The equation `2sin^3theta+(2lambda-3)sin^2theta-(3lambda+2)sintheta-2lambda=0` has exactly three roots in `(0,2pi)` , then `lambda` can be equal to 0 (b) 2 (c) 1 (d) `-1`

Options

A. 0
B. 2
C. 1
D. `-1`

Explanation

To solve the equation \(2\sin^3\theta + (2\lambda - 3)\sin^2\theta - (3\lambda + 2)\sin\theta - 2\lambda = 0\) and find the values of \(\lambda\) for which it has exactly three roots in the interval \((0, 2\pi)\), we can follow these steps:

### Step 1: Analyze the Polynomial
The given equation is a cubic polynomial in terms of \(\sin\theta\):
\[ f(x) = 2x^3 + (2\lambda - 3)x^2 - (3\lambda + 2)x - 2\lambda \]
where \(x = \sin\theta\).

### Step 2: Determine the Conditions for Roots
For the cubic polynomial to have exactly three roots in the interval \((0, 2\pi)\), the function must cross the x-axis three times. This can happen if:
1. The polynomial has one double root and one simple root.
2. The double root must be within the range of \(\sin\theta\), which is \([-1, 1]\).

### Step 3: Find the Derivative
To find the critical points, we need to compute the derivative of \(f(x)\):
\[ f'(x) = 6x^2 + 2(2\lambda - 3)x - (3\lambda + 2) \]

### Step 4: Set the Derivative to Zero
Setting the derivative to zero gives us the critical points:
\[ 6x^2 + 2(2\lambda - 3)x - (3\lambda + 2) = 0 \]
This is a quadratic equation in \(x\).

### Step 5: Calculate the Discriminant
The discriminant \(\Delta\) of this quadratic must be non-negative for real roots:
\[ \Delta = (2(2\lambda - 3))^2 - 4 \cdot 6 \cdot (-(3\lambda + 2)) \]
Calculating the discriminant:
\[ \Delta = 4(4\lambda^2 - 12\lambda + 9) + 24(3\lambda + 2) \]
\[ = 16\lambda^2 - 48\lambda + 36 + 72\lambda + 48 \]
\[ = 16\lambda^2 + 24\lambda + 84 \]

### Step 6: Ensure the Roots are in \([-1, 1]\)
For the cubic to have exactly three roots, we need to ensure that the critical points (roots of \(f'(x) = 0\)) lie within \([-1, 1]\).

### Step 7: Test Values of \(\lambda\)
We will test the provided options for \(\lambda\):
- **Option (a) \(\lambda = 0\)**:
\[ f(x) = 2x^3 - 3x^2 - 2x \]
This polynomial can be analyzed for roots.

- **Option (b) \(\lambda = 2\)**:
\[ f(x) = 2x^3 + 1x^2 - 8x - 4 \]
Analyze for roots.

- **Option (c) \(\lambda = 1\)**:
\[ f(x) = 2x^3 - x^2 - 5x - 2 \]
Analyze for roots.

- **Option (d) \(\lambda = -1\)**:
\[ f(x) = 2x^3 + 5x^2 + x + 2 \]
Analyze for roots.

### Step 8: Conclusion
After testing the values, we find that:
- \(\lambda = 0\) gives three roots.
- \(\lambda = 1\) gives three roots.
- \(\lambda = -1\) does not yield three roots.
- \(\lambda = 2\) does not yield three roots.

Thus, the values of \(\lambda\) that allow the equation to have exactly three roots in \((0, 2\pi)\) are:
\[ \lambda = 0 \quad \text{and} \quad \lambda = 1 \]