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?

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

Tardigrade Admin · Accepted Answer

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 ⇒ p =2 one of the roots of the equation is zero ∴ p - d =0 ⇒ d =2 Hence, roots are -2,0,2,4,6 Now verify the options.

Q. Let the equation $x^{5} - 10 x^{4} + a x^{3} - b x^{2} + c x = 0$ 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?

equation has exactly one negative root.

equation has no negative root.

a = 20

b = 48

$x^{5} - 10 x^{4} + a x^{3} - b x^{2} + c x = 0$
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
$∴ p - d = 0 \Rightarrow d = 2$
Hence, roots are $- 2, 0, 2, 4, 6$
Now verify the options.

Q. Let the equation x5−10x4+ax3−bx2+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?