Function `x=Asin^(2)omegat+Bcos^(2)omegat+Csinomegat cos omegat` represents simple harmonic motion,

Options

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

Explanation

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 given function step by step.

### Step 1: Rewrite the function using trigonometric identities
We start with the given function:
\[ x = A \sin^2(\omega t) + B \cos^2(\omega t) + C \sin(\omega t) \cos(\omega t) \]

Using the trigonometric identities:
- \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \)
- \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)
- \( \sin(\theta) \cos(\theta) = \frac{1}{2} \sin(2\theta) \)

We can rewrite the function:
\[ x = A \left(\frac{1 - \cos(2\omega t)}{2}\right) + B \left(\frac{1 + \cos(2\omega t)}{2}\right) + C \left(\frac{1}{2} \sin(2\omega t)\right) \]

### Step 2: Simplify the equation
Now, simplifying the equation:
\[ x = \frac{A}{2} - \frac{A}{2} \cos(2\omega t) + \frac{B}{2} + \frac{B}{2} \cos(2\omega t) + \frac{C}{2} \sin(2\omega t) \]
\[ = \left(\frac{A + B}{2}\right) + \left(\frac{B - A}{2}\right) \cos(2\omega t) + \left(\frac{C}{2}\right) \sin(2\omega t) \]

### Step 3: Identify the form of SHM
The equation can be expressed in the standard form of SHM:
\[ x = D + E \cos(2\omega t) + F \sin(2\omega t) \]
where \( D = \frac{A + B}{2} \), \( E = \frac{B - A}{2} \), and \( F = \frac{C}{2} \).

### Step 4: Determine conditions for SHM
For the function to represent SHM, the coefficients \( E \) and \( F \) must not both be zero simultaneously. This means:
- If \( C \neq 0 \), then \( F \neq 0 \).
- If \( B \neq A \), then \( E \neq 0 \).

### Step 5: Analyze the options
Now we analyze the options provided in the question:

1. **Option A**: For any value of \( A, B, C \) except \( C = 0 \) - This is correct because if \( C = 0 \) and \( B = A \), the equation does not represent SHM.

2. **Option B**: If \( A = -B \) and \( C = 2B \), the amplitude is \( B\sqrt{2} \) - This is also correct as shown in the video.

3. **Option C**: If \( A = B \) and \( C = 0 \), it does not represent SHM - This is correct as it simplifies to a constant.

4. **Option D**: If \( A = B \) and \( C = 2B \), the amplitude is \( |B| \) - This is also correct as shown in the video.

### Final Conclusion
Thus, the correct options are A, B, and D.