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...

The maximum volume of a parallelopiped (in cubic units) with vectors $(2a\hat{i} + \hat{k}), (a\hat{j} - a\hat{k})$, and $(3\hat{i} + a\hat{j})$, where $a \in [0, 1]$, as its coterminous edges is...