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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Self Evaluation Test - Determinats

  • question_answer
    If \[\alpha .\beta .\gamma \in R,\]then the determinant\[\Delta =\left| \begin{matrix}    {{({{e}^{i\alpha }}+{{e}^{-i\alpha }})}^{2}} & {{({{e}^{i\alpha }}-{{e}^{-i\alpha }})}^{2}} & 4  \\    {{({{e}^{i\beta }}+{{e}^{-i\beta }})}^{2}} & {{({{e}^{i\beta }}-{{e}^{-i\beta }})}^{2}} & 4  \\    {{({{e}^{i\gamma }}+{{e}^{-i\gamma }})}^{2}} & {{({{e}^{i\gamma }}-{{e}^{-i\gamma }})}^{2}} & 4  \\ \end{matrix} \right|\] 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

    Correct Answer: A

    Solution :

    [a] \[{{C}_{1}}\to {{C}_{1}}-{{C}_{2}}\] \[\Rightarrow \left| \begin{matrix}    4 & {{({{e}^{i\alpha }}-{{e}^{-i\alpha }})}^{2}} & 4  \\    4 & {{({{e}^{i\beta }}-{{e}^{-i\beta }})}^{2}} & 4  \\    4 & {{({{e}^{i\gamma }}-{{e}^{-i\gamma }})}^{2}} & 4  \\ \end{matrix} \right|=0\]


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