If $A$ and $B$ are two independent events such that $P(B)=\frac{2}{7}$ and $P(A \cup B^c)=0.8$, then $P(A)$ is equal to:

An insulated container contains $4 \, \text{moles}$ of an ideal diatomic gas at temperature $T$. Heat $Q$ is supplied to this gas, due to which $2 \, \text{moles}$ of gas are dissociated into atoms, but the temperature of the gas remains constant. Then:

The vectors $\vec{AB} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{BC} = \hat{i} - 2\hat{k}$ are the adjacent sides of a parallelogram. The angle between its diagonals is

Product $(X)$ is

The joint equation of two lines passing through the origin,each making an angle of $30^{\circ}$ with the $Y$-axis,is

If $y=y(x)$ is the solution of the differential equation $\frac{5+e^x}{2+y} \cdot \frac{dy}{dx} + e^x = 0$ satisfying $y(0)=1$,then a value of $y(\log_e 13)$ is