a) $$\frac{{10}}{{\sqrt 3 + 1}}\,{\text{cm}}$$
b) $$20\sqrt 3 \,{\text{cm}}$$
c) $$20\,{\text{cm}}$$
d) $$\frac{{10}}{{\sqrt 3 – 1}}\,{\text{cm}}$$

Answer: d
Explanation:

Let PQ be three and M and N be positions where Mphan stands.
$$\eqalign{ & {\text{Now,}}\,{\text{tan}}{45^ \circ } = 1 = \frac{{PQ}}{{MQ}} \cr & PQ = MQ \cr & \tan {30^ \circ } = \frac{1}{{\sqrt 3 }} = \frac{{PQ}}{{NQ}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{PQ}}{{10 + MQ}} \cr & 10 + MQ = \sqrt 3 \,PQ \cr} $$
$$ 10 + PQ = \sqrt 3 \,PQ$$     (As, PQ = MQ)
$$ PQ = \frac{{10}}{{\sqrt 3 – 1}}\,{\text{cm}}$$