(C) Let $f(x) = x^2 - 3x + a$.Since $\alpha < 1 < \beta$,the value of the quadratic expression $f(x)$ at $x = 1$ must be negative,i.e.,$f(1) < 0$.Subs

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(C) Let $f(x) = x^2 - 3x + a$.Since $\alpha Substituting $x = 1$ into the equation:$f(1) = (1)^2 - 3(1) + a $1 - 3 + a $-2 + a $a Since the roots are real and distinct (as $1$ lies between them),the discriminant $D$ must be greater than $0$:$D = b^2 - 4ac = (-3)^2 - 4(1)(a) = 9 - 4a > 0$$9 > 4a \Rightarrow a Combining the conditions $a Thus,$a \in (-\infty, 2)$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then which of the following is true?

A
$a \in (-\infty, 9/4)$
B
$a \in (2, 9/4)$
C
$a \in (-\infty, 2)$
D
None of these

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

Number of integral values of $a$ for which both roots of the quadratic equation $x^2 - (2a + 3)x + a^2 + 3a = 0$ lie in the interval $(0, 4)$ is:

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Let $f(x) = (1 + b^2)x^2 + 2bx + 1$ and $m(b)$ be the minimum value of $f(x)$. If $b$ can take any real value,what is the range of $m(b)$?

Difficult

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The quadratic expression $(2x+1)^{2} - px + q \neq 0$ for any real $x$, if

MediumWBJEE 2015

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Let $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$,where $x \in R$. If $b$ and $c$ are nonzero real numbers such that $\min f(x) > \max g(x)$,then $\left|\frac{c}{b}\right|$ lies in the interval:

MediumAP EAMCET 2025

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If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then which of the following is true?

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