If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2 + cx$ on the interval $x \in [-1, 1]$ at the point $x = \frac{1}{2}$,then $(2b + c)$ is equal to:

$Na^+$,$Mg^{2+}$,$Al^{3+}$,and $Si^{4+}$ are isoelectronic. What is the order of their ionic radii?

Which of the following is $NOT$ included in $In-situ$ conservation?

The solution of $x \frac{dy}{dx} = y + x e^{y/x}$ with $y(1) = 0$ is

$A$ coil of wire of radius $r$ has $600$ turns and a self-inductance of $108 \ mH$. The self-inductance of a coil with the same radius and $500$ turns is (in $mH$)

The objective and eyepiece of an astronomical telescope are double convex lenses with refractive index $1.5$. When the telescope is adjusted to infinity,the separation between the two lenses is $16 ~cm$. If the space between the lenses is now filled with water (refractive index $4/3$) and the telescope is again adjusted for infinity,then the present separation between the lenses is: (in $~cm$)