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JEE Main & Advanced AIEEE Solved Paper-2008

  • question_answer
    Directions: Questions number 16 to 20 are Assertion-Reason type questions. Each of these questions contains two statements: Statement-I (Assertion) and Statement-2 (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice. Let A be a \[2\times 2\] matrix with real entries. Let I be the \[2\times 2\] identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that \[{{A}^{2}}=I\]. Statement-1: If \[A\ne I\] and \[A\ne -I\], then det\[A=-I\]. Statement-2: If \[A\ne I\] and \[A\ne -I\], then \[tr\left( A \right)\ne 0\].       AIEEE  Solved  Paper-2007

    A) Statement-1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1.

    B) Statement-1 is true, Statement-2 is false.

    C) Statement-1 is false, Statement-2 is true.

    D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

    Correct Answer: B

    Solution :

    Let \[A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\]                 \[{{A}^{2}}=I\Rightarrow {{a}^{2}}+bc=1,\,bc+{{d}^{2}}=1,\,\]    \[\,\left( a+b \right)b=0,\,\left( a+b \right)c=0\] Out of all possible matrices if we consider \[A=\left[ \begin{matrix}    1 & 0  \\    0 & -1  \\ \end{matrix} \right]\], then tr A = 0. \[\Rightarrow \] Statement-2 is wrong. Again if \[A\ne \pm I\], then \[\left| A \right|=-1\]                 \[\Rightarrow \] Statement-1 is correct.


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