Question

Assertion (A)
An arc of a circle of length 5π cm bounds a sector whose area is 20π cm². Then, the radius of the circle is 4 cm.

Reason (R)
A chord of a circle of radius 12 cm subtends an angle of 60° at the centre of the circle. The area of the minor segment of the circle is 13.08 cm².

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) Reason (R) true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

(d) Assertion (A) is false and reason (R) is true.
Assertion (A):
Let r be the radius of the circle.
We have:
Arc length$=\frac{\mathrm{\pi }r\theta }{180}$
Now,

$$\begin{gathered} \frac{\mathrm{\pi }r\theta }{180}=5\mathrm{\pi } \\ \Rightarrow \frac{r\mathrm{\theta }}{180}=5 \\ \Rightarrow r\mathrm{\theta }=\left(5\times 180\right) \end{gathered}$$

Area of the sector$=\frac{{\mathrm{\pi r}}^2\mathrm{\theta }}{360}$
Thus, we have:
$\frac{\mathrm{\pi }{r}^2\theta }{360}=20\mathrm{\pi }\Rightarrow \frac{{\mathrm{r}}^2\mathrm{\theta }}{360}=20$

$\Rightarrow r^2\mathrm{\theta }=\left(20\times 360\right)$

Now,
$\frac{r^2\mathrm{\theta }}{r\mathrm{\theta }}=\frac{\left(20\times 360\right)}{\left(5\times 180\right)}$
∴ r = 8 cm

Hence, assertion (A) is false.

Reason (R):
Let r be the radius of the circle.
Now,
Area of the minor segment$=\left(\frac{\mathrm{\pi }{r}^2\theta }{360}-\frac{1}{2}r\right)$²sin θ
$$\begin{gathered} =\left|\left(3.14\times 12\times 12\times {\displaystyle \frac{60}{360}}\right)-\left({\displaystyle \frac{1}{2}}\times 12\times 12\times \sin 60°\right)\right|{\mathrm{cm}}^2 \\ =\left|\left(75.36\right)-\left(36\times 1.73\right)\right|{\mathrm{cm}}^2 \\ =\left(75.36-62.28\right){\mathrm{cm}}^2 \\ =13.08{\mathrm{cm}}^2 \end{gathered}$$

Hence, reason (R) is true.