Question

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

• 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^3-3 a b c=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 $ax ^2+ bx + c =0$

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