of a circle with centre \(O\) and radius \(16\text{ cm}\). Angle \(AOB\) is \(\dfrac{2\pi}{7}\) radians. The point \(C\) lies on \(OB\) such that \(OC\) is \(7.5\text{ cm}\), and \(AC\) is a straight line.

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

In this question all lengths are in centimetres.

The diagram shows a shaded shape. The arc \(AB\) is the major arc of a circle, centre \(O\), radius \(10\). The line \(AB\) is of length \(15\), the line \(OC\) is of length \(25\) and the lengths of \(AC\) and \(BC\) are equal.

(a) Show that the angle \(AOB\) is \(1.70\) radians correct to 2 decimal places.

(b) Find the perimeter of the shaded shape.

(c) Find the area of the shaded shape.

The diagram shows a circle, centre \(O\), radius \(2a\). The points \(A\) and \(B\) lie on the circumference of the circle. The points \(C\) and \(D\) are the mid-points of the lines \(OB\) and \(OA\) respectively. The arc \(DC\) is part of a circle centre \(O\). The chord \(AB\) is of length \(2a\).

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

(b) Find, in terms of \(a\) and \(\pi\), the perimeter of the shaded region \(ABCD\).

(c) Find, in terms of \(a\) and \(\pi\), the area of the shaded region \(ABCD\).

\(DAB\) is a sector of a circle, centre \(A\), radius \(18\) cm. The lines \(CB\) and \(CD\) are tangents to the circle. Angle \(DAB\) is \(\frac{7\pi}{9}\) radians.

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(5\text{ cm}\). The lines \(AOB\) and \(COD\) are diameters of this circle. The line \(AC\) has length \(6\text{ cm}\).

(a) Show that angle \(AOC=1.287\) radians, correct to 3 decimal places.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(12\text{ cm}\), and a rectangle \(ABCD\). The diagonals \(AC\) and \(BD\) intersect at \(O\). The sides \(AB\) and \(AD\) of the rectangle have lengths \(6\text{ cm}\) and \(4\text{ cm}\) respectively. The points \(M\) and \(N\) lie on the circumference of the circle such that \(MAC\) and \(NDB\) are straight lines.

(a) Show that angle \(AOD\) is \(1.176\) radians correct to 3 decimal places.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(10\text{ cm}\). The points \(A\), \(B\) and \(P\) lie on the circumference of the circle. The chord \(AB\) is of length \(14\text{ cm}\). The point \(Q\) lies on \(AB\) and the line \(POQ\) is perpendicular to \(AB\).

(a) Show that angle \(POA\) is \(2.366\) radians, correct to 3 decimal places.

(b) Find the area of the shaded region.

(c) Find the perimeter of the shaded region.

(a) A sector of a circle has radius \(6\) cm and perimeter \(2(6+5\pi)\) cm. Find the area of the sector.

(b) The diagram shows a sector \(AOB\) of a circle with centre \(O\), radius \(7\) cm and angle \(\angle AOB=\frac14\pi\). Find the perimeter of the shaded region.

The diagram shows an isosceles triangle \(OAB\) such that \(OA=OB\) and angle \(AOB=\theta\) radians. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively. \(CD\) is an arc of length \(9.6\) cm of the circle, centre \(O\), radius \(12\) cm. The arc \(CD\) touches the line \(AB\) at the point \(M\).

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

(b) Find the total area of the shaded regions.

(c) Find the total perimeter of the shaded regions.

The diagram shows an isosceles triangle \(OAB\) such that \(OA=OB=12\) cm and angle \(AOB=\theta\) radians. Points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively such that \(CD\) is an arc of the circle, centre \(O\), radius \(10\) cm. The area of the sector \(OCD\) is \(35\text{ cm}^2\).

(a) Show that \(\theta=0.7\).

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The circles with centres \(C_1\) and \(C_2\) have equal radii of length \(r\) cm. The line \(C_1C_2\) is a radius of both circles. The two circles intersect at \(A\) and \(B\).

(a) Given that the perimeter of the shaded region is \(4\pi\) cm, find the value of \(r\).

(b) Find the exact area of the shaded region.

The diagram shows a shaded region in a rectangle. The arcs \(AB\) and \(CD\) are parts of circles with centres \(O\), and \(BC\) is an arc of a circle with centre \(O\). The radius is \(r\), and the length of arc \(BC\) is \(1.5r\).

(a) Find an expression for the perimeter of the shaded region in terms of \(r\).

(b) Find an expression for the area of the shaded region in the form \(kr^2\), where \(k\) is a constant to be found correct to 3 significant figures.

In this question all lengths are in centimetres.

The diagram shows the figure \(ABC\). The arc \(AB\) is part of a circle, centre \(O\), radius \(r\), and is of length \(1.45r\). The point \(O\) lies on the straight line \(CB\) such that \(CO=0.5r\).

(a) Find, in radians, the angle \(AOB\).

(b) Find the area of \(ABC\), giving your answer in the form \(kr^2\), where \(k\) is a constant.

(c) Given that the perimeter of \(ABC\) is \(12\) cm, find the value of \(r\).