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Question

Assertion (A)

The area of the quadrant of a circle having a circumference of 22 cm is $9\frac{5}{8}{\mathrm{cm}}^{2}.$

Reason (R)

The area of a sector of a circle of radius r with central angle x° is $\left(\frac{x\times \mathrm{\pi}{r}^{2}}{360}\right).$

(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.

The area of the quadrant of a circle having a circumference of 22 cm is $9\frac{5}{8}{\mathrm{cm}}^{2}.$

Reason (R)

The area of a sector of a circle of radius r with central angle x° is $\left(\frac{x\times \mathrm{\pi}{r}^{2}}{360}\right).$

(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.

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Solution

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Assertion (A):

Let r be the radius of the circle.

Now,

Circumference of the circle $=2\mathrm{\pi}r\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

We have:

$2\mathrm{\pi}r=22\phantom{\rule{0ex}{0ex}}\Rightarrow 2\times \frac{22}{7}\times r=22\phantom{\rule{0ex}{0ex}}\Rightarrow r=\left(22\times \frac{7}{44}\right)\mathrm{cm}\phantom{\rule{0ex}{0ex}}\Rightarrow r=\frac{7}{2}\mathrm{cm}$

Area of the quadrant$=\frac{90\xb0}{360\xb0}\mathrm{\pi}{r}^{2}$

$=\left(\frac{1}{4}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\right){\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=\frac{77}{8}{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=9\frac{5}{8}{\mathrm{cm}}^{2}$

Hence, assertion (A) is true.

Reason (R):

The given statement is true.

Assertion (A) is true and reason (R) is the correct explanation of assertion (A).

Assertion (A):

Let r be the radius of the circle.

Now,

Circumference of the circle $=2\mathrm{\pi}r\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

We have:

$2\mathrm{\pi}r=22\phantom{\rule{0ex}{0ex}}\Rightarrow 2\times \frac{22}{7}\times r=22\phantom{\rule{0ex}{0ex}}\Rightarrow r=\left(22\times \frac{7}{44}\right)\mathrm{cm}\phantom{\rule{0ex}{0ex}}\Rightarrow r=\frac{7}{2}\mathrm{cm}$

Area of the quadrant$=\frac{90\xb0}{360\xb0}\mathrm{\pi}{r}^{2}$

$=\left(\frac{1}{4}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\right){\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=\frac{77}{8}{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}=9\frac{5}{8}{\mathrm{cm}}^{2}$

Hence, assertion (A) is true.

Reason (R):

The given statement is true.

Assertion (A) is true and reason (R) is the correct explanation of assertion (A).

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