Let $a$ be an integer selected at random from the set $\{0, 1, 2, 3, ..., 9\}$. The probability that the equation $ax^2 - ax + 1 = 0$ has real roots is ...

Let $S = \{ \alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2 \}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2(\sum_{\alpha \in S} \alpha)^2 x + \sum_{\alpha \in S} (\alpha+1)^2 \beta = 0$ has real roots,is $...........$

When are the roots of the quadratic equation $ax^2 + bx + c = 0$ imaginary?

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:

If $x \in \mathbb{R}$,then one of the solutions of $\sqrt{x+1}-|\sqrt{x-1}|=\sqrt{4x-1}$ among the following is

The positive integer $n$, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+...+(x+2n-2)(x+2n) = \frac{8n}{3}$ are two consecutive even integers, is: