STATEMENT-1 : `y = e^(x)` is a particular solution of `(dy)/(dx) = y`.
STATEMENT-2 : The differential equation representing family of curve `y = a cos omega t + b sin omega t`, where a and b are parameters, is `(d^(2)y)/(dt^(2)) - omega^(2) y = 0`.
STATEMENT-3 : `y = (1)/(2)x^(3)+c_(1)x+c_(2)` is a general solution of `(d^(2)y)/(dx^(2)) = 3x`.

Options

A. a) T F T
B. b) T T T
C. c) F F F
D. d) F F T

Answer

To solve the problem, we will evaluate each statement one by one to determine whether they are true or false.

### Statement 1:
**Statement:** \( y = e^x \) is a particular solution of \( \frac{dy}{dx} = y \).

**Solution:**
1. Differentiate \( y = e^x \):
\[ \frac{dy}{dx} = e^x \]
2. Since \( y = e^x \), we can substitute \( y \) into the equation:
\[ \frac{dy}{dx} = y \]
Therefore, we have:
\[ e^x = e^x \]
3. This confirms that \( y = e^x \) satisfies the differential equation.

**Conclusion:** Statement 1 is **True**.

### Statement 2:
**Statement:** The differential equation representing the family of curves \( y = a \cos(\omega t) + b \sin(\omega t) \) is \( \frac{d^2y}{dt^2} - \omega^2 y = 0 \).

**Solution:**
1. Differentiate \( y = a \cos(\omega t) + b \sin(\omega t) \) once:
\[ \frac{dy}{dt} = -a \omega \sin(\omega t) + b \omega \cos(\omega t) \]
2. Differentiate again to find \( \frac{d^2y}{dt^2} \):
\[ \frac{d^2y}{dt^2} = -a \omega^2 \cos(\omega t) - b \omega^2 \sin(\omega t) \]
This can be rewritten as:
\[ \frac{d^2y}{dt^2} = -\omega^2 (a \cos(\omega t) + b \sin(\omega t)) = -\omega^2 y \]
3. Rearranging gives:
\[ \frac{d^2y}{dt^2} + \omega^2 y = 0 \]
Thus, the correct form of the differential equation is:
\[ \frac{d^2y}{dt^2} + \omega^2 y = 0 \]
which is different from the statement.

**Conclusion:** Statement 2 is **False**.

### Statement 3:
**Statement:** \( y = \frac{1}{2}x^3 + c_1 x + c_2 \) is a general solution of \( \frac{d^2y}{dx^2} = 3x \).

**Solution:**
1. Differentiate \( y = \frac{1}{2}x^3 + c_1 x + c_2 \) once:
\[ \frac{dy}{dx} = \frac{3}{2}x^2 + c_1 \]
2. Differentiate again to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = 3x \]
3. This matches the given differential equation.

**Conclusion:** Statement 3 is **True**.

### Final Summary:
- Statement 1: True
- Statement 2: False
- Statement 3: True

Thus, the correct option is **Option A: TFT**.

---