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JEE Main & Advanced Sample Paper JEE Main Sample Paper-21

  • question_answer
    If the system of equations \[2x-y+z\text{ }=\text{ }0;\text{ }x-2y+z=0;\] \[\lambda x-y+2z=0\]has infinitely many solution and \[f(x)\]be a continuous function such that \[f(x)+f(x+5)=100,\]then the value  of \[\int\limits_{0}^{2\lambda }{f(x)}\,dx\]is equal to

    A)  100                             

    B)  200

    C)  500                             

    D)  1000

    Correct Answer: C

    Solution :

    \[D=0\] \[\left| \begin{matrix}    2 & -1 & 1  \\    1 & -2 & 1  \\    \lambda  & -1 & 2  \\ \end{matrix} \right|=0\] \[2(-4+1)\,+1(2-\lambda )\,+1(2\lambda -1)=0\] \[-6+2-\lambda +2-1=0\Rightarrow \,\lambda =5\] \[I=\int\limits_{0}^{5}{f(x)dx+\int\limits_{5}^{10}{f(x)dx}}\]    Put x = 5 + t \[I=\int\limits_{0}^{5}{f(x)dx+\int\limits_{0}^{5}{f(5+t)dt}}\] \[=\int\limits_{0}^{5}{(f(x)+f(5+x)\,)dx=100\times 5=500}\]


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