The function `x = A sin^2 (omega)t + B cos^2 (omega)t + Csin (omega)t cos (omega)t` represent (SHM) for which of the option(s).

Options

A. For all value of `A,B and C` (except C = 0)
B. `A = -B, C = 2 B, amplitude = |B sqrt(2)|`
C. `A = B, C = 0`
D. `A = B, C = 2 B, amplitude = |B|`

Explanation

To determine for which conditions the function \( x = A \sin^2(\omega t) + B \cos^2(\omega t) + C \sin(\omega t) \cos(\omega t) \) represents Simple Harmonic Motion (SHM), we can simplify the expression using trigonometric identities.

### Step-by-Step Solution:

1. **Start with the given function:**
\[ x = A \sin^2(\omega t) + B \cos^2(\omega t) + C \sin(\omega t) \cos(\omega t) \]

2. **Use trigonometric identities:**
We can use the identities:
- \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \)
- \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)
- \( \sin(\theta) \cos(\theta) = \frac{1}{2} \sin(2\theta) \)

Applying these identities:
\[ \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2}, \quad \cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2}, \quad \sin(\omega t) \cos(\omega t) = \frac{1}{2} \sin(2\omega t) \]

3. **Substituting the identities into the equation:**
\[ x = A \left(\frac{1 - \cos(2\omega t)}{2}\right) + B \left(\frac{1 + \cos(2\omega t)}{2}\right) + C \left(\frac{1}{2} \sin(2\omega t)\right) \]

Simplifying this gives:
\[ x = \frac{A}{2} (1 - \cos(2\omega t)) + \frac{B}{2} (1 + \cos(2\omega t)) + \frac{C}{2} \sin(2\omega t) \]

\[ x = \frac{A + B}{2} + \left(\frac{B - A}{2}\right) \cos(2\omega t) + \frac{C}{2} \sin(2\omega t) \]

4. **Rearranging the equation:**
\[ x = \frac{A + B}{2} + \frac{B - A}{2} \cos(2\omega t) + \frac{C}{2} \sin(2\omega t) \]

5. **Identifying the form of SHM:**
The equation can be expressed in the form:
\[ x = D + E \cos(2\omega t) + F \sin(2\omega t) \]
where \( D = \frac{A + B}{2} \), \( E = \frac{B - A}{2} \), and \( F = \frac{C}{2} \).

6. **Condition for SHM:**
For \( x \) to represent SHM, the coefficients of \( \cos(2\omega t) \) and \( \sin(2\omega t) \) must not both be zero simultaneously. This means:
\[ E^2 + F^2 > 0 \]
or equivalently:
\[ \left(\frac{B - A}{2}\right)^2 + \left(\frac{C}{2}\right)^2 > 0 \]

### Conclusion:
The function represents SHM under the conditions that \( B - A \) and \( C \) are not both zero simultaneously, which leads us to the options provided in the question.