Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$, such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $\mathcal{R}$ denote the region lying in the first quadrant, enclosed by the parabola $y^2=x$, the curve $S$, and the lines $x=1$ and $x=4$.

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

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ be two distinct points on the ellipse

$$ \frac{x^2}{9}+\frac{y^2}{4}=1 $$

such that $y_1>0$, and $y_2>0$. Let $C$ denote the circle $x^2+y^2=9$, and $M$ be the point $(3,0)$.

Suppose the line $x=x_1$ intersects $C$ at $R$, and the line $x=x_2$ intersects C at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle R O M=\frac{\pi}{6}$ and $\angle S O M=\frac{\pi}{3}$, where $O$ denotes the origin $(0,0)$. Let $|X Y|$ denote the length of the line segment $X Y$.

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

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by

$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$

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

Let $y(x)$ be the solution of the differential equation

$$ x^2 \frac{d y}{d x}+x y=x^2+y^2, \quad x>\frac{1}{e} $$

satisfying $y(1)=0$. Then the value of $2 \frac{(y(e))^2}{y\left(e^2\right)}$ is ____________.