A) if \[{{p}^{2}}-2q<0\] , then the equation has one real and two imaginary root.
B) If \[{{p}^{2}}-2q\ge 0\] then the equation has all real roots.
C) If \[{{p}^{2}}-2q>0\], then the equation has all real and distinct roots.
D) If \[4{{p}^{3}}-27{{q}^{2}}>0\], then the equation has real and distinct roots.
Correct Answer: A
Solution :
Let \[f(x)={{x}^{3}}+p{{x}^{2}}+qx+r\] \[f'(x)=3{{x}^{2}}+2px+q\] Disc. \[=4{{p}^{2}}-12q=4({{p}^{2}}-3q)=4({{p}^{2}}-2p-q)\] \[\therefore \] If \[{{p}^{2}}<2q\Rightarrow {{p}^{2}}<3q\] So, the equation \[f(x)=0\]has one real and two imaginary roots.
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