Location of Roots

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 value atleast $4$, 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 $p<q<r<s$ then the equation $\left(x-p\right)\left(x-r\right)+5\left(x-q\right)\left(x-s\right)=0$ has

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

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

If the roots of the quadratic equation $\left(4p-p^2-5\right)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

The values of $a$ for which $2x^2-2\left(2a+1\right)x+a\left(a+1\right)=0$ may have one root less than $a$ and other root greater than $a$ is

If $x^2-2x+2\sin \alpha =0$ has unique root in $(-1,1),$ then length of largest continuous interval of $\alpha$ in $[0,2\pi ]$ is

If $e_1$ and $e_2$ are the roots of the equation $x^2-ax+2=0$ where $e_1,e_2$ are the eccentricities of an ellipse and hyperbola respectively then the value of $a$ belongs to

If roots of equation $\left(k^2-4\mathrm{k}+5\right)x^2+\left(2\mathrm{k}-1\right)x-3\mathrm{k}=0$ lie on either sides of unity, then number of integral values of $k$ is

Let $f\left(x\right)=x^2+ax+a.$ If the equation $f\left(x\right)=0$ has exactly one root in $[1,2],$ then number of possible integral values of $a$ is

Number of integral values of $\text{'}a\text{'}$ for which $(\mathrm{a}-5){\mathrm{x}}^2-2\mathrm{ax}+\left(\mathrm{a}-4\right)=0,$ has exactly one root in $\left(0,2\right),$ is

Complete set of values of $a$ for which quadratic equation $x^2-\left(5-a\right)x+a^2-11a+22=0$ has one root greater than $2$ and another less than $2$ is