Consider the cubic equation in `x , x ^(3) - x^(2) + (x- x ^(2)) sin theta + (x -x ^(2)) cos theta + (x-1) sin theta cos theta =0` whose roots are `alpha, beta , gamma. `
Number of value of `theta` in `[0,2pi]` for which at least two roots are equal, is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the given cubic equation \( x^3 - x^2 + (x - x^2) \sin \theta + (x - x^2) \cos \theta + (x - 1) \sin \theta \cos \theta = 0 \) and find the number of values of \( \theta \) in the interval \( [0, 2\pi] \) for which at least two roots are equal, we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We can factor out common terms:

\[ x^3 - x^2 + (x - x^2)(\sin \theta + \cos \theta) + (x - 1) \sin \theta \cos \theta = 0 \]

### Step 2: Group terms
Notice that we can group the terms involving \( x \):

\[ x^3 - x^2 + x(\sin \theta + \cos \theta - x(\sin \theta + \cos \theta)) + (\sin \theta \cos \theta - \sin \theta \cos \theta) = 0 \]

This simplifies to:

\[ x^3 - x^2(1 + \sin \theta + \cos \theta) + x(\sin \theta + \cos \theta) + \sin \theta \cos \theta = 0 \]

### Step 3: Identify conditions for equal roots
For a cubic equation to have at least two equal roots, the discriminant must be zero. However, we can also analyze when the roots can be equal based on values of \( \sin \theta \) and \( \cos \theta \).

### Step 4: Set conditions for equal roots
1. **Case 1:** When \( \sin \theta = 1 \):
- This occurs at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{5\pi}{2} \) (but only \( \frac{\pi}{2} \) is in \( [0, 2\pi] \)).

2. **Case 2:** When \( \cos \theta = 1 \):
- This occurs at \( \theta = 0 \) and \( \theta = 2\pi \).

3. **Case 3:** When \( \sin \theta = \cos \theta \):
- This occurs at \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

### Step 5: Count the values of \( \theta \)
Now we can count the distinct values of \( \theta \) that give us at least two equal roots:

- From Case 1: \( \frac{\pi}{2} \) (1 value)
- From Case 2: \( 0, 2\pi \) (2 values)
- From Case 3: \( \frac{\pi}{4}, \frac{5\pi}{4} \) (2 values)

### Total Values
Adding these values together, we have:

- \( 1 + 2 + 2 = 5 \)

### Final Answer
Thus, the number of values of \( \theta \) in the interval \( [0, 2\pi] \) for which at least two roots are equal is **5**.

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