If the sum of the roots of the equation `sin^(2)theta=k,(0ltk lt 1)` lying in `[0,2pi]` is equal to the angles of a n-sided regular polygon, then the value of n is (a) 6 (b) 4 (c) 2 (d) none of these

Options

A. 6
B. 4
C. 2
D. none of these

Explanation

To solve the problem, we need to find the value of \( n \) such that the sum of the roots of the equation \( \sin^2 \theta = k \) (where \( 0 < k < 1 \)) lying in the interval \( [0, 2\pi] \) is equal to the angles of an \( n \)-sided regular polygon.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation given is \( \sin^2 \theta = k \). This can be rewritten as:
\[ \sin \theta = \pm \sqrt{k} \]

2. **Finding the Roots**:
The solutions for \( \theta \) in the interval \( [0, 2\pi] \) can be found as follows:
- For \( \sin \theta = \sqrt{k} \):
- The first solution is \( \theta = \arcsin(\sqrt{k}) \).
- The second solution is \( \theta = \pi - \arcsin(\sqrt{k}) \).
- For \( \sin \theta = -\sqrt{k} \):
- The third solution is \( \theta = \pi + \arcsin(\sqrt{k}) \).
- The fourth solution is \( \theta = 2\pi - \arcsin(\sqrt{k}) \).

Thus, the four roots in the interval \( [0, 2\pi] \) are:
\[ \theta_1 = \arcsin(\sqrt{k}), \quad \theta_2 = \pi - \arcsin(\sqrt{k}), \quad \theta_3 = \pi + \arcsin(\sqrt{k}), \quad \theta_4 = 2\pi - \arcsin(\sqrt{k}) \]

3. **Calculating the Sum of the Roots**:
Now, we need to find the sum of these roots:
\[ \text{Sum} = \theta_1 + \theta_2 + \theta_3 + \theta_4 \]
Substituting the values:
\[ \text{Sum} = \arcsin(\sqrt{k}) + (\pi - \arcsin(\sqrt{k})) + (\pi + \arcsin(\sqrt{k})) + (2\pi - \arcsin(\sqrt{k})) \]
Simplifying this:
\[ \text{Sum} = \arcsin(\sqrt{k}) + \pi - \arcsin(\sqrt{k}) + \pi + \arcsin(\sqrt{k}) + 2\pi - \arcsin(\sqrt{k}) \]
\[ \text{Sum} = 4\pi \]

4. **Relating to the Polygon**:
The sum of the interior angles of an \( n \)-sided regular polygon is given by:
\[ \text{Sum of angles} = (n - 2) \cdot 180^\circ = (n - 2) \cdot \pi \text{ (in radians)} \]
We set the sum of the roots equal to the sum of the angles:
\[ 4\pi = (n - 2) \cdot \pi \]

5. **Solving for \( n \)**:
Dividing both sides by \( \pi \):
\[ 4 = n - 2 \]
Adding 2 to both sides gives:
\[ n = 6 \]

### Conclusion:
The value of \( n \) is \( 6 \), which corresponds to a hexagon.