The displacement equation of a particle is `x=3 sin 2t+4cos2t`. The amplitude and maximum velocity will be respectively

Options

A. 5,10
B. 3,2
C. 4,2
D. 3,4

Explanation

To solve the problem, we need to find the amplitude and maximum velocity of the particle described by the displacement equation \( x = 3 \sin(2t) + 4 \cos(2t) \).

### Step-by-Step Solution:

1. **Identify the Components**:
The displacement equation is given as:
\[ x = 3 \sin(2t) + 4 \cos(2t) \]
Here, we can identify:
- \( a_1 = 3 \) (coefficient of \( \sin \))
- \( a_2 = 4 \) (coefficient of \( \cos \))

2. **Calculate the Amplitude**:
The resultant amplitude \( A \) of the combined simple harmonic motion can be calculated using the formula:
\[ A = \sqrt{a_1^2 + a_2^2} \]
Substituting the values:
\[ A = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Determine the Angular Frequency**:
The angular frequency \( \omega \) can be identified from the sine and cosine terms. In this case, since both terms have \( 2t \), we have:
\[ \omega = 2 \]

4. **Calculate Maximum Velocity**:
The maximum velocity \( V_{\text{max}} \) in simple harmonic motion is given by the formula:
\[ V_{\text{max}} = A \cdot \omega \]
Substituting the values we found:
\[ V_{\text{max}} = 5 \cdot 2 = 10 \]

5. **Final Answer**:
Thus, the amplitude and maximum velocity are:
\[ \text{Amplitude} = 5, \quad \text{Maximum Velocity} = 10 \]
Therefore, the answer is \( (5, 10) \).