1. (10p.) Using the mathematical induction show that 133 divides \( 11^{n+2}+12^{2 n+1} \) for all \( n \geqslant 1 \).
2. (10p.) Using Euclidean algorithm find integers \( a, b \) such that \( \operatorname{gcd}(-14,4091)=(-14) \cdot a+4091 \cdot b \).
3. (10p.) Let \( X=[0,1] \), where \( [0,1] \) is the set of all \( x \in \mathbb{R} \) such that \( -1 \leqslant x \leqslant 1 \). On the set \( X \) define a binary relation \( \sim \) as follows
\[
x \sim y \Longleftrightarrow x=y \text { or }(0<x<1 \text { and } 0<y<1) .
\]
Show that \( \sim \) is an equivalence relation on \( X \) and find all the equivalence classes.
4. (10p.) Solve the following matrix equation:
\[
4 X \cdot\left[\begin{array}{lll}
5 & 0 & 1 \\
0 & 3 & 0 \\
3 & 0 & 1
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 4 \\
0 & 5 & 0 \\
2 & 0 & 3
\end{array}\right]+X \cdot\left[\begin{array}{lll}
3 & 0 & 5 \\
0 & 4 & 0 \\
8 & 0 & 2
\end{array}\right]
\]
5. (10p.) Calculate the following determinant:
\[
\left|\begin{array}{cccc}
1 & 0 & -2 & -1 \\
-3 & 1 & 2 & 0 \\
1 & 2 & 1 & 4 \\
-1 & 3 & 2 & -1
\end{array}\right|
\]
