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 $\sqrt{3}$ be the radius and $\frac{\pi}{3}$ be the semi-vertical angle of a given cone. Then the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is

$N$ characters of information are held on magnetic tape,in batches of $x$ characters each; the batch processing time is $\alpha + \beta x^2$ seconds; $\alpha$ and $\beta$ are constants. The optimal value of $x$ for fast processing is

Let a function $f:[0,5] \rightarrow R$ be continuous. $f(1)=3$ and $F$ be defined as $F(x)=\int_{1}^{x} t^{2} g(t) dt$,where $g(t)=\int_{1}^{t} f(u) du$. Then for the function $F$,the point $x=1$ is

An extreme value of $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $(0, \pi / 2)$ is

$A$ manufacturer can sell $x$ items at a price of rupees $\left(5 - \frac{x}{100}\right)$ each. The cost price of $x$ items is Rs. $\left(\frac{x}{5} + 500\right)$. The number of items that the manufacturer should sell to earn the maximum profit is