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KVPY Sample Paper KVPY Stream-SX Model Paper-22

  • question_answer
    If a, \[{{\alpha }_{1}},\]\[{{\alpha }_{2}},\]\[...{{\alpha }_{2n\,-1}},\] b are in A.P., a, \[{{\beta }_{1}},\]\[{{\beta }_{2}},\]?, \[{{\beta }_{2n\,-1}},\]b are in G.P. and a, \[{{\gamma }_{1}},\]\[{{\gamma }_{2}},\]\[...{{\gamma }_{2n\,-1}},\] b are in H.P., where a, b are positive, then the equation \[{{\alpha }_{n}}\,\,{{x}^{2}}-{{\beta }_{n}}\,\,x+{{\gamma }_{n}}=0\] has

    A) real and equal, roots

    B) real and unequal roots

    C) imaginary roots

    D)  roots which are in A.P.

    Correct Answer: C

    Solution :

    \[{{\alpha }_{n}}=\frac{a+b}{2},\] \[{{\beta }_{n}}\sqrt{ab},\] \[{{\gamma }_{n}}=\frac{2ab}{a+b}\]
    \[\therefore \]      the equation is
    \[\frac{a+b}{2}{{x}^{2}}-\sqrt{ab}\,x+\frac{2ab}{a+b}=0\]
    its discriminant D < 0
    \[\therefore \]      the roots are imaginary


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