First slide

Theory of expressions

Question

If $α, β$ are the roots of $x^{2} - 3 x + a = 0, a \in R$ and $α < 1 < β,$ then

Moderate

A

$a \in (- \infty, 2)$
B

$a \in (- \infty, 9 / 4]$
C

$a \in (2, 9 / 4]$
D

none of these

Solution

Let $f (x) = x^{2} - 3 x + a .$ Clearly, $y = f (x)$ represents a parabola opening upward
It is give that 1 lies between the roots of $f (x) = 0$
Discriminant $> 0$ and $f (1) < 0$
$\Rightarrow 9 - 4 a > 0$ and $1 - 3 + a < 0$
$\Rightarrow a < \frac{9}{4}$ and $a < 2 \Rightarrow a < 2 \Rightarrow a \in (- \infty, 2)$

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