If $f(x) \in \mathbb{Q}[x]$ is a non-zero polynomial such that all its roots are irrational,then the degree of $f(x)$ is

Consider the equation $x^2+4x-n=0$,where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$,for which the given equation has integral roots,is equal to

The roots of the quadratic equation $2x^2 + 3x + 1 = 0$ are:

Let $\alpha_1, \alpha_2, \ldots, \alpha_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$. Then $\alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6$ is equal to $..................$.

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^2 + z + 1$ is real. Then $a$ cannot take the value

Consider two numbers whose arithmetic mean is $9$ and geometric mean is $4$. These numbers are the roots of which quadratic equation?