Location of roots

If the quadratic equation $(\lambda+2)x^2-3\lambda x+4\lambda=0$, $\lambda\ne-2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of hyperbolas}\}=(\alpha,\beta)$, and
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of an ellipse and a hyperbola, respectively}\}=(\gamma,\infty)$.
Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

Let $\displaystyle\lim_{x\to2}\frac{\left(\tan(x-2)\right)\left(rx^2+(p-2)x-2p\right)}{(x-2)^2}=5$ for some $r,p\in\mathbf R$. If the set of all possible values of q, such that the roots of the equation $rx^2-px+q=0$ lie in $(0,2)$, be the interval $(\alpha,\beta]$, then $4(\alpha+\beta)$ equals :

Let $|M|$ denote the determinant of a square matrix $M$. Let $g : \left[0, \dfrac\pi2\right] \to \mathbb{R}$ be the function defined by
$$g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac\pi2 - \theta\right) - 1}$$
where
$$f(\theta) = \frac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta\\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta + \frac\pi4\right) & \tan\left(\theta - \frac\pi4\right)\\ \sin\left(\theta - \frac\pi4\right) & -\cos\frac\pi2 & \log_e\left(\frac4\pi\right)\\ \cot\left(\theta + \frac\pi4\right) & \log_e\left(\frac\pi4\right) & \tan\pi\end{vmatrix}.$$
Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2) = 2 - \sqrt2$. Then, which of the following is/are TRUE?

If $a\in\mathbf R$ and the equation
$$-3(x-[x])^2+2(x-[x])+a^2=0$$
(where $[x]$ denotes the greatest integer $\le x$) has no integral solution, then all possible values of $a$ lie in the interval :

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 at least 4, is