Let $y = f(x)$ be the real valued function defined on the interval $(0, \infty)$, satisfying $y(1) = 0$ and the differential equation

$$ x \frac{dy}{dx} = y - x^3. $$

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

Let $\mathbb{R}$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices

$$ S=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right] \quad \text { and } \quad T=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] . $$

Let $a, b, c, d$ be real numbers such that

$$ S T=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] $$

Let

$$ H=\{x+i y: \quad x, y \in \mathbb{R} \text { and } y>0\} . $$

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

Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets

$$ A=\{1,2,3,4,5\} \text { and } B=\{1,2,3,4,5,6,7\} . $$

Let $S$ be the set of all functions $f: A \rightarrow B$ such that $f(2) \neq 2$ and $f(4) \neq 4$. Consider the set $T=\left\{f \in S:\right.$ there exists a function $g: B \rightarrow \mathbb{N}$ such that $g(f(x))=2^x$ for all $\left.x \in A\right\}$.

Then the number of elements in the set $T$ is $\_\_\_\_$ .

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $X$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $X$, then the value of $77 \alpha$ is $\_\_\_\_$ .