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.You need to login to perform this action.
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