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Date None Specimen Marks available 5 Reference code SPNone.2.sl.TZ0.7
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 7 Adapted from N/A

Question

A circle centre O and radius \(r\) is shown below. The chord [AB] divides the area of the circle into two parts. Angle AOB is \(\theta \) .


Find an expression for the area of the shaded region.

[3]
a.

The chord [AB] divides the area of the circle in the ratio 1:7. Find the value of \(\theta \) .

[5]
b.

Markscheme

substitution into formula for area of triangle     A1

e.g. \(\frac{1}{2}r \times r\sin \theta \)

evidence of subtraction     M1

correct expression     A1     N2

e.g. \(\frac{1}{2}{r^2}\theta  - \frac{1}{2}{r^2}\sin \theta \) , \(\frac{1}{2}{r^2}(\theta  - \sin \theta )\)

[3 marks]

a.

evidence of recognizing that shaded area is \(\frac{1}{8}\) of area of circle     M1

e.g. \(\frac{1}{8}\) seen anywhere

setting up correct equation     A1

e.g. \(\frac{1}{2}{r^2}(\theta - \sin \theta ) = \frac{1}{8}\pi {r^2}\)

eliminating 1 variable     M1

e.g. \(\frac{1}{2}(\theta - \sin \theta ) = \frac{1}{8}\pi \) , \(\theta - \sin \theta  = \frac{\pi }{4}\)

attempt to solve     M1

e.g. a sketch, writing \(\sin x - x + \frac{\pi }{4} = 0\)

\(\theta  = 1.77\)  (do not accept degrees)     A1     N1

[5 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.5 » Solving trigonometric equations in a finite interval, both graphically and analytically.
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