Three sinusoidal waves having same frequency have amplitudes in the ratio 6:3:2 and their phases are `0, pi/2 ,pi` . If 6a be the amplitude of the first wave , then which of the following is not correct

Options

A. the amplitude of the resultant wave of the three waves is 5a
B. phase of resultant wave is `tan^(-1) (3/4)`
C. the amplitude of resultant wave is 5a/6
D. the frequency of the resultant wave will be that of given wave

Answer

To solve the problem, we will analyze the three sinusoidal waves given their amplitudes and phases, and then find the resultant wave.

### Step 1: Define the waves
Given:
- Amplitude of the first wave: \(6a\)
- Amplitude of the second wave: \(3a\)
- Amplitude of the third wave: \(2a\)
- Phases:
- First wave: \(0\)
- Second wave: \(\frac{\pi}{2}\)
- Third wave: \(\pi\)

The equations for the waves can be written as:
1. \(y_1 = 6a \sin(\omega t + 0) = 6a \sin(\omega t)\)
2. \(y_2 = 3a \sin(\omega t + \frac{\pi}{2}) = 3a \cos(\omega t)\) (since \(\sin(\theta + \frac{\pi}{2}) = \cos(\theta)\))
3. \(y_3 = 2a \sin(\omega t + \pi) = -2a \sin(\omega t)\) (since \(\sin(\theta + \pi) = -\sin(\theta)\))

### Step 2: Write the resultant wave
Now, we can write the resultant wave \(y\) as:
\[ y = y_1 + y_2 + y_3 \]
Substituting the expressions we found:
\[ y = 6a \sin(\omega t) + 3a \cos(\omega t) - 2a \sin(\omega t) \]
Combining the sine terms:
\[ y = (6a - 2a) \sin(\omega t) + 3a \cos(\omega t) = 4a \sin(\omega t) + 3a \cos(\omega t) \]

### Step 3: Find the amplitude of the resultant wave
To find the amplitude of the resultant wave, we can use the formula for the amplitude \(A\) of the resultant wave when combining sine and cosine:
\[ A = \sqrt{(4a)^2 + (3a)^2} \]
Calculating this:
\[ A = \sqrt{16a^2 + 9a^2} = \sqrt{25a^2} = 5a \]

### Step 4: Determine the phase of the resultant wave
The phase \(\phi\) can be determined using:
\[ \tan(\phi) = \frac{\text{Coefficient of } \cos(\omega t)}{\text{Coefficient of } \sin(\omega t)} = \frac{3a}{4a} = \frac{3}{4} \]
Thus, \(\phi = \tan^{-1}\left(\frac{3}{4}\right)\).

### Conclusion
- The amplitude of the resultant wave is \(5a\).
- The phase of the resultant wave is \(\tan^{-1}\left(\frac{3}{4}\right)\).

The question asks which statement is not correct. Based on our calculations:
1. The amplitude of the resultant wave is \(5a\) (correct).
2. The phase of the resultant wave is \(\tan^{-1}\left(\frac{3}{4}\right)\) (correct).
3. If any statement claims the amplitude is \(5/6\) (incorrect).
4. The frequency of the resultant wave is the same as the original waves (correct).

Thus, the statement that the amplitude of the resultant wave is \(5/6\) is not correct.