Arrange the following compounds in decreasing order of reactivity towards nucleophilic addition reactions:
$(I) HCHO$
$(II) CH_3CHO$
$(III) CH_3COCH_3$
$(IV) CH_3CH_2COCH_2CH_3$

Along the $x$-axis,three charges $\frac{q}{2}, -q$ and $\frac{q}{2}$ are placed at $x=0, x=a$ and $x=2a$ respectively. The resultant electric potential at a point $P$ located at a distance $r$ from the charge $-q$ (where $r > a$) is ($\varepsilon_0$ is the permittivity of free space):

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the number of ways to select at least one book is $63$,then the value of $n$ is:

The Earth is assumed to be a sphere of radius $R$. $A$ platform is arranged at a height $R$ from the surface of the Earth. The escape velocity of a body from this platform is $fv$,where $v$ is the escape velocity from the surface of the Earth. The value of $f$ is

Which one of the following equations represents the variation of viscosity coefficient $(\eta)$ with temperature $(T)$?

$A$ particle is tied to a $20\, cm$ long string. It performs circular motion in a vertical plane. What is the angular velocity of the string when the tension in the string at the top is zero? (Take $g = 9.8\, m/s^2$)