The figure shows the vertical section of a frictionless surface. $A$ block of mass $2 \, kg$ is released from position $A$. Its kinetic energy $(KE)$ as it reaches position $C$ is ............ $J$.

If one of the lines $2x^2 - xy + by^2 = 0$ passes through the point $(-4, -2)$,then $b^2 =$

$2 \tanh^{-1} \frac{1}{2}$ is equal to

Three non-zero non-collinear vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $\vec{a}+3\vec{b}$ is collinear with $\vec{c}$,and $3\vec{b}+2\vec{c}$ is collinear with $\vec{a}$. Then $\vec{a}+3\vec{b}+2\vec{c}$ equals to

If $x > 0$ and $x^y = e^{x-y}$,then $\frac{dy}{dx}$ is equal to

The wavelengths of $K_{\alpha}$ $X$-rays of two metals $A$ and $B$ are $\frac{4}{1875R}$ and $\frac{1}{675R}$ respectively,where $R$ is the Rydberg constant. The number of elements lying between $A$ and $B$ according to their atomic numbers is: