If ${\left( \frac{2}{3} \right)^{x + 2}} = {\left( \frac{3}{2} \right)^{2 - 2x}}$,then $x =$

$(x^2+y^2) dy = xy dx$. If $y(x_0) = e$ and $y(1) = 1$,then the value of $x_0$ is equal to

If $x = \frac{1 - t^2}{1 + t^2}$ and $y = \frac{2at}{1 + t^2}$,then $\frac{dy}{dx} = $

$\int \frac{1+\cos 4 x}{\cot x-\tan x} d x$ is equal to

$A$ and $B$ are ideal gases. The molecular weights of $A$ and $B$ are in the ratio of $1: 4$. The pressure of a gas mixture containing equal weights of $A$ and $B$ is $P \ atm$. What is the partial pressure (in $atm$) of $B$ in the mixture?

$A$ ball of mass $m_1$ and another ball of mass $m_2$ are dropped from the same height. If the time taken by the balls to reach the ground is $t_1$ and $t_2$ respectively,then: