CAIE IGCSE Additional Math 9 Circular Measure

[Maximum number: 7]

(a)

Figure for Question (a) — CAIE IGCSE Additional Math

The diagram shows an isosceles triangle.
Find the value of $\theta$ in radians.

[ 2 ]

$2\sin^{-1}\left(\frac{4}{5}\right)$ or $\cos^{-1}\left(\frac{64-25-25}{-2(5)(5)}\right)$ oe
1.85 or 1.854 to 1.855

(b)

Figure for Question (b) — CAIE IGCSE Additional Math

This diagram shows a shape made of two arcs.
Each arc is part of a circle with radius 5 cm.
The height of the shape is 8 cm.
Use your answer to part (a) to find

[ 5 ]

(i)

the perimeter of the shape

[ 2 ]

$5\times\text{their }1.85\times 2$
M1 FT their 1.85 providing it is less than $2\pi$ .
18.5 or 18.54 to 18.55
A1

(ii)

the area of the shape.

[ 3 ]

$\frac{1}{2}\times5^2\times\text{their }1.85$
M1 FT their 1.85 providing it is less than $2\pi$ .
Correct plan, e.g. $[2\times\frac{1}{2}\times]5^2(\text{their }1.85-\sin(\text{their }1.85))$
M1 dep on previous M1.
22.4 or 22.21 to 22.38 nfww
A1

[Maximum number: 8]

Figure for Question 2 — CAIE IGCSE Additional Math

The diagram shows the shape OABCDEF.
AOF is a straight line.
OAB and OEF are sectors of a circle with centre O and radius r.
Angle B O A= angle EOF.
OCD is a sector of a circle with centre O and radius $\frac{4 r}{3}$ .
Angle COD is $\theta$ radians.
The point B lies on the line OC and the point E lies on the line OD. The line BE is parallel to the line AOF.

(a)

Find, in terms of r and $\theta$ , the area of the shaded region BCDE.

[ 3 ]

$\frac{1}{2}\left(\frac{4 r}{3}\right)^{2} \theta-\frac{1}{2} r^{2} \sin \theta$ ое

B1 for $\frac{1}{2}\left(\frac{4 r}{3}\right)^{2} \theta$

B1 for $\frac{1}{2} r^{2} \sin \theta$ or $r^{2} \sin \frac{\theta}{2} \cos \frac{\theta}{2}$ or $\frac{r^{2} \sin ^{2} \frac{\theta}{2}}{\tan \frac{\theta}{2}}$ oe

Allow final answer unsimplified

(b)

The diagram shows the shape from part (a) with region OABEF shaded. Find, in terms of r and $\theta$ , the perimeter of the shaded region.

[ 5 ]

Arc length $=r \frac{(\pi-\theta)}{2}$

B1

$B E=\sqrt{2 r^{2}(1-\cos \theta)}$ or $2 r \sin \frac{\theta}{2}$ or
$\frac{r \sin \theta}{\sin \left(\frac{\pi-\theta}{2}\right)}$ oe

M1 for complete method to find BE or $B E^{2}$ , using either cosine rule, sine rule or basic trig using r and a correct angle.

A0 if an error in rearranging to obtain BE is made.

Allow unsimplified

Perimeter $=r(\pi-\theta)+2 r+\sqrt{2 r^{2}(1-\cos \theta)}$
oe

M1 for a correct plan using their lengths

2 arc lengths +2 r+ their BE Allow A1 for an unsimplified answer

[Maximum number: 6]

Figure for Question 3 — CAIE IGCSE Additional Math

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

(a)

Find the perimeter of the shaded region.

[ 3 ]

25.9 cm

M1 for A B=12 soi

M1 for arc length $=5 \times$ their AOB in radians

(b)

Find the area of the shaded region.

[ 3 ]

Area $=15.3 \mathrm{~cm}^{2}$

M1 for area of triangle = 30 soi

M1 for area of sector = $\frac{1}{2} \times 25 \times$ theirAOB in radians

CAIE IGCSE Additional Math 9 Circular Measure Topic Practice

Covered syllabus points

CAIE IGCSE Additional Math 9 Circular Measure Topic Practice

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)