Graph of Quadratic Expression

The least value of the quadratic polynomial, $f\left(x\right)=\left(2p^2+1\right)x^2+2\left(4p^2-1\right)x+4\left(2p^2+1\right)$ for real values of $p$ and $x$ is

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

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12x+\left[x\right]+31=0$ and $x^2-5|x+2|-4=0$ respectively, where $\left[x\right]$ denotes the greatest integer $\le x$. Then $m^2+mn+n^2$ is equal to

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

If $f:\left[0,2\right]\to \left[0,2\right]$ is bijective function defined by $f\left(x\right)=a{x}^2+bx+c,$ where $a, b \& c$ are non-zero real numbers, then $f\left(2\right)$ is equal to

Let $f\left(x\right)=k{x}^2+x\left(3-4k\right)-12$. If the set of values of $k$ for which $f\left(x\right)<0\forall x\in \left(-3,3\right)$ and $f\left(-4\right)>0$ is $(p,q]$ then find the value of $\left(4p+3q\right)$.

If the range of $f\left(x\right)=\frac{x^2+ax+b}{x^2+2x+3}$ is $\left[-5,4\right] a,b\in N$, then the value of $a+b$ is

The range of the function  $f\left(x\right)=\frac{3{\mathrm{x}}^2+9\mathrm{x}+17}{3{\mathrm{x}}^2+9\mathrm{x}+7}$ for all $x\in R$  is

For $x\in R$, find the set of values attainable by

(i) $\frac{x^2-3x+4}{x^2+3x+4}$, (ii) $\frac{x^2+2x+1}{x^2+2x+7}$, (iii) $\frac{x^2+x-3}{x^2+x}$

The range of value of $\lambda$ for which the expression$\frac{2x^2-5x+3}{4x-\lambda }$ can take all real values for $x\in R-\left\{\frac{\lambda }{4}\right\}$, is

The range of the expression $\frac{\left(2+x^2\right)}{\left(4-x^2\right)}$ is

The range of the function $f\left(x\right)=\frac{3x^2+9x+17}{3x^2+9x+7}$ for all $x\in R$ is 

The graph of quadratic polynomial $f\left(x\right)=(a-x)(x-b)$ where, $a, b>0$ and $a\ne b$, does not pass through

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 $x\in R.$

Find the range of the following function:

$f\left(x\right)=\frac{1}{x-2}$.

Determine whether the graph of function is a parabola that opens upward or downward: $f\left(x\right)=6x^2+7x-9$.

Determine whether the graph of function is a parabola that opens upward or downward: $f\left(x\right)=-3x^2+2x-4$.

For real number $x$, if the minimum value of $f\left(x\right)=x^2+2bx+2c^2$ is greater than the maximum value of $g\left(x\right)=-x^2-2cx+b^2$, then