Let P be the plane such that it contains the straight line $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to P.

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

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be an arbitrary function and let $g : \mathbb{R} \to \mathbb{R}$ be the function defined by

$$g(x) = x f(x), \quad \text{for all } x \in \mathbb{R}.$$

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

Consider the matrix

$$ M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}. $$

Let $p, q, r, s, a, b, c$ and $d$ be integers such that

$$ M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \quad \text{and} \quad \sum\limits_{k=1}^{26} M^k = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. $$

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

Let $S = \{1, 2, 3, \ldots, 10\}$. Consider the set

$X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly 42 elements}\}$.

Then the number of elements in $X$ is ____________.