cat 2022 Question 9

Question

Let r and c be real numbers. If r and −r are roots of $5 x^3+c x^2-10 x+9=0$, then c equals

Accepted Answer

Let the roots of the given equation $5x^3 + cx^2 - 10x + 9 = 0$ be r, −r and p

r − r + p = $-\frac{c}{5}$

p = $-\frac{c}{5}$ …… (1)

$-r^2 - pr + pr = -2$

$r^2 = 2$ …… (2)

$-r^2 p = -\frac{9}{5}$

$p = \frac{9}{10}$ …… (3)

Substituting p in (1), we get

$\frac{9}{10} = -\frac{c}{5}$

$-\frac{9}{2} = c$

Question 9.