The area in square units of the region bounded by the curve $y = \sqrt{16 - x^2}$ and lines $x = 0, x = 4$ above the X-axis is

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

The order and degree of the differential equation $\sqrt{1 + \dfrac{1}{\left(\frac{dy}{dx}\right)^2}} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac{3}{2}}$, respectively are

Let $\vec{a} = 2\hat{i} + \hat{k}, \vec{b} = \hat{i} + \hat{j} + \hat{k}$, and $\vec{c} = 4\hat{i} - 3\hat{j} + 7\hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$, Then $\vec{r} \cdot \vec{c} =$