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4-2.Quadratic Equations and Inequations
hard
Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.
A
$I$ and $II$ only
B
$I$ and $III$ only
C
$II$ and $III$ only
D
All of them
(KVPY-2013)
Solution
(b)
We have,
$\Rightarrow a x^2+(a+b) x+b =0$
$\Rightarrow a x^2+a x+b x+b =0$
$\Rightarrow (a x+b)(x+1) =0$
$\Rightarrow =-\frac{b}{a},-1$
It has at least one negative root, i.e. $-1$. So, it has both roots are real.
$\therefore$ Option $(b)$ is correct.
Standard 11
Mathematics
