If the solution of the differential equation $(1 + x^3)\dfrac{dy}{dx} + 6x^2y = 1 + x^2$ is $y = \dfrac{1}{(1 + x^3)^s}\left[x + \dfrac{x^p}{p} + \dfrac{x^q}{q} + \dfrac{x^r}{r} + c\right]$, then the LCM of $p, q, r$ and $s$ is...

The solution of the differential equation $\dfrac{dy}{dx} = \cos(x + y)$ is...

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $\vec{a} \perp (\vec{b} + \vec{c}), \vec{b} \perp (\vec{c} + \vec{a}),$ and $\vec{c} \perp (\vec{a} + \vec{b})$ and $|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 3$, then $|\vec{a} + \vec{b} + \vec{c}|$ is...

Two adjacent sides of a parallelogram ABCD are given by $\overline{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ and $\overline{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$. The side $AD$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $AD$ becomes $AD'$. If $AD'$ makes a right angle with the side $AB$, then the cosine of the angle $\alpha$ is...