KVPY Sample Paper KVPY Stream-SX Model Paper-21

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

Let \[{{z}_{1}}=a+i\,\,b\] \[{{z}_{2}}=c+i\,\,d\]

\[\operatorname{Re}\,\,({{z}_{1}}+{{z}_{2}})=0\] \[\Rightarrow \] \[a+c=0\] i.e. \[c=-\,a\]

\[\text{lm(}{{z}_{1}}{{z}_{2}}\text{)=0}\] \[\Rightarrow \] \[ad+bc=0\] i.e. \[a\,(d-b)=0\]\[\Rightarrow \] \[d=b\] \[[\because a=-c\ne 0]\]

\[{{z}_{1}}=a+ib=-\,c+id=-\,(c-id)\]\[\Rightarrow \] \[{{z}_{1}}=-\,{{\bar{z}}_{2}}\]