Two liquids $A$ and $B$ have $P_A^o$ and $P_B^o$ in the ratio of $1 : 3$ and the ratio of number of moles of $A$ and $B$ in the liquid phase are $1 : 3$. The mole fraction of $A$ in the vapour phase in equilibrium with the solution is equal to:

If $l_1 = \lim_{x \rightarrow 2^{+}} (x + [x])$,$l_2 = \lim_{x \rightarrow 2^{-}} (2x - [x])$ and $l_3 = \lim_{x \rightarrow \pi / 2} \frac{\cos x}{x - \pi / 2}$,then:

If $N$ is the number of turns in a coil,the value of self-inductance varies as

Which one of the following is wrong for fungi?

$ABC$ is a triangular plate of uniform thickness. Its sides are in the ratio as shown in the figure. Let $I_{AB}$,$I_{BC}$,and $I_{CA}$ be the moments of inertia of the plate about the sides $AB$,$BC$,and $CA$ respectively. Which of the following relations is true for this arrangement?

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?