Assertion (A): Area of shaded region = \frac{154}{3}\mathrm{\thinsp}{\mathrm{cm}}^2 Reason (R): \angle MAN={120}^\circand area

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Solution

Q.
Assertion (A): Area of shaded region $= \frac{154}{3} {cm}^{2}$

Reason (R): $∠ M A N = 120 ° a r e a o f s e c t o r = \frac{θ}{360 °} π r^{2} .$

1 Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2 Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3 Assertion (A) is true but Reason (R) is false.

4 Assertion (A) is false but Reason (R) is true.

Ans.
(1)

From the figure,

$∠ D C B + ∠ C D A + ∠ M A B + ∠ A B C = 360 ° (Sum of angles of a quadrilateral) \Rightarrow 90 ° + 90 ° + ∠ M A B + ∠ A B C = 360 ° \Rightarrow ∠ M A B + ∠ A B C = 180 ° \Rightarrow ∠ M A N = 180 ° - ∠ A B C = 120 ° ∴ Area of the shaded region = \frac{120 °}{360 °} \times \frac{22}{7} \times 7 \times 7 = \frac{154}{3} {cm}^{2}$

Thus, Assertion (A) and Reason (R) both are true and Reason (R) is correct explanation of Assertion (A).

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