Location of Roots

For how many integral values of $k$, the equation $x^2-4x+k=0$, where $k$ is an integer has real roots and both of them lie in the interval $\left(0, 5\right)$?

If $\alpha +\frac{1}{\alpha }, \beta +\frac{1}{\beta }$ are zeros of $f\left(x\right)=x^2-5x-a,$ where $\alpha , \beta \in \left(0,\infty \right)$ for all $x\in R$, then the complete set of values of $a$ is

Let $a,b,c\in R$ and $a\ne 0$ be such that ${\left(a+c\right)}^2<b^2$, then the quadratic equation $a{x}^2+bx+c=0$ has 

The value of $a$ for which the quadratic expression $a{x}^2+\left|2a-3\right|x-6$ is positive for exactly three integral values of $x$ is

If the quadratic equation $f\left(x\right)=p{x}^2-qx+r=0$ has two distinct roots in $(0, 2)$ where $p, q, r\in N$ and $f\left(1\right)=-1$then the minimum value of $p$ is

The values of $k$ for which each root of the equation $x^2-6kx+2-2k+9k^2=0$ is greater than $3$ , always satisfy the inequality

Given $f\left(x\right)=x^2+\left(a+2\right)x+a^2-a+2$ and $f\left(-2\right)<0$.  Then $a$ lies in the interval:

The values of $\text{'}\mathrm{a}\text{'}$ for which the roots of the equation ${\mathrm{x}}^2+\mathrm{x}+\mathrm{a}=0$ are real and exceed $\text{'}\mathrm{a}\text{'}$ are -

For the given equation $2x^3+3x+k=0$, what are the values of $k$ so that it contains two distinct real roots in the interval $\left[0,1\right]:$

The range of $a$ for which the equation $x^2+ax-4=0$ has its smaller root in the interval $\left(-1,\mathrm{ }2\right)$ is

Let the values of $a$ for which one root of the equation $\left(a-5\right)x^2-2ax+\left(a-4\right)=0$ is smaller than $1$ and the other greater than $2$ be $(m, n)$. Find $n-m.$

If $2$ lies between both the roots of $x^2-\left(\lambda +1\right)x+{\lambda }^2+\lambda -8=0$, then 

Consider the function $f\left(x\right)=\frac{x^3}{4}-\sin \pi x+3$

The smallest value of $k,$ for which both the roots of the equation $x^2-8kx+16\left(k^2-k+1\right)=0$ are real, distinct and have values atleast $4,$ is $\dots \dots$

If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$ then $m$ lies in the interval 

Consider the quadratic equation, $(c-5)x^2-2cx+(c-4)$ $=0,c\ne 5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0,2)$ and its other root lies in the interval $(2,3) .$ Then, the number of elements in $S$ is 

Consider the inequality $x^2-2x+a-5<0$, for at least one negative value of $x$, the complete set of values of ‘$a$’ is:

If both roots of the quadratic equation $x^2=mx-4$ are real and distinct and they lie in the interval $\left(1,5\right),$ then $m\in$_____

Note: In the actual JEE paper interval was $\left[1, 5\right]$

If the roots of the quadratic equation $x^2-2ax+a^2=3-a$ are real and less than $3$, then

Let $S$ denotes the set of all real values of $a$ for which the roots of the equation $x^2-2ax=1-a^2$ lie between $5$ and $10$, then $S$ equals to :