The solution of the differential equation $x\dfrac{dy}{dx} + y = x^3 y^6$, is...

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\bar{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k}$ are three vectors in which $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$, then $x = \cdots$

If $|\bar{a}| = 4, |\bar{b}| = 3$ and $\bar{a} \cdot \bar{b} = 8$ then $[\bar{a}\ \ \bar{a} + \bar{b}\ \ \bar{a} \times \bar{b}] = $

Let $\bar{a}, \bar{b}, \bar{c}$ be the unit vectors such that $\bar{a}$ is perpendicular to $\bar{b}$ and the angle between $\bar{b}$ and $\bar{c}$ is $120^\circ$. If $\bar{a} + \bar{c}$ is perpendicular to $\bar{b} + \bar{c}$ then