The displacement (x) of a particle as a function of time (t) is given by
`x=asin(bt+c)`
Where a,b and c are constant of motion. Choose the correct statemetns from the following.

Options

A. The motion repeats itself in a time interval of `(2pi)/(b)`
B. The energy of the particle remains constant.
C. The velocity of the particle zero at `x=+-a`
D. The acceleration of the particle is zero at `x=+-a`

Explanation

To solve the question, we need to analyze the given displacement function of a particle in simple harmonic motion (SHM), which is represented as:

\[ x = a \sin(bt + c) \]

Where:
- \( a \) is the amplitude,
- \( b \) is the angular frequency,
- \( c \) is the phase constant.

Now, let's evaluate each of the statements provided in the question.

### Step 1: Evaluate the first statement
**Statement A:** The motion repeats itself in a time interval of \( \frac{2\pi}{b} \).

**Solution:**
In SHM, the period \( T \) is the time taken for one complete cycle of motion. The angular frequency \( b \) relates to the period by the formula:

\[ T = \frac{2\pi}{b} \]

Thus, the motion indeed repeats itself every \( \frac{2\pi}{b} \) seconds. Therefore, **Statement A is correct**.

### Step 2: Evaluate the second statement
**Statement B:** The energy of the particle remains constant.

**Solution:**
In SHM, the total mechanical energy (E) is conserved and is given by:

\[ E = \frac{1}{2} k A^2 \]

Where \( k \) is the spring constant and \( A \) is the amplitude. Since there are no non-conservative forces doing work (like friction), the energy remains constant throughout the motion. Hence, **Statement B is correct**.

### Step 3: Evaluate the third statement
**Statement C:** The velocity of the particle is zero at \( x = \pm a \).

**Solution:**
In SHM, the velocity \( v \) is given by the derivative of displacement with respect to time:

\[ v = \frac{dx}{dt} = a b \cos(bt + c) \]

The velocity is zero when the displacement is at its maximum or minimum, which occurs at \( x = \pm a \). Therefore, **Statement C is correct**.

### Step 4: Evaluate the fourth statement
**Statement D:** The acceleration of the particle is zero at \( x = \pm a \).

**Solution:**
The acceleration \( a \) in SHM is given by:

\[ a = -\omega^2 x \]

Where \( \omega = b \). At \( x = \pm a \), the acceleration is:

\[ a = -b^2(\pm a) \]

This means the acceleration is at its maximum magnitude (not zero) at the extreme positions \( x = \pm a \). Therefore, **Statement D is incorrect**.

### Conclusion:
The correct statements are A, B, and C.

### Final Answer:
The correct statements are:
- Statement A: Correct
- Statement B: Correct
- Statement C: Correct
- Statement D: Incorrect