These questions consists of two statements each printed as Assertion and Reason. While answering these question you are required to choose any one of the following five reponses
(a) If both Assertion and Reason arecorrect and Reason is the correct explanation of Asserrtion.
(b) If both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
(c ) If Assertion is true but Reason is false.
(d) If Assertion is false but Reason is true.
Assertion. Velocity of a block changes from `(2hati+3hatj) ms^(-1)` to `(-4hati-6hatj) ms^(-1)`. Then, work done by all the forces during this interval of time is positive.
Reason Speed of block is increasing.

Explanation

To solve the question, we need to analyze both the Assertion and the Reason provided.

**Assertion:** The velocity of a block changes from \( (2\hat{i} + 3\hat{j}) \, \text{m/s} \) to \( (-4\hat{i} - 6\hat{j}) \, \text{m/s} \). Then, the work done by all the forces during this interval of time is positive.

**Reason:** The speed of the block is increasing.

### Step-by-Step Solution:

1. **Understanding the Assertion:**
- The initial velocity of the block is \( \mathbf{v_i} = 2\hat{i} + 3\hat{j} \).
- The final velocity of the block is \( \mathbf{v_f} = -4\hat{i} - 6\hat{j} \).
- To determine if the work done is positive, we need to calculate the change in kinetic energy.

2. **Calculating Initial and Final Speeds:**
- The initial speed \( v_i \) can be calculated as:
\[ v_i = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \, \text{m/s} \]
- The final speed \( v_f \) can be calculated as:
\[ v_f = \sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \, \text{m/s} \]

3. **Calculating Change in Kinetic Energy:**
- The initial kinetic energy \( KE_i \) is given by:
\[ KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} m (13) = \frac{13m}{2} \]
- The final kinetic energy \( KE_f \) is given by:
\[ KE_f = \frac{1}{2} m v_f^2 = \frac{1}{2} m (52) = 26m \]
- The change in kinetic energy \( \Delta KE \) is:
\[ \Delta KE = KE_f - KE_i = 26m - \frac{13m}{2} = \frac{52m - 13m}{2} = \frac{39m}{2} \]
- Since \( \Delta KE \) is positive, the work done by all forces is also positive.

4. **Understanding the Reason:**
- The reason states that the speed of the block is increasing.
- Since the final speed \( v_f = \sqrt{52} \) is greater than the initial speed \( v_i = \sqrt{13} \), it confirms that the speed is indeed increasing.

5. **Conclusion:**
- Both the Assertion and Reason are correct.
- The Reason correctly explains the Assertion because the increase in speed leads to positive work done by the forces.

### Final Answer:
(a) If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

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