Idea of Intervals and Signs of Inequations

If $x^2+2ax+10-3a >0$ for all $x \in R$, then

If $l, m\left(l<m\right)$ are roots of $a{x}^2+bx+c=0$, then $\underset{x\to a}{\lim }\frac{\left|a{x}^2+bx+c\right|}{a{x}^2+bx+c}=$

$f\left(x\right)=a{x}^2+bx+2;a,b\in R$. If $f\left(x\right)\ge 0;\forall x\in R$, then the greatest possible value of $b-4a$ is

The following quadratic expression $f\left(x\right)=-x^2+2x-4$ is always

The following quadratic expression $f\left(x\right)=2x^2+4x+4$ is always

The quadratic equations $x^2-6x+4=0$ has

The quadratic equations $x^2-5x+3=0$ has

If the equation $a{x}^2+2bx-3c=0$ has no real roots and $4\left(a+b\right)>3c$, then

Find the real values of $x$ for which $(x^2-4x+1)$ is always negative.

If $a\in \left(m,n\right)$, then the sign of $x^2-(a+2)x+4$ is always positive. Find the value of $-m+n$?

Find the least integral value of $k$ for which, $\left(k-2\right)x^2+8x+k+4>0$ for all $x\in R.$

If $x^2+2\mathrm{px}-2\mathrm{p}+8>0$ for all real values of $x$ then the set of all possible values of $p$ is

The exhaustive range of $\mathrm{\lambda }$ for which $\mathrm{\lambda }{x}^2>7x-\lambda , \forall x\in R$ , is:

For all $x\in R,$ if  $m{x}^2+\left(5m+1\right)>9mx,$ then $m$ lies in the interval _____

If $\alpha \in R^{-}$ and $\left(\alpha –2\right)x^2+8x+\alpha +4<0,\forall x\in R,$ then $\alpha$ cannot be ____

Consider the quadratic equation: $x^2-2\left(4k-1\right)x+15k^2-2k-7>0 \forall x ϵ R$ .Find the integral value of $k$

The values of $a$ such that  $\left(a^2-1\right)x^2+2\left(a-1\right)x+2$ is positive for any $x$ is

Find the number of integral values of $k$ for which ${\mathrm{x}}^2-2(4\mathrm{k}-1)\mathrm{x}+15{\mathrm{k}}^2-2\mathrm{k}-7\ge 0$ hold for all real $x$.

Find least integral value of $k$ such that $(k-2)x^2+8x+k+4$ is positive for all real values of $x$.

Find the largest integral value of $m$ for which the quadratic expression $y=x^2+\left(2m+4\right)x+7m+8$ is positive for all $x\in R$