If $A$ and $G$ are arithmetic and geometric means and ${x^2} - 2Ax + {G^2} = 0$,then

Let $f(x) = (x - a)(x - b) - (\frac{a + b}{2})$. If $f(x) = 0$ has both non-negative roots,then the minimum value of $f(x)$ is:

The number of real roots of the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$ is

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

Let $f(x) = x^3 + a x^2 + b x + c$ be a polynomial with integer coefficients. If the roots of $f(x)$ are integers and are in Arithmetic Progression,then '$a$' cannot take the value:

If $\alpha$ and $\beta$ are the roots of $x^2+3(a+3)x-9a=0$ such that the roots are equal for different values of $a$ (where $\alpha > \beta$ is not applicable as roots are equal,but let $\alpha$ be the root for $a=-9$ and $\beta$ be the root for $a=-1$),then the minimum value of the expression $x^2+\alpha x-\beta$ is: