JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

If a, b, c, d\[\in \]R, then the equation \[({{x}^{2}}+ax-3b)\]\[({{x}^{2}}-cx+b)\]\[({{x}^{2}}-dx+2b)\]=0 has

A) 6 real roots

B) at least 2 real roots

C) 4 real toots

D) 3 real roots

Correct Answer: B

Solution :

[b] The discriminants of the given equations are \[{{D}_{1}}={{a}^{2}}+12b\], \[{{D}_{2}}={{c}^{2}}-4b\] and \[{{D}_{3}}={{d}^{2}}-8b\].

\[\therefore {{D}_{1}}+{{D}_{2}}+{{D}_{3}}={{a}^{2}}+{{c}^{2}}+{{d}^{2}}\ge 0\]

Hence, at least one \[{{D}_{1}}\],\[{{D}_{2}}\],\[{{D}_{3}}\] is non-negative. Therefore, the equation has at least two real roots.