(C) Given $f(x)=ax^2-bx-a$ is a quadratic expression such that $f(x) \leq K, \forall x \in R$. This implies $ax^2-bx-a-K \leq 0, \forall x \in R$. For

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(C) Given $f(x)=ax^2-bx-a$ is a quadratic expression such that $f(x) \leq K, \forall x \in R$. This implies $ax^2-bx-a-K \leq 0, \forall x \in R$. For a quadratic expression $Ax^2+Bx+C$ to be less than or equal to $0$ for all $x$,the coefficient of $x^2$ must be negative $(a Here,$D = (-b)^2 - 4(a)(-a-K) \leq 0$. $b^2 + 4a(a+K) \leq 0$. $b^2 + 4a^2 + 4aK \leq 0$. Since $b^2 + 4a^2 \geq 0$,for the expression to be $\leq 0$,$4aK$ must be negative. Given $a Thus,$K > 0$.

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

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$f(x)=ax^2-bx-a$ is a quadratic expression. If $K$ is the least real number such that $f(x) \leq K, \forall x \in R$,then

TS EAMCET 2021 All TS EAMCET

A
$K=0$
B
$K < -2$
C
$K > 0$
D
$-1 < K < 0$

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

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For what possible values of $a$ does $6$ lie between the roots of the equation $x^2 + 2(a - 3)x + 9 = 0$?

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Match the following: Consider the equation $x^2 + 2(a - 1)x + a + 5 = 0$. Match the real values of $a$ with the conditions on the roots of the given equation.

Column-$I$ Column-$II$

$A$. Imaginary roots $P$. $a \in (-1, 4)$

$B$. One root less than $3$ and other greater than $3$ $Q$. $a \in (-\infty, -1)$

$C$. One root less than $1$ and other greater than $3$ $R$. $a \in (-\infty, -4/3)$

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

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DifficultJEE MAIN 2023

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If the quadratic equation $ax^2+bx+c=0$ $(a>0)$ has two roots $\alpha$ and $\beta$ such that $\alpha < -2$ and $\beta > 2$, then which of the following is true?

EasyWBJEE 2024

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If the roots of ${x^2} + x + a = 0$ exceed $a$,then

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$f(x)=ax^2-bx-a$ is a quadratic expression. If $K$ is the least real number such that $f(x) \leq K, \forall x \in R$,then

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For what possible values of $a$ does $6$ lie between the roots of the equation $x^2 + 2(a - 3)x + 9 = 0$?

The number of integral values of $k$,for which one root of the equation $2x^2-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$,is:

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