11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer If \[a<b<c<d\], then the roots of the equation \[(x-a)(x-c)+2(x-b)(x-d)=0\] are [IIT 1984]

    A) Real and distinct

    B) Real and equal

    C) Imaginary

    D) None of these

    Correct Answer: A

    Solution :

    Given equation can be rewritten as \[3{{x}^{2}}-(a+c+2b+2d)x+(ac+2bd)=0\] Its discriminant D \[={{(a+c+2b+2d)}^{2}}-4.3(ac+2bd)\] \[={{\left\{ (a+2d)+(c+2b) \right\}}^{2}}-12(ac+2bd)\]      \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+4(a+2d)(c+2b)-12(ac+2bd)\]     \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}-8ac+8ab+8dc-8bd\] \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+8(c-b)(d-a)\] which is +ve, since \[a<b<c<d\]. Hence roots are real and distinct.

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