JEE MAIN - Mathematics (2013 (Offline) - No. 19)
If z is a complex number of unit modulus and argument $$\theta $$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals :
$$ - \theta \,\,$$
$${\pi \over 2} - \theta \,$$
$$\theta \,$$
$$\,\pi - \theta \,\,$$
Explanation
Given $$\,\,\,\,\left| z \right| = 1,\,\,\arg \,z = \theta $$
As we know, $$\,\,\,\,\overrightarrow z = {1 \over z}$$
$$\therefore$$ $$\,\,\,\,\arg \left( {{{1 + z} \over {1 + \overrightarrow z }}} \right) = \arg \left( {{{1 + z} \over {1 + {1 \over z}}}} \right)$$
$$ = \arg \left( z \right) = \theta .$$
As we know, $$\,\,\,\,\overrightarrow z = {1 \over z}$$
$$\therefore$$ $$\,\,\,\,\arg \left( {{{1 + z} \over {1 + \overrightarrow z }}} \right) = \arg \left( {{{1 + z} \over {1 + {1 \over z}}}} \right)$$
$$ = \arg \left( z \right) = \theta .$$
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