• question_answer If the equation ${{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1=\text{ }0$ have four positive roots, then choose the incorrect option A) Roots are necessarily integers B) $a+b=2~$ C) $ab=-24~$ D) None of these

 let the roots be ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},$then ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=4$ and ${{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}=1$ $\Rightarrow$$A.M.\,of\,{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}$ $=\operatorname{G}.M.\,of\,{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}$$\Leftrightarrow$${{x}_{1}}={{x}_{2}}={{x}_{3}}={{x}_{4}}$ $\therefore$${{x}_{1}}={{x}_{2}}={{x}_{3}}={{x}_{4}}=1$$\Rightarrow$${{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}$in A.P. as well as G.P. and in H.P. Also ${{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1={{\left( x-1 \right)}^{4}}$ $\Rightarrow$$a=6,b=-4$