Funcation `x=A sin^(2) omegat+B cos^(2) omegat+ C sin omegat cos omegat` Represents SHM.

Options

A. For any value of A, B and C (except C = 0).
B. If A=B,C=2B and amplitude= `|Bsqrt2|`
C. If A=B,C=0
D. If A=B,C=2B amplitude=|B|.

Answer

To determine whether 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 will analyze the expression step by step.

### Step 1: Rewrite the trigonometric identities
We can use the trigonometric identities to rewrite the terms in the function:

1. \( \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2} \)
2. \( \cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2} \)
3. \( \sin(\omega t) \cos(\omega t) = \frac{1}{2} \sin(2\omega t) \)

### Step 2: Substitute the identities into the function
Substituting these identities into the function, we get:

\[ x = A \left(\frac{1 - \cos(2\omega t)}{2}\right) + B \left(\frac{1 + \cos(2\omega t)}{2}\right) + C \left(\frac{1}{2} \sin(2\omega t)\right) \]

### Step 3: Simplify the expression
Now, we can simplify the expression:

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

Combining the terms gives:

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

### Step 4: Identify the form of SHM
The expression now has the form:

\[ x = D + E \cos(2\omega t) + F \sin(2\omega t) \]

where:
- \( D = \frac{A + B}{2} \)
- \( E = \frac{B - A}{2} \)
- \( F = \frac{C}{2} \)

This represents SHM if \( E \) and \( F \) are not both zero.

### Step 5: Analyze the conditions for SHM
1. If \( C = 0 \) and \( A \neq B \), then \( x \) represents SHM.
2. If \( A = -B \) and \( C = 2B \), we find that the terms will still allow for SHM.
3. If \( A = B = C = 0 \), then there is no motion, hence not SHM.

### Conclusion
The function represents SHM under the conditions:
- For any values of \( A \) and \( B \) except when \( C = 0 \).
- If \( A = -B \) and \( C = 2B \), the amplitude will be \( b\sqrt{2} \).
- If \( A = B = C = 0 \), it does not represent SHM.