$\int \frac{d x}{7+5 \cos x}$ is equal to

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is

Two thin lenses of focal lengths $f_1$ and $f_2$ are in contact and coaxial. The power of the combination is

An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity $C$ remains constant. If during this process the relation of pressure $P$ and volume $V$ is given by $PV^n = \text{constant}$, then $n$ is given by (Here $C_P$ and $C_V$ are molar specific heat at constant pressure and constant volume, respectively).

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

The atomic weights of two elements $A$ and $B$ are $40$ and $80$ respectively. If $x \ g$ of $A$ contains $y$ atoms,how many atoms are present in $2x \ g$ of $B$?