JEE Main & Advanced Sample Paper JEE Main Sample Paper-5

  • question_answer
    The value of \[\theta \] in \[[0,2\pi ]\] such that the matrix \[\left[ \begin{matrix}    2\sin \theta -1 & \sin \theta  & \cos \theta   \\    \sin (\theta +\pi ) & 2\cos \theta -\sqrt{3} & \tan \theta   \\    \cos (\theta -\pi ) & \tan (\pi -\theta ) & 0  \\ \end{matrix} \right]\] is a skew symmetric is

    A)  \[\frac{\pi }{2}\]                                             

    B)  \[\frac{\pi }{3}\]

    C)  \[\frac{\pi }{4}\]                                             

    D)  \[\frac{\pi }{6}\]

    Correct Answer: D

    Solution :

     Idea A square matrix \[A=[{{a}_{ij}}]\] is skew symmetric matrix if \[{{a}_{ij}}=-{{a}_{ji}}\] for all the diagonal elements of a skew symmetric matrix is zero. e.g., \[A=\left| \begin{matrix}    o & h & g  \\    -h & o & f  \\    -g & -f & o  \\ \end{matrix} \right|\] is a skew symmetric matrix A'= -A The given matrix is\[\left[ \begin{matrix}    2\sin \theta -1 & \sin \theta  & \cos \theta   \\    \sin (\theta +\pi ) & 2\cos \theta -\sqrt{3} & \tan \theta   \\    \cos (\theta -\pi ) & \tan (\pi -\theta ) & 0  \\ \end{matrix} \right]\] It is given that the above matrix is skew symmetric. So, its all diagonal entries are zero. \[\Rightarrow \]\[2\sin \theta -1=0\]and\[2\cos \theta -\sqrt{3}=0\] \[\Rightarrow \]\[\sin \theta =\frac{1}{2}\]and\[\cos \theta =\frac{\sqrt{3}}{2}\] \[\Rightarrow \]\[\sin \theta =\sin \frac{\pi }{6}\]and\[\cos \theta =\cos \frac{\pi }{6}\] \[\Rightarrow \]\[\theta =\frac{\pi }{6}\in [0,2\pi ]\] TEST Edge Transpose matrix, symmetric matrix related question are asked. To solve these types of question, students are advised to understand the concept of these matrices and learn the properties of matrix.

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