Question

Assertion (A) and Reason (R)

Assertion (A):
\[ (\cos\theta+i\sin\theta)^{p/q} \] has \(q\) and only \(q\) different roots, where \(q\) is a positive integer.

Reason (R):
\[ (-1)^{1/3} \] has three different roots.

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

A complex number generally has \(q\) distinct \(q\)th roots.

Answer 1

The Correct Option is B

Concept:
In complex numbers, fractional powers generally give multiple roots.

Step 1: Check Assertion.
For: \[ (\cos\theta+i\sin\theta)^{p/q} \] there are \(q\) distinct values or roots. So: \[ A \text{ is correct} \]

Step 2: Check Reason.
The cube roots of \(-1\) are three distinct complex roots. So: \[ R \text{ is correct} \]

Step 3: Check explanation.
The reason is only an example of multiple roots. It does not directly explain the general result in Assertion. \[ R \text{ is not the correct explanation of } A \] \[ \therefore \text{Correct Answer is (B)} \]