Question

Assertion (A) : In a circle of radius 21 cm, an arc of length 22 cm subtends an angle of 60\(^\circ\) at the centre.
Reason (R) : The length of arc of a sector of a circle of radius r and central angle \(\theta\) is \(\frac{2\pi r\theta}{360}\).

• A. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
• B. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

For Assertion-Reason questions, always write down the verification calculation explicitly.
This establishes the logical link between both statements clearly!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to evaluate the truth value of the given Assertion (A) and Reason (R), and decide if the Reason correctly explains the Assertion.

Step 2: Key Formula or Approach:
The formula for the length of an arc is:
\[ l = \frac{\theta}{360} \times 2\pi r \]

Step 3: Detailed Explanation:

• Evaluate Reason (R):
The formula for the arc length of a circle of radius \(r\) with angle \(\theta\) is indeed \(\frac{\theta}{360} \times 2\pi r = \frac{2\pi r \theta}{360}\). Thus, Reason (R) is True.

• Evaluate Assertion (A):
Let us calculate the arc length for \(r = 21 \text{ cm}\) and \(\theta = 60^\circ\):
\[ l = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21 \]
\[ l = \frac{1}{6} \times 2 \times 22 \times 3 \]
\[ l = \frac{1}{6} \times 132 = 22 \text{ cm} \]
This matches the given arc length of 22 cm. Thus, Assertion (A) is True.

• Since the assertion was verified using the exact formula stated in the reason, Reason (R) is the correct explanation of Assertion (A).


Step 4: Final Answer:
Both statements are true and R is the correct explanation of A.