If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$,where $a > 0$,attains its maximum and minimum at $p$ and $q$ respectively such that $p^2 = q$,then $a$ equals:

Let $f: R \to R$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leqslant -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$,then what is the possible value of $k$?

An enemy fighter jet is flying along the curve given by $y = x^2 + 2$. $A$ soldier is placed at $(3, 2)$ and wants to shoot down the jet when it is nearest to him. The nearest distance is:

$x$ and $y$ are two positive integers such that $2x + 3y = 50$. If $x^2 y^3$ is maximum for $x = \alpha$ and $y = \beta$, then $\frac{\alpha}{2} + \frac{\beta}{5} =$

The equations of motion of two stones thrown vertically upwards simultaneously are $s_1 = 19.6t - 4.9t^2$ and $s_2 = 9.8t - 4.9t^2$ respectively. If the maximum height attained by the first stone is $h$,then when the first stone is at its maximum height,the height of the second stone will be:

If $x$ and $y$ are two variables such that $x > 0$ and $xy = 1$,what is the minimum value of $x + y$?