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 ____________.

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that

$$ \left(1+\frac{2}{5} x\right)^{23}=\sum\limits_{i=0}^{23} a_i x^i $$

for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \leq j \leq 23$. Then the value of $r$ is ____________.

A factory has a total of three manufacturing units, $M_1, M_2$, and $M_3$, which produce bulbs independent of each other. The units $M_1, M_2$, and $M_3$ produce bulbs in the proportions of $2: 2: 1$, respectively. It is known that $20 \%$ of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M_1, 15 \%$ are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M_2$ is $\frac{2}{5}$.

If a bulb is chosen randomly from the bulbs produced by $M_3$, then the probability that it is defective is __________.