\[(a)\;\tan ^{-1}\bigg(\frac{1}{2}\bigg)\quad (b)\;\tan ^{-1}\bigg(\frac{1}{3}\bigg) \quad (c)\;\tan ^{-1}\bigg(\frac{4}{5}\bigg) \quad (d)\;\tan ^{-1}(3)\]

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Velocity of rain $=v_r=20 j$

Velocity of man $=v_m=5 i$

Velocity of wind $=15 i$

Velocity of rain with respect to man when wind is blowing is

$\quad=15 i-5i+20j$

$\quad=20j+10 i$

$\tan \theta=\large\frac{20}{10}$

$\qquad=2$ from the horizontal.

Hence the person should hold is umbrella at $ \alpha$ (with vertical)

$\quad=\tan ^{-1}\bigg(\large\frac{1}{2}\bigg)$

Hence a is the correct answer

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