The nucleus of an element contains $9$ protons. Its valency would be

If $u=f(r)$,where $r^2=x^2+y^2$,then $\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)$ is equal to

Which of the following reactions will yield $2-$propanol? Select the correct answer from $a, b, c$ and $d$.
$(I)$ $CH_2=CH-CH_3 + H_2O \xrightarrow{H^+} $
$(II)$ $CH_3CHO \xrightarrow[(ii) H_2O]{(i) CH_3MgI}$
$(III)$ $CH_2O \xrightarrow[(ii) H_2O]{(i) C_2H_5MgI}$
$(IV)$ $CH_2=CH-CH_3 \xrightarrow{\text{neutral } KMnO_4}$

$(x^2+y^2) dy = xy dx$. If $y(x_0) = e$ and $y(1) = 1$,then the value of $x_0$ is equal to

Three identical bodies of equal mass $M$ each are moving along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each body is

Mass $M$ is divided into two parts $xM$ and $(1-x)M$. For a given separation,the value of $x$ for which the gravitational attraction between the two pieces becomes maximum is