Which of the following is not simple harmonic function ?

Options

A. `y = a sin 2omegat + b cos^(2) omegat`
B. `y = a sin omega t + b cos 2 omega t`
C. `y = 1 - 2 sin^(2) omega t`
D. `y = (sqrt(a^(2) +b^(2)))sin omega t cos omega t`

Explanation

To determine which of the given equations is not a simple harmonic function (SHM), we can analyze each equation to see if it can be expressed in the standard form of SHM, which is:

\[ x = a \sin(\omega t + \phi) \]

where:
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time,
- \( \phi \) is the phase constant.

Let's analyze each equation step by step.

### Step 1: Analyze Equation A
**Equation A:** \( y = a \sin(2 \omega t) + b \cos^2(\omega t) \)

1. **Convert \( \cos^2(\omega t) \):**
- We can use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).
- Thus, \( \cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2} \).

2. **Substituting into Equation A:**
- \( y = a \sin(2\omega t) + b \left(\frac{1 + \cos(2\omega t)}{2}\right) \)
- This can be simplified to:
\[ y = a \sin(2\omega t) + \frac{b}{2} + \frac{b}{2} \cos(2\omega t) \]

3. **Conclusion for A:**
- This equation can be expressed in terms of sine and cosine with the same frequency (2ω), hence it represents SHM.

### Step 2: Analyze Equation B
**Equation B:** \( y = a \sin(\omega t) + b \cos(2\omega t) \)

1. **Different Frequencies:**
- Here, we have two different frequencies: \( \omega \) and \( 2\omega \).

2. **Conclusion for B:**
- Since the terms have different frequencies, this equation cannot be expressed in the standard SHM form. Therefore, this is **not SHM**.

### Step 3: Analyze Equation C
**Equation C:** \( y = 1 - 2 \sin^2(\omega t) \)

1. **Convert \( \sin^2(\omega t) \):**
- We can use the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \).
- Thus, \( \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2} \).

2. **Substituting into Equation C:**
- \( y = 1 - 2 \left(\frac{1 - \cos(2\omega t)}{2}\right) \)
- This simplifies to:
\[ y = 1 - (1 - \cos(2\omega t)) = \cos(2\omega t) \]

3. **Conclusion for C:**
- This can be expressed in the standard form of SHM, hence it represents SHM.

### Step 4: Analyze Equation D
**Equation D:** \( y = \sqrt{a^2 + b^2} \sin(2\omega t) \)

1. **Standard Form:**
- This is already in the form of \( A \sin(\omega t) \) where \( A = \sqrt{a^2 + b^2} \) and \( \omega = 2\omega \).

2. **Conclusion for D:**
- This equation can be expressed in the standard form of SHM, hence it represents SHM.

### Final Conclusion
From our analysis:
- **Equation A:** SHM
- **Equation B:** Not SHM
- **Equation C:** SHM
- **Equation D:** SHM

Thus, the equation that is **not a simple harmonic function** is:

**Answer:** \( y = a \sin(\omega t) + b \cos(2\omega t) \) (Equation B)