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JEE Main & Advanced Sample Paper JEE Main Sample Paper-15

  • question_answer
    Let \[\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}.\]Then the value of the  determinant \[\left| \begin{matrix}    1 & 1 & 1  \\    1 & -1-{{\omega }^{2}} & {{\omega }^{2}}  \\    1 & {{\omega }^{2}} & {{\omega }^{4}}  \\ \end{matrix} \right|\]is

    A) \[3\omega \]                                    

    B) \[3\omega (\omega -1)\]

    C) \[3{{\omega }^{2}}\]                                     

    D) \[3\omega (1-\omega )\]

    Correct Answer: B

    Solution :

    \[D=\left| \begin{matrix}    1 & 1 & 1  \\    1 & -1-{{w}^{2}} & {{w}^{2}}  \\    1 & {{w}^{2}} & {{w}^{4}}  \\ \end{matrix} \right|=\left| \begin{matrix}    1 & 1 & 1  \\    1 & +w & {{w}^{2}}  \\    1 & {{w}^{2}} & w  \\ \end{matrix} \right|\] \[=\left| \begin{matrix}    3 & 0 & 0  \\    1 & w & {{w}^{2}}  \\    1 & {{w}^{2}} & w  \\ \end{matrix} \right|\] \[=3[{{w}^{2}}-{{w}^{4}}]=3({{w}^{2}}-w)=3w(w-1)\]


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