$CH_4$ diffuses two times faster than a gas $X$. The number of molecules present in $32 \ g$ of gas $X$ is ($N$ is Avogadro number).

$A$ homogeneous solid cylinder of length $L$ $(L < H/2)$ and cross-sectional area $A'$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L/4$ in the denser liquid (density $2d$) and $3L/4$ in the lighter liquid (density $d$) as shown in the figure. The density $D$ of the solid is given by:

Identify the correct statements from the following: $(i)$ The cryptozoite ingests food and becomes spherical. $(ii)$ Sporozoites enter the human blood. $(iii)$ Two stages of the life cycle are observed in Plasmodium. $(iv)$ The cryptozoite transforms into cryptomerozoites.

Evaluate $\int_{-2}^1 f(x) dx$,where $f(x)= \begin{cases} 1-2x, & x \leq 0 \\ 1+2x, & x > 0 \end{cases}$

Among the following, the $CORRECT$ combinations are :
$A.$ $IF_3$ $\rightarrow$ $T$-shaped $(sp^3d)$
$B.$ $IF_5$ $\rightarrow$ Square pyramidal $(sp^3d^2)$
$C.$ $IF_7$ $\rightarrow$ Pentagonal bipyramidal $(sp^3d^3)$
$D.$ $ClO_4^-$ $\rightarrow$ Square planar $(sp^3d)$
Choose the correct answer from the options given below :

The differential of ${e^{{x^3}}}$ with respect to $\log x$ is