If a `SHM` is given by `y = (sinomegat + cos omegat)m`, which of the following statement are ture

Options

A. The amplitude is `1m`
B. The amplitude is `(sqrt(2))m`
C. Time is considered from `y = 1 m`
D. Time is considered from `y = 0`

Answer

To solve the problem, we need to analyze the given equation of simple harmonic motion (SHM) and derive the necessary parameters to determine which statements are true. The equation given is:

\[ y = \sin(\omega t) + \cos(\omega t) \]

### Step 1: Rewrite the Equation
We start by rewriting the equation in a more manageable form. We can factor out a common term:

\[ y = \sin(\omega t) + \cos(\omega t) \]

To combine these terms, we can use the identity for the sum of sine and cosine functions. We can express this as:

\[ y = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(\omega t) + \frac{1}{\sqrt{2}} \cos(\omega t) \right) \]

### Step 2: Use the Angle Addition Formula
Using the angle addition formula, we can express the combination of sine and cosine as a single sine function:

\[ y = \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]

This shows that the motion is indeed simple harmonic motion.

### Step 3: Determine the Amplitude
From the rewritten equation, we can identify the amplitude \( A \):

\[ A = \sqrt{2} \, \text{meters} \]

### Step 4: Calculate Position at \( t = 0 \)
Next, we calculate the position \( y \) at \( t = 0 \):

\[ y(0) = \sqrt{2} \sin\left(0 + \frac{\pi}{4}\right) \]
\[ y(0) = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 \, \text{meter} \]

### Step 5: Evaluate Statements
Now that we have the amplitude and the position at \( t = 0 \), we can evaluate which of the statements (A, B, C, D) are true based on our findings:

1. **Amplitude**: The amplitude is \( \sqrt{2} \, \text{meters} \).
2. **Position at \( t = 0 \)**: The position at \( t = 0 \) is \( 1 \, \text{meter} \).

Based on this analysis, we can conclude that:
- Statement B (Amplitude is \( \sqrt{2} \, \text{meters} \)) is true.
- Statement C (Position at \( t = 0 \) is \( 1 \, \text{meter} \)) is true.

### Final Answer
The true statements are B and C.

---