Let $ \vec{a}, \vec{b} $ be two vectors, and let P, Q and R be the points with position vectors $ \vec{a}, \vec{b} $ and $ \vec{a} + \vec{b} $, respectively, with respect to the origin O. If $ |\vec{a} + \vec{b}| = \sqrt{21} $, $ |\vec{a} - \vec{b}| = 3 $, and $ \vec{a} $ and $ (\vec{a} - \vec{b}) $ are perpendicular to each other, then the area of the triangle OPR is :

Let T be the tangent to the parabola $y^2 = 16x$ at the point $(64, 32)$. Let L be the tangent to the same parabola at another point $(x_1, y_1)$ on the parabola. If L and T are perpendicular to each other, then the distance between the point $(x_1, y_1)$ and the focus of the parabola, is :

Let $y : (-\infty, \infty) \to (0, \infty)$ be the solution of the differential equation

$$\frac{dy}{dx} = \frac{e^{5x} y^3 + y^3}{e^x + e^x y^4},$$

satisfying $y(0) = \frac{1}{\sqrt{2}}$. Then the value of $y(\log_e 2)$ is

The value of the definite integral

$$\int\limits_{0}^{2} \frac{1}{3^x + 3} dx$$

is