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

Let $\mathbb{R}$ denote the set of all real numbers. Consider the polynomial function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$$ f(x)=\frac{d^{10}}{d x^{10}}\left(\left(x^2-1\right)^{10}\right), \quad \text { for all } x \in \mathbb{R} $$

Here $\frac{d^{10}}{d x^{10}}\left(\left(x^2-1\right)^{10}\right)$ is the $10^{\text {th }}$ order derivative of the function $\left(x^2-1\right)^{10}$.

Then which of the following statements is (are) TRUE ?