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JEE Main & Advanced AIEEE Solved Paper-2010

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    Directions: Questions number 90 are Assertion - Reason type questions. Each of these questions contains two statements: Statement - 1 (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 non zero entries and let\[{{A}^{2}}=I,\]where I is\[2\times 2\]identity matrix. Define Tr(A) = sum of diagonal elements of A and \[|A|=\]determinant of matrix A. Statement - 1 : Tr(A) = 0 Statement - 2 : \[|A|=1\] Statement - 1 (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.     AIEEE  Solved  Paper-2010

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

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

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

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

    Correct Answer: C

    Solution :

    Let \[\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\] \[{{A}^{2}}=I\] \[\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] \[\left[ \begin{matrix}    {{a}^{2}}+bc & ab+bd  \\    ac+dc & bc+{{d}^{2}}  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] \[ab+bd=0\] \[b(a+d)=0\] \[b\ne 0\] so, \[a=d\] \[A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\]                    \[a+b=0\] \[{{T}_{r}}(A)=0\] But \[|A|\ne 1.\] So, statement I is true and statement 2 is false.


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