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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Standard |
The following diagram shows a circle with centre O and radius 9 cm.
The points A, B and C lie on the circumference of the circle, and AÔC = 1.2 radians.
(a) Find the length of the arc ABC.
(b) Find the perimeter of the minor sector OAC.
(c) Find the area of the minor sector OAC.
2. | IB Analysis and Approaches |
The perimeter of the sector is 8 cm and the area of the sector is 3 cm2.
(a) Show that \(r^2 - 4r + 3 = 0\).
(b) Hence, or otherwise, find the value of \(r\) and the value of \(\theta\).
3. | IB Analysis and Approaches |
The diagram shows a circle with a minor sector shaded blue and a major sector shaded yellow.
The angle subtended by the major sector at the centre of the circle is six radians.
The perimeter of the major sector is 32.
(a) Find the value of r.
(b) Hence find the exact area of the minor sector.
4. | IB Analysis and Approaches |
The following diagram (not to scale) shows a circle with centre O and radius 7 cm.
The points A, B and C lie on the circumference of the circle and reflex angle \(A\hat{O}C = \theta\) where \( \theta\) is measured in radians.
The length of arc ABC is 30cm.
(a) Find the perimeter of the sector shaded red.
(b) Find \( \theta \).
(c) Find the area of the sector shaded red.
5. | IB Analysis and Approaches |
The following diagram shows part of a circle with centre O and radius 10cm.
Chord PQ has a length of 12cm and angle PÔQ = θ.
(a) Find the value of θ, giving your answer in radians.
(b) Find the area of the sector shaded green.
6. | IB Analysis and Approaches |
The following diagram shows a semicircle with centre O. Points A. B and C lie on the circumference of the circle and AC is a diameter. Angle BOC = \(\theta\), where \(0 < \theta < \frac{\pi}{2}\).
Given that the area of the triangle shaded in red is twice the size of the area of the segment shaded in blue, find the value of \(\theta\).
7. | IB Analysis and Approaches |
An Asian water buffalo is tethered in a rectangular field by a rope of length \(r\) metres. One end of the rope is securely tied to point \(P\) as shown in this diagram (not drawn to scale).
Points \(Q\) and \(R\) on the fence enclosing the field, are directly opposite each other and are the furthest points the buffalo can reach on the edge of the field. \(PQ =PR = r\) and the angle \(Q\hat{P}R = \theta\) radians. The length of arc QR shown is 38m.
(a) Write down an expression for \(r\) in terms of \(\theta\).
(b) Show that the area of the field that the buffalo can reach is \( \frac{1444}{\theta^2} ( \frac{\theta}{2} + \sin \theta) \)
(c) The area of the field that the buffalo can reach is 900m2. Find the value of \(\theta\).
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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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