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

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

we need to find the number of values of \(\theta\) in the interval \([0, 2\pi]\) for which at least two roots are equal.

### Step 1: Simplifying the Equation
We can rewrite the equation as:

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

### Step 2: Factoring Out Common Terms
Notice that we can factor out \(x - 1\):

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

This gives us:

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

### Step 3: Finding Roots
We can observe that \(x = 1\) is a root of the equation. To find the conditions under which at least two roots are equal, we need to analyze the quadratic part of the equation.

### Step 4: Analyzing the Quadratic
The remaining part of the equation can be expressed as a quadratic in \(x\):

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

### Step 5: Condition for Equal Roots
For the quadratic equation \(ax^2 + bx + c = 0\) to have equal roots, the discriminant must be zero:

\[ b^2 - 4ac = 0 \]

In our case, \(a = 1\), \(b = -1\), and \(c = \sin \theta + \cos \theta\):

\[ (-1)^2 - 4(1)(\sin \theta + \cos \theta) = 0 \]

This simplifies to:

\[ 1 - 4(\sin \theta + \cos \theta) = 0 \]

### Step 6: Solving for \(\theta\)
Rearranging gives us:

\[ 4(\sin \theta + \cos \theta) = 1 \]

Thus,

\[ \sin \theta + \cos \theta = \frac{1}{4} \]

### Step 7: Finding Values of \(\theta\)
Using the identity \(\sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right)\):

\[ \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{4} \]

This leads to:

\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{4\sqrt{2}} \]

### Step 8: Finding \(\theta\)
The general solutions for \(\sin x = k\) are given by:

\[ x = \arcsin(k) + 2n\pi \quad \text{or} \quad x = \pi - \arcsin(k) + 2n\pi \]

Thus, we have:

\[ \theta + \frac{\pi}{4} = \arcsin\left(\frac{1}{4\sqrt{2}}\right) + 2n\pi \quad \text{or} \quad \theta + \frac{\pi}{4} = \pi - \arcsin\left(\frac{1}{4\sqrt{2}}\right) + 2n\pi \]

### Step 9: Finding Values in \([0, 2\pi]\)
We need to solve for \(\theta\) in the interval \([0, 2\pi]\). Each equation will yield two solutions, leading to a total of four possible values for \(\theta\) in the interval \([0, 2\pi]\).

### Conclusion
Thus, the number of values of \(\theta\) in \([0, 2\pi]\) for which at least two roots are equal is:

\[ \boxed{4} \]