JEE MAIN - Mathematics (2014 (Offline) - No. 18)
If z is a complex number such that $$\,\left| z \right| \ge 2\,$$, then the minimum value of $$\,\,\left| {z + {1 \over 2}} \right|$$ :
is strictly greater that $${{5 \over 2}}$$
is strictly greater that $${{3 \over 2}}$$ but less than $${{5 \over 2}}$$
is equal to $${{5 \over 2}}$$
lie in the interval (1, 2)
Explanation
We know minimum value of
$$\,\,\,\left| {{Z_1} + {Z_2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| {{Z_1}} \right| - \left| {{Z_2}} \right|} \right|$$
Thus minimum value of
$$\,\,\,\left| {Z + {1 \over 2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| Z \right| - {1 \over 2}} \right|$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \left| {Z + {1 \over 2}} \right| \le \left| Z \right| + {1 \over 2}$$
Since, $$\,\,\,\left| Z \right| \ge 2$$
$$\therefore$$ $$\,\,\,2 - {1 \over 2} < \left| {Z + {1 \over 2}} \right| < 2 + {1 \over 2}$$
$$ \Rightarrow {3 \over 2} < \left| {Z + {1 \over 2}} \right| < {5 \over 2}$$
$$\,\,\,\left| {{Z_1} + {Z_2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| {{Z_1}} \right| - \left| {{Z_2}} \right|} \right|$$
Thus minimum value of
$$\,\,\,\left| {Z + {1 \over 2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| Z \right| - {1 \over 2}} \right|$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \left| {Z + {1 \over 2}} \right| \le \left| Z \right| + {1 \over 2}$$
Since, $$\,\,\,\left| Z \right| \ge 2$$
$$\therefore$$ $$\,\,\,2 - {1 \over 2} < \left| {Z + {1 \over 2}} \right| < 2 + {1 \over 2}$$
$$ \Rightarrow {3 \over 2} < \left| {Z + {1 \over 2}} \right| < {5 \over 2}$$
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