Question

Statement-1: If $a+b+c>0$ and $a<0<b<c$, then both roots of the quadratic equation $a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0$ are real and unequal.
Statement-2: If both roots of the quadratic equation $p{x}^2+qx+r=0$ are of opposite sign then product of roots is negative and sum of roots is positive.

• 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: (C)

We have $(a+b+c)x^2-2(ab+bc+ca)x+3abc=0$
$\therefore$ Discriminant $=D=4\left[(ab+bc+ca{)}^2-3(a+b+c)abc\right]($ Since $a+b+c>0$ and $abc<0)$
$\Rightarrow D>0$
Hence roots are real and distinct.
Product of roots $=\frac{3abc}{a+b+c}<0$
$\Rightarrow$ Roots are of opposite sign.
Statement - 1 is True and Statement - 2 is False.