Question

Statement-1: If $a>b>c$ and $a^3+b^3+c^3=3abc$ then the quadratic equation $a^2+bx+c$ $=0$ has roots of opposite sign.
Statement-2: If roots of a quadratic equation $a{x}^2+bx+c=0$ are of opposite sign then product of roots $<0$ and $\mid$ sum of roots $\mid \ge 0$

• A. Statement-1 is true, statement- 2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement- 2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement- 2 is false.
• D. Statement-1 is false, statement- 2 is true.

Answer 1

Answer: (A)

$a>b>c\Rightarrow a,b,c$ are distinct real
$\text{ also }{a}^3+b^3+c^3-3abc=0$
$\left(\frac{a+b+c}{2}\right)\left[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\right]=0\text{ as a, b, c are distinct }$
$\therefore a+b+c=0$
hence $x=1$ is a root of $a{x}^2+bx+c=0$

$a+b+c=0$ and $a>b>c\Rightarrow a$ and $c$ are of oppsite sign otherwise $a+b+c\ne 0$ therefore $\frac{c}{a}$ negative