Radians (Edexcel IGCSE Further Pure Maths): Exam Questions

Figure 1 shows the sector $AOB$ of a circle with centre $O$ and radius $3r$ cm

A circle with radius $r$ cm touches $OA$ and $OB$ and the arc $AB$

Angle $AOB$is $\theta$ radians, where $0<\theta <\frac{\mathrm{\pi }}{2}$

Find the exact value of $\theta$

The area of the region shown shaded in Figure 1 is $8\pi$ cm²

Find the value of $r$

Figure 1 shows sector $ROS$ of a circle with centre $O$ and radius 2 cm
The size of angle $ROS$ is $\theta$ radians.

The area of sector $ROS$ is $\frac{\mathrm{\pi }}{2}{\mathrm{cm}}^2$. Find the exact value of $\theta$

Figure 1 shows sector $ROS$ of a circle with centre $O$ and radius 2 cm
The size of angle $ROS$ is $\theta$ radians.

The perimeter of sector $ROS$ is $P$ cm. Find the exact value of $P$

Figure 1 shows a circle, centre $O$, with radius $r$ cm.

The points $A$, $P$ and $B$ lie on the circle.

The obtuse angle $AOB=\theta$ radians.

The area of the sector $APBO$, shown shaded, is 372.4 cm² and the length of the arc $APB$ is 53.2cm.

Find, to 3 significant figures where appropriate, the value of

(i) $r$

(ii) $\theta$

A logo, $AEBCD$, is shown shaded in Figure 2.

The straight line $ABC$ is the diameter of the semicircle $ADC$
$AEB$ is an arc of a circle with centre $O$
All angles are measured in radians.

• $BC=2x\mathrm{cm}$

• $OA=OB=x\mathrm{cm}$

• $\mathrm{length}\mathrm{of}\mathrm{arc}AEB\mathit{=}1.8x\mathrm{cm}$

The perimeter of the logo is $P$

Show that $P=ax(\pi +\pi \sin 0.9+b)$ where $a$ and $b$ are constants to be found.

Given that $x=10\mathrm{cm}$,

find, in cm² to 3 significant figures, the area of the logo.

Figure 1 shows a badge, shown shaded, made from two identical rectangles, $ABCD$ and $DEFG$, and a sector $DCG$ of a circle with centre $D$.

Each rectangle measures $x$ cm by $y$ cm.
The radius of the sector is $x$ cm and the angle $CDG$ is 0.5 radians.

The area of the badge is 50 cm²
The perimeter of the badge is $P$ cm.

Show that

$P=2x+\frac{100}{x}$

Given that $x$ can vary,

use calculus, to find the exact value of $x$ for which $P$ is a minimum.
Justify that this value of $x$ gives a minimum value for $P$

Find the minimum value of $P$
Give your answer in the form $k\sqrt{2}$, where $k$ is an integer to be found.