A) Roots are necessarily integers
B) \[a+b=2~\]
C) \[ab=-24~\]
D) None of these
Correct Answer: D
Solution :
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 \[{{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1={{\left( x-1 \right)}^{4}}\] |
\[\Rightarrow \]\[a=6,b=-4\] |
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