If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$,then $\alpha$ is equal to:

Find the potential difference across the $24\,\Omega$ resistor in the given circuit.

Two point charges $+9e$ and $+e$ are placed at a distance of $16\, cm$ apart. Where should a third charge $q$ be placed between them so that it remains in equilibrium?

Consider the following reaction: $N_{2(g)} + 3H_{2(g)} \longrightarrow 2NH_{3(g)}$. The rate of this reaction in terms of $N_2$ at $T \ K$ is $-\frac{d[N_2]}{dt} = 0.02 \ mol \ L^{-1} \ s^{-1}$. What is the value of $-\frac{d[H_2]}{dt}$ (in units of $mol \ L^{-1} \ s^{-1}$) at the same temperature?

The circle $4x^2 + 4y^2 - 12x - 12y + 9 = 0$

$A$ large tank is filled with water to a height $H$. $A$ small hole is made at the base of the tank. It takes $T_1$ time to decrease the height of water to $H/\eta$ (where $\eta > 1$); and it takes $T_2$ time to drain the rest of the water. If $T_1 = T_2$,then the value of $\eta$ is: