Number of integral values of $a$ for which both roots of the quadratic equation $x^2 - (2a + 3)x + a^2 + 3a = 0$ lie in the interval $(0, 4)$ is:

If the graph of $y = ax^2 + bx + c$ is as follows,where $\Delta ABC$ is a right-angled isosceles triangle with hypotenuse $AC = 4\sqrt{2} \text{ units}$,then the minimum value of $ax^2 + bx + c$ is:

For what minimum value of $k$ do both roots of the equation $x^2 - 8kx + 16(k^2 - k + 1) = 0$ exist,are real,distinct,and at least $4$?

The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

The number of integral values of $k$,for which one root of the equation $2x^2-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$,is:

If both the roots of the equation $x^2-4ax+1-3a+4a^2=0$ exceed $1$,then $a$ lies in the interval