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JEE Main & Advanced JEE Main Paper (Held On 11 April 2014)

  • question_answer
    Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in \[x,p{{'}_{i}}(x)\]and \[p'{{'}_{i}}(x)\]be the first and second order derivatives of \[{{p}_{i}}(x)\] respectively. Let,\[A(x)=\left[ \begin{matrix}    {{p}_{1}}(x) & {{p}_{1}}'(x) & {{p}_{1}}''(x)  \\    {{p}_{2}}(x) & {{p}_{2}}'(x) & {{p}_{2}}''(x)  \\    {{p}_{3}}(x) & {{p}_{3}}'(x) & {{p}_{3}}''(x)  \\ \end{matrix} \right]\]and \[B(x)={{[A(x)]}^{T}}A(x).\]Then determinant of B(x):   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

    A) is a polynomial of degree 6 in x.

    B) is a polynomial of degree 3 in x.

    C) is a polynomial of degree 2 in x.

    D) does not depend on x.

    Correct Answer: A

    Solution :

                    let\[{{p}_{1}}x={{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}\] \[{{p}_{2}}x={{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}\] and\[{{p}_{3}}x={{a}_{3}}{{x}^{2}}+{{b}_{3}}x+{{c}_{3}}\] where \[{{a}_{1}},{{a}_{2}},{{a}_{3}},{{b}_{1}},{{b}_{2}},{{b}_{3}},{{c}_{1}},{{c}_{2}},{{c}_{3}}\]are real numbers. \[\therefore \]\[A(x)=\left[ \begin{matrix}    {{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}} & 2{{a}_{1}}x+{{b}_{1}} & 2{{a}_{1}}  \\    {{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}} & 2{{a}_{2}}x+{{b}_{2}} & 2{{a}_{2}}  \\    {{a}_{3}}{{x}^{2}}+{{b}_{3}}x+{{c}_{3}} & 2{{a}_{3}}x+{{b}_{3}} & 2{{a}_{3}}  \\ \end{matrix} \right]\] \[B(x)=\left[ \begin{matrix}    {{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}} & {{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}} & {{a}_{3}}{{x}^{2}}+{{b}_{3}}x+{{c}_{3}}  \\    2{{a}_{1}}x+{{b}_{1}} & 2{{a}_{2}}x+{{b}_{2}} & 2{{a}_{3}}x+{{b}_{2}}  \\    2{{a}_{1}} & 2{{a}_{2}} & 2{{a}_{3}}  \\ \end{matrix} \right]\]    \[\times =\left[ \begin{matrix}    {{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}} & 2{{a}_{1}}x+{{b}_{1}} & 2{{a}_{1}}  \\    {{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}} & 2{{a}_{2}}x+{{b}_{2}} & 2{{a}_{2}}  \\    {{a}_{3}}{{x}^{2}}+{{b}_{3}}x+{{c}_{3}} & 2{{a}_{3}}x+{{b}_{3}} & 2{{a}_{3}}  \\ \end{matrix} \right]\] It is clear from the above multiplication, the degree of determinant of B(x) can not be less than 4.


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