1. Use the interference intensity formula

For two coherent beams of intensities $I_1$ and $I_2$, the resultant intensity is

$I_r = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$

where $\phi$ is the phase difference.

Here,

$I_1 = I, \qquad I_2 = 9I$

So,

$\sqrt{I_1 I_2} = \sqrt{I\cdot 9I} = 3I$

Hence,

$I_r = I + 9I + 2(3I)\cos\phi = 10I + 6I\cos\phi$

2. Intensity at point $P$

At point $P$, phase difference is

$\phi_P = \frac{\pi}{2}$

Since

$\cos\frac{\pi}{2} = 0$

therefore,

$I_P = 10I + 6I(0) = 10I$

3. Intensity at point $Q$

At point $Q$, phase difference is

$\phi_Q = \pi$

Since

$\cos\pi = -1$

therefore,

$I_Q = 10I + 6I(-1) = 4I$

4. Find the difference

$I_P - I_Q = 10I - 4I = 6I$

So, the difference between the resultant intensities at $P$ and $Q$ is

$\boxed{6I}$

5. Option check

• A: $2I$ ❌
• B: $6I$ ✅
• C: $5I$ ❌
• D: $7I$ ❌

Therefore, the correct option is B.