Question

Let the equation $x ^5-10 x ^4+ ax ^3- bx ^2+ cx =0$ has 5 distinct real roots which are in A.P. If the sum of the positive roots is maximum then which of the following must be true?

• A. equation has exactly one negative root.
• B. equation has no negative root.
• C. a = 20
• D. b = 48

Answer 1

Answer: (A), (C)

$ x ^5-10 x ^4+ ax ^3- bx ^2+ cx =0$
Let the roots be $p-2 d, p-d, p, p+d, p+2 d$
$p -2 d + p - d + p + p + d + p +2 d =10 \Rightarrow p =2$
one of the roots of the equation is zero
$\therefore p - d =0 \Rightarrow d =2$
Hence, roots are $-2,0,2,4,6$
Now verify the options.