The general solution of the differential equations $\dfrac{dy}{dx} = (9x + y + 5)^2$ is...

Let $\vec{a}$ and $\vec{b}$ be linearly independent vectors such that
$|\vec{a}| = \sqrt{3}, |\vec{b}| = 3$ and $|\vec{a} - \vec{b}| = 4$.
If $\vec{a} \times (2\hat{i} + 2\hat{j} - \hat{k}) = (2\hat{i} + 2\hat{j} - \hat{k}) \times \vec{b}$ and $|(\vec{a} + \vec{b}) \cdot (3\hat{i} + 4\hat{j} + 2\hat{k})| = \sqrt{\lambda}$, then $\lambda = \ldots$

The value of $b$ such that the scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with the unit vector parallel to the sum of the vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $b\hat{i} + 2\hat{j} + 3\hat{k}$ is one, is...

If $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, $\vec{a} \cdot \vec{b} < 0$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\sin\theta + \tan\theta$ is...