Let $a$ be an integer such that all the real roots of the polynomial $2x^{5}+5x^{4}+10x^{3}+10x^{2}+10x+10$ lie in the interval $(a, a+1)$. Then,$|a|$ is equal to ...... .

The minimum value of $(x-\alpha)(x-\beta)$ is

If the roots of the equation $3x^2 + 4kx + 3 = 0$ are non-real,then $k$ lies in the interval

Let $f(x) = ax^2 + bx + c$,where $a, b, c$ are integers. Suppose $f(1) = 0$,$40 < f(6) < 50$,$60 < f(7) < 70$,and $1000t < f(50) < 1000(t+1)$ for some integer $t$. Then,the value of $t$ is

Consider a quadratic equation $ax^2 + 2bx + c = 0$ where $a, b, c$ are positive real numbers. If the equation has no real root,then which of the following is true?

If $1-i$ is a root of the equation $x^2+ax+b=0$,then $b$ is equal to