STATEMENT - 1 : A ball of mass m is moving towards a batsman at a speed v. The batsman strikes the ball and deflects it by an angle `theta` without changing its speed. The impulse imparted to the ball is zero.
because
STATEMENT - 2 : Impulse = change in momentum.

Options

A. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1.
B. Statement - 1 is True, Statement - 2 is True, Statement - 2 is NOT a correct explanation for statement - 1.
C. Statement - 1 is True, Statement - 2 is False.
D. Statement - 1 is False, Statement - 2 is True.

Answer

To solve the problem, we need to analyze the statements given in the question regarding the collision of a ball with a batsman.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- A ball of mass \( m \) is moving towards the batsman with a speed \( v \).
- The batsman strikes the ball and deflects it by an angle \( \theta \) without changing its speed.

2. **Analyzing Momentum Before and After Collision**:
- Before the collision, the momentum of the ball can be expressed as:
\[ \text{Initial Momentum} = m \vec{v} = m v \hat{i} \]
- After the collision, the ball is deflected by an angle \( \theta \) but retains the same speed \( v \). Therefore, the final momentum can be expressed as:
\[ \text{Final Momentum} = m \vec{v'} = m v (\cos \theta \hat{i} + \sin \theta \hat{j}) \]

3. **Calculating Change in Momentum**:
- The change in momentum \( \Delta \vec{p} \) is given by:
\[ \Delta \vec{p} = \text{Final Momentum} - \text{Initial Momentum} \]
- Substituting the expressions for initial and final momentum:
\[ \Delta \vec{p} = m v (\cos \theta \hat{i} + \sin \theta \hat{j}) - m v \hat{i} \]
- Simplifying this gives:
\[ \Delta \vec{p} = m v (\cos \theta - 1) \hat{i} + m v \sin \theta \hat{j} \]

4. **Magnitude of Change in Momentum**:
- The magnitude of the change in momentum can be calculated using the Pythagorean theorem:
\[ |\Delta \vec{p}| = \sqrt{(m v (\cos \theta - 1))^2 + (m v \sin \theta)^2} \]
- This expression is not equal to zero unless \( \theta = 0 \) (which means no deflection).

5. **Impulse and Its Relation to Change in Momentum**:
- Impulse \( J \) is defined as the change in momentum:
\[ J = \Delta \vec{p} \]
- Since we have shown that \( \Delta \vec{p} \) is not zero, the impulse imparted to the ball is also not zero.

6. **Conclusion**:
- Therefore, Statement 1 is false because the impulse imparted to the ball is not zero.
- Statement 2 is true as impulse is indeed defined as the change in momentum.

### Final Answer:
- Statement 1 is false, and Statement 2 is true.