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SSC Sample Paper SSC-CGL TIER - I Sample Test Paper-6

  • question_answer
    Consider the following statements in     respect of the quadratic equation  \[a{{x}^{2}}+bx+b=0\], where \[a\ne 0:\]
    1. The product of the roots is equal to the sum of the roots.            
    2. The roots of the equation are always unequal and real.          
    Which of the above statements is/are correct?                            

    A) Only                

    B) Only

    C) Both 1 and 2                 

    D) Neither 1 nor 2

    Correct Answer: D

    Solution :

    \[~a{{x}^{2}}+bx+b=0\]\[\Rightarrow \,\,{{x}^{2}}+\frac{b}{a}x+\frac{b}{a}=0\] So, sum of roots, \[\alpha +\beta =\frac{-b}{a}\]and products of roots, \[\alpha \beta =\frac{b}{a}\] Hence, product of roots is not equal to the sum of roots, so Statement I is not correct. Now, for roots to be real and unequal, Determinant D > 0 \[\Rightarrow ~{{b}^{2}}-4ac>0\Rightarrow ~{{b}^{2}}-4a\times b>0\] \[\Rightarrow ~{{b}^{2}}-4ab>0\](here, b = b, a = a and c = b) \[\Rightarrow {{b}^{2}}>4ab\Rightarrow ~b>4a\] So, if b > 4 a, then roots are unequal and real, so Statement II is not always true. It will depend on values of 'a' and 'b'


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