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 ?

Let a, b, c be positive integers in arithmetic progression such that the equation

$$ax^2 + bx + c = 0$$

has only integer solutions.

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

Let L be the straight line joining the points P(1, 2, –1) and Q(2, 3, 1). Let S be the foot of the perpendicular drawn from the point R(4, –1, 5) to the line L. Another line passing through R intersects L at a point T such that the point S divides the line segment PT internally in the ratio $|PS| : |ST| = 1 : 2$, where $|PS|$ and $|ST|$ are the lengths of the line segments PS and ST, respectively.

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