The number of values of $x$ where the function $f(x) = \cos x + \cos (\sqrt{2} x)$ attains its maximum is

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:

If the derivative of a function $f$ is given by $f'(x) = (x - a)^{2m} (x - b)^{2n + 1}$,where $m$ and $n$ are positive integers and $a > b$,then which of the following is true?

The minimum value of the function $f(x) = x^{10} + x^2 + \frac{1}{x^{12}} + \frac{1}{1 + \sec^{-1} x}$ is

An open tank with a square bottom,to contain $4000 \ cm^3$ of liquid,is to be constructed. The dimensions of the tank,so that the surface area of the tank is minimum,are

If $0 < x < \frac{\pi}{2}$,then the maximum area (in sq. units) of the triangle whose vertices are $(0,0)$,$(x, \cos x)$ and $(\sin^3 x, 0)$ is