If $f(x) = x^3 - 10x^2 + 200x - 10$,then

In an interference experiment,the third bright fringe is obtained at a point on the screen with light of wavelength $700 \, nm$. What should be the wavelength of the light source in order to obtain the $5$th bright fringe at the same point (in $, nm$)?

Three concentric spherical shells have radii $a, b,$ and $c$ $(a < b < c)$ and have surface charge densities $\sigma, -\sigma,$ and $\sigma$ respectively. If the potentials on the surfaces of these shells are $V_A, V_B,$ and $V_C$ respectively,then for $c = a + b$,which of the following is true?

If the distance of the point $P(1, -2, 1)$ from the plane $x + 2y - 2z = \alpha$,where $\alpha > 0$,is $5$,then the foot of the perpendicular from $P$ to the plane is:

Let $a, b, x, y$ be real numbers such that $a^2+b^2=81$,$x^2+y^2=121$ and $ax+by=99$. Then,the set of all possible values of $ay-bx$ is

The number of proper subsets of the set ${1, 2, 3}$ is