Question
If $$\alpha ,\beta ,\gamma \in R,$$ then the determinant \[\Delta = \left| {\begin{array}{*{20}{c}}
{{{\left( {{e^{i\alpha }} + {e^{ - i\alpha }}} \right)}^2}}&{{{\left( {{e^{i\alpha }} - {e^{ - i\alpha }}} \right)}^2}}&4\\
{{{\left( {{e^{i\beta }} + {e^{ - i\beta }}} \right)}^2}}&{{{\left( {{e^{i\beta }} - {e^{ - i\beta }}} \right)}^2}}&4\\
{{{\left( {{e^{i\gamma }} + {e^{ - i\gamma }}} \right)}^2}}&{{{\left( {{e^{i\gamma }} - {e^{ - i\gamma }}} \right)}^2}}&4
\end{array}} \right|\] is
A.
independent of $$\alpha ,\beta $$ and $$\gamma $$
B.
dependent on $$\alpha ,\beta $$ and $$\gamma $$
C.
independent of $$\alpha ,\beta $$ only
D.
independent of $$\alpha ,\gamma $$ only
Answer :
independent of $$\alpha ,\beta $$ and $$\gamma $$
Solution :
$${C_1} \to {C_1} - {C_2}$$
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
4&{{{\left( {{e^{i\alpha }} - {e^{ - i\alpha }}} \right)}^2}}&4\\
4&{{{\left( {{e^{i\beta }} - {e^{ - i\beta }}} \right)}^2}}&4\\
4&{{{\left( {{e^{i\gamma }} - {e^{ - i\gamma }}} \right)}^2}}&4
\end{array}} \right| = 0\]