First slide

Theory of equations

Question

Let $f (x) = {ax}^{2} - bx + c^{2}, b \neq 0$ and $f (x) \neq 0$ for all $x \in R$ . Then

Easy

A

$a + c^{2} < b$
B

$4 a + c^{2} > 2 b$
C

$9 a - 3 b + c^{2} < 0$
D

none of these

Solution

Here, ${ax}^{2} - bx + c^{2} = 0$ does not have real roots. So,
$D < 0 \Rightarrow b^{2} - 4 {ac}^{2} < 0 \Rightarrow a > 0$
Therefore, f(x) is always positive. So,
$f (2) > 0 \Rightarrow 4 a - 2 b + c^{2} > 0$

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