KVPY Sample Paper KVPY Stream-SX Model Paper-4

If \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers such that \[\operatorname{Re}\,\,({{z}_{2}})\ne 0,\] \[Re\,\,({{z}_{1}}+{{z}_{2}})=0\] and \[\operatorname{Im}\,({{z}_{1}}{{z}_{2}})=0\]then -

A) \[{{z}_{1}}={{z}_{2}}\]

B) \[{{z}_{1}}={{\bar{z}}_{2}}\]

C) \[{{z}_{1}}=-\,{{\bar{z}}_{2}}\]

D) none of these

Correct Answer: C

Solution :

[C]

Let \[{{z}_{1}}=a+i\,b\] \[{{z}_{2}}=c+i\,d\]
\[\operatorname{Re}\,\,({{z}_{1}}+{{z}_{2}})=0\Rightarrow a+c=0\]	\[i.e.\,\,\,c=-\,a\]
\[\operatorname{Im}\,\,({{z}_{1}}{{z}_{2}})=0\]
\[\Rightarrow ad+bc=0\]	\[6\Omega \]
\[\Rightarrow d=b\]	\[[\because a=-c\ne 0)\]
\[{{z}_{1}}=a+ib\]
\[=c+id\]
\[=-\,(c-id)\]
\[\Rightarrow {{z}_{1}}=-\,{{\bar{z}}_{2}}\]