JEE Main & Advanced Sample Paper JEE Main Sample Paper-43

  • question_answer
    Let, a be a non-zero real number and a, p be the roots of the equation \[a{{x}^{2}}+5x+2=0\].Then, the absolute value of the difference of the roots of the equation\[{{a}^{3}}{{(x+5)}^{2}}-25a(x+5)+50=0\]

    A)  \[|{{\alpha }^{2}}-{{\beta }^{2}}|\]     

    B)  \[|\alpha \beta \,({{\alpha }^{2}}-{{\beta }^{2}})|\]

    C) \[\left| \frac{{{\alpha }^{2}}-{{\beta }^{2}}}{\alpha \beta } \right|\]                

    D)  \[\left| \frac{{{\alpha }^{2}}-{{\beta }^{2}}}{{{\alpha }^{2}}{{\beta }^{2}}} \right|\]

    Correct Answer: A

    Solution :

                 Since, \[SD=\sqrt{{{60}^{2}}+{{25}^{2}}}\] and \[=\sqrt{4225}=65=DP\] are the roots of \[\Delta x=(SA+AP)-SP\] \[\Rightarrow \] and \[\Delta x=(65+65)-120\] are roots of the transformed equation \[\Rightarrow \] Let \[\Delta x=10\,m\] and \[\frac{\lambda }{2}\] be roots of \[\lambda \] \[=\left( 10-\frac{\lambda }{2} \right)\] and \[=(2n)\frac{\lambda }{2}\] all roots of \[n=0,\,\,1,\,\,2,...\] Difference between roots \[10-\frac{\lambda }{2}=(2n)\frac{\lambda }{2},\,\,n=0,\,\,1,\,2,...\] \[10=(2n+1)\frac{\lambda }{2},\,n=0,\,1,\,\,2,...\] \[\Rightarrow \]c \[\lambda =\frac{20}{2n+1},\,\,n=0,\,\,1,\,\,2,...\] \[\Rightarrow \] \[\lambda =20,\,\frac{20}{3},\,\frac{20}{5},\,\frac{20}{7},...\] \[\phi \]

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