Question 10 of 295

A light ray is incident, at an incident angle ${\theta }_{1}$, on the system of two plane mirrors ${M}_{1}$ and ${M}_{2}$ having an inclination angle $75^{\circ}$ between them (as shown in figure). After reflecting from mirror ${M}_{1}$ it gets reflected back by the mirror ${M}_{2}$ with an angle of reflection $30^{\circ}$. The total deviation of the ray will be _____ degree.

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Answer: 210

From the above figure, ${i}_{2}=30^{\circ}$,
Now,
$\angle OBA+\angle ABR+\angle RBP=180^{\circ} \Rightarrow 2\angle OBA+60^{\circ}=180^{\circ} \Rightarrow \angle OBA=60^{\circ}$
In triangle $OAB$,
$\angle OBA+\angle AOB+\angle OAB=180^{\circ} \Rightarrow 60^{\circ}+75^{\circ}+\angle OAB=180^{\circ} \Rightarrow \angle OAB=45^{\circ}$
And,
$\angle OAB+\angle BAI+\angle IAQ=180^{\circ} \Rightarrow 2\angle OAB+2{i}_{1}=180^{\circ} \Rightarrow {i}_{1}=45^{\circ}$
Now the total deviation of the ray will be,
$\delta =360^{\circ}-2\left({i}_{1}+{i}_{2}\right) \Rightarrow \delta =360^{\circ}-2\left(30^{\circ}+45^{\circ}\right) \delta =210^{\circ}$