If $x=e^{y+e^{y+e^{y+\ldots}}}$,then $\frac{d y}{d x}=$

$A$ gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Neglecting all vibrational modes,the total internal energy of the system is: (in $RT$)

Two capacitors of capacities $1 \mu F$ and $C \mu F$ are connected in series and the combination is charged to a potential difference of $120 \ V$. If the charge on the combination is $80 \mu C$,the energy stored in the capacitor of capacity $C$ in $\mu J$ is

If $x y \neq 0, x+y \neq 0$ and $x^m y^n=(x+y)^{m+n}$, where $m, n \notin N$, then $\frac{d y}{d x}$ is equal to

If the position vectors of $A, B, C, D$ are $\hat{i}+2\hat{j}+2\hat{k}, 2\hat{i}-\hat{j}, \hat{i}+\hat{j}+3\hat{k}$ and $4\hat{j}+5\hat{k}$ respectively,then the quadrilateral $ABCD$ is a

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x+6=0$,then $\Sigma \alpha^2 \beta+\Sigma \alpha \beta^2$ is equal to :