A system of three polarizers P$$_1$$, P$$_2$$, P$$_3$$ is set up such that the pass axis of P$$_3$$ is crossed with respect to that of P$$_1$$. The pass axis of P$$_2$$ is inclined at 60° to the pass axis of P$$_3$$. When a beam of unpolarized light of intensity I$$_0$$ is incident on P$$_1$$, the intensity of light transmitted by the three polarizers is I. The ratio (I$$_0$$/I) equals (nearly):

Let the incident unpolarized light have intensity $$I_0$$. For unpolarized light passing through a single polarizer, the well-known result is that exactly one half of the incident intensity is transmitted. Mathematically,

$$I_1=\dfrac{I_0}{2}.$$

Here $$I_1$$ is the intensity just after the first polarizer $$P_1$$.

Now the second polarizer $$P_2$$ makes an angle of $$30^{\circ}$$ with respect to $$P_1$$ because $$P_1$$ is at $$90^{\circ}$$ from $$P_3$$ while $$P_2$$ is at $$60^{\circ}$$ from $$P_3$$, giving

$$\theta_{12}=90^{\circ}-60^{\circ}=30^{\circ}.$$

Whenever already plane-polarized light of intensity $$I$$ meets another polarizer at an angle $$\theta$$, Malus’s law states:

$$I_{\text{after}} = I_{\text{before}}\cos^2\theta.$$

Applying this law to the passage from $$P_1$$ to $$P_2$$, we obtain

$$I_2 = I_1\cos^2 30^{\circ}.$$

Because $$\cos 30^{\circ}= \dfrac{\sqrt3}{2},$$ we have

$$I_2=\dfrac{I_0}{2}\left(\dfrac{\sqrt3}{2}\right)^2=\dfrac{I_0}{2}\cdot\dfrac{3}{4}=\dfrac{3I_0}{8}.$$

Next the third polarizer $$P_3$$ is crossed with $$P_1$$, so its axis differs from that of $$P_2$$ by the given $$60^{\circ}$$. Applying Malus’s law once more with $$\theta_{23}=60^{\circ},$$

$$I_3 = I_2\cos^2 60^{\circ}.$$

Because $$\cos 60^{\circ}= \dfrac12,$$ this gives

$$I_3 = \dfrac{3I_0}{8}\left(\dfrac12\right)^2=\dfrac{3I_0}{8}\cdot\dfrac14=\dfrac{3I_0}{32}.$$

The intensity emerging from the complete set of three polarizers is therefore

$$I=\dfrac{3I_0}{32}.$$

Taking the ratio asked for in the problem,

$$\frac{I_0}{I}= \frac{I_0}{\dfrac{3I_0}{32}} = \frac{32}{3}\approx 10.67.$$

Hence, the correct answer is Option D.