$A$ point charge $+Q$ is positioned at the centre of the base of a square pyramid as shown. The flux through one of the four identical upper faces of the pyramid is

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) =$ the sum of positive divisors of $n$,then $f(2^k \cdot 3)$,where $k$ is a positive integer,is

$A$ baby has been born with a small tail. It is a case exhibiting:

Consider an ionic solid $M X$ with $NaCl$ structure. Construct a new structure $( Z )$ whose unit cell is constructed from the unit cell of $M X$ following the sequential instructions given below. Neglect the charge balance.

$(i)$ Remove all the anions ( $X$ ) except the central one

$(ii)$ Replace all the face centered cations $(M)$ by anions $(X)$

$(iii)$ Remove all the corner cations $(M)$

$(iv)$ Replace the central anion ( $X$ ) with cation $(M)$

The value of $\left(\frac{\text { number of anions }}{\text { number of cations }}\right)$ in $Z$ is. . . . .

In which of the following are vessels and companion cells found?

The integral $\int \frac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4} dx$ is equal to (where $C$ is a constant of integration)