Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$.

Define the functions $f: \mathbb{R} \to \mathbb{R}$, $g: \mathbb{R} \to \mathbb{R}$, and $h: \mathbb{R} \to \mathbb{R}$ by

$f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4$

$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4$

$h(x) = f(x + 1) - g(x + 2)$

If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is

Three students $S_1, S_2,$ and $S_3$ are given a problem to solve. Consider the following events:

U: At least one of $S_1, S_2,$ and $S_3$ can solve the problem,

V: $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem,

W: $S_2$ can solve the problem and $S_3$ cannot solve the problem,

T: $S_3$ can solve the problem.

For any event $E$, let $P(E)$ denote the probability of $E$. If

$P(U) = \dfrac{1}{2}$ , $P(V) = \dfrac{1}{10}$ , and $P(W) = \dfrac{1}{12}$,

then $P(T)$ is equal to

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by

$f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$

Then which one of the following statements is TRUE?

Consider the matrix

$$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that

$$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$

is