Assertion: Velocity and acceleration of a particle in circular motion at some instant are:
`v=(2hat(i))ms^(-1)` and `a=(-hat(i)+2hat(j))ms^(-2)` , then radius of circle is `2m` .
Reason: Speed of particle is decreasing at a rate of `1ms^(-2)` .

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, true but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the problem, we need to analyze both the assertion and the reason provided in the question.

### Step 1: Analyze the Assertion
The assertion states that the velocity \( \mathbf{v} \) and acceleration \( \mathbf{a} \) of a particle in circular motion are given as:
- \( \mathbf{v} = 2 \hat{i} \, \text{m/s} \)
- \( \mathbf{a} = -\hat{i} + 2\hat{j} \, \text{m/s}^2 \)

We need to determine if the radius of the circle is indeed \( 2 \, \text{m} \).

### Step 2: Calculate the Radius
For a particle in circular motion, the radial (centripetal) acceleration \( a_r \) is given by the formula:
\[ a_r = \frac{v^2}{r} \]
where \( v \) is the speed of the particle and \( r \) is the radius of the circular path.

From the velocity given:
\[ |\mathbf{v}| = 2 \, \text{m/s} \]

Now, substituting the values into the radial acceleration formula:
\[ a_r = \frac{(2)^2}{r} = \frac{4}{r} \]

### Step 3: Find the Total Acceleration
The total acceleration \( \mathbf{a} \) is given as:
\[ \mathbf{a} = -\hat{i} + 2\hat{j} \, \text{m/s}^2 \]
To find the magnitude of the total acceleration:
\[ |\mathbf{a}| = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \, \text{m/s}^2 \]

### Step 4: Identify the Radial Component of Acceleration
The radial component of acceleration \( a_r \) can be identified as the component of acceleration that is directed towards the center of the circular path. In this case, the radial component is:
\[ a_r = -\hat{i} \, \text{m/s}^2 \quad \text{(which has a magnitude of 1 m/s}^2\text{)} \]

### Step 5: Equate the Radial Acceleration
We can now equate the radial acceleration to the expression we derived:
\[ 1 = \frac{4}{r} \]
Solving for \( r \):
\[ r = 4 \, \text{m} \]

### Step 6: Conclusion on the Assertion
The assertion states that the radius is \( 2 \, \text{m} \), but our calculation shows that the radius is \( 4 \, \text{m} \). Therefore, the assertion is **false**.

### Step 7: Analyze the Reason
The reason states that the speed of the particle is decreasing at a rate of \( 1 \, \text{m/s}^2 \).

From the acceleration components:
- The x-component of acceleration is \( -1 \, \text{m/s}^2 \), which indicates that the speed in the x-direction is decreasing at that rate.

### Step 8: Conclusion on the Reason
The reason is **true** because the negative x-component of acceleration indicates a decrease in speed in that direction.

### Final Conclusion
- Assertion: **False**
- Reason: **True**