The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

If $A(\vec{a})$, $B(\vec{b})$ and $C(\vec{c})$ are vertices of $\triangle ABC$. Point D divides segment BC internally in the ratio $2 : 1$. Point E divides segment AD internally in the ratio $1 : 2$, then the position vector of E is ____

If $|\vec{a}| = 3$, $|\vec{b}| = 4$, $|\vec{c}| = 5$ such that each vector is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

The value of $x$ so that the volume of the parallelopiped formed by the vectors $\hat{i} + x\hat{j} + \hat{k}$, $\hat{j} + x\hat{k}$ and $x\hat{i} + \hat{k}$ is minimum, is