Let $S = \{x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$ and $T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$. Then the number of elements in $S \cap T$ is $....$

If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0 < \alpha_1 < \beta_1 < \gamma_1 < \delta_1$,$0 < \alpha_2 < \beta_2 < \gamma_2 < \delta_2$,$\alpha_1-\delta_2=2$,$\beta_1-\gamma_2=2$,$\gamma_1-\beta_2=4$,and $\delta_1-\alpha_2=4$,then $a+b+c+d+e=$

If the extreme value of $3x - 2x^2 + 1$ is $k$, then the set of all real values of $x$ for which $kx^2 + 2x + 1 > 0$ is

Let $a_1, a_2, a_3, a_4$ be real numbers such that $a_1+a_2+a_3+a_4=0$ and $a_1^2+a_2^2+a_3^2+a_4^2=1$. Then,the smallest possible value of the expression $(a_1-a_2)^2+(a_2-a_3)^2+(a_3-a_4)^2+(a_4-a_1)^2$ lies in the interval

Consider the quadratic equation $nx^2 + 7\sqrt{n}x + n = 0$ where $n$ is a positive integer. Which of the following statements are necessarily correct?
$I$. For any $n$,the roots are distinct.
$II$. There are infinitely many values of $n$ for which both roots are real.
$III$. The product of the roots is necessarily an integer.

If $y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ for all $x \in R$,then the interval of maximum length in which $y$ lies is