16. Sea \( A \in \mathcal{L}\left(\mathbf{R}^{3}\right) \) cuya matriz con respecto a la base canónica es \[ A=\left(\begin{array}{rrr}0 & x & 2 \\ -2 & 0 & -2 \\ 2 & -2 & 0\end{array}\right) \] con \( x \in \mathbf{R} \). a) Encontrar \( x \) pará que \( A \) sea diagonalizable.

12. Given set \( \mathrm{A}=\{\emptyset,\{\emptyset\}, 0\} \), which of the following is true? C. \( \emptyset \subseteq A \) B. \( \{\emptyset\} \in A \) C. \( \{\{\emptyset\}\} \subseteq A \) D. All are correct 13. For any sets \( A \) and \( B \) which of the following is true. A. If \( B \subseteq A \), then \( B^{\prime} \subseteq A^{\prime} \) C. \( A-B=A \cap B^{\prime} \) B. If \( A \subseteq B \), then \( A \cup B=B \) D. all except A. Part Three: solve the following problems. 1. For any propositions \( p \) and \( q \), determine whether \( (p \wedge q) \Rightarrow(q \wedge \neg p) \) is tautology, contradiction or neither? 2. Show that the argument: \( p \Rightarrow q, \neg p \Rightarrow h, h \Rightarrow k \vdash \neg q \Rightarrow k \) is valid. 3. Let \( \mathrm{U}= \) the set of one digit counting numbers \( A=\{x: x \) is an even number in \( U\} \) \( B=\{x: x \) is a square number in \( U\} \) \( \mathrm{C}=\{x: x+2<10 \), where \( x \in U\} \), then find the following. A. \( (A \cup B) / C \) C. \( \left(A \cap B^{\prime}\right) \Delta C \) B. \( (A \Delta B) / C \) D. \( \left(A \cap B^{\prime}\right)^{\prime} \) 4. Write the converse, inverse and contrapositive of the conditional statement. "If it is cold or rainy, I will not take a walk."

11 If \( \overrightarrow{\mathrm{A}}=(4,3,2) \quad, \overrightarrow{\mathrm{B}}=(-2,5,-1), \overrightarrow{\mathrm{C}}=(8,19,4) \) (1) Find in terms of fundamental unit vectors each of : \( \vec{B}-\overrightarrow{\mathrm{C}}, 2(\overrightarrow{\mathrm{~B}}+\overrightarrow{\mathrm{C}}) \) (2) Find : \( \overrightarrow{\mathrm{C}} \) in terms of \( \overrightarrow{\mathrm{A}} \) and \( \overrightarrow{\mathrm{B}} \)

Partie A: Évaluation des ressources (15,5points) Exercice 1: ( 3,5 points) Le plan complexe \( P \) est muni d'un repère orthonormé direct \( (O ; \vec{u} ; \vec{v}) \). A et \( B \) sont des points du plan d'affixes respectives \( i \) et \( -2+i . F \) est l'application de \( P-\{A\} \) vers \( P \) qui, à tout point \( M(z) \) associe le point \( M^{\prime}\left(z^{\prime}\right) \) tel que \( z^{\prime}=i\left(\frac{z+2-i}{z-i}\right) \). Pour tout entier naturel \( n \), on désigne par \( C_{n} \) le point d'affixe \( (\sqrt{3}-i)^{n} \). On considère dans \( \mathbb{C} \) l'équation \( (E): z^{2} \cos ^{2} \alpha-2 z \sin 2 \alpha+1=0 \) où \( \left.\alpha \in\right] 0 ; \frac{\pi}{2}[ \). 1) Pour quelles valeurs de \( n \) les points \( C_{n} \) appartiennent-ils à l'axe des réels ? 2) Déterminent \( n \) pour que les points \( C_{n} \) appartiennent à la première bissectrice. \( 0,75 p t \) 3) Déterminer l'ensemble des points invariants par \( F \). 0,5pt 4) Déterminer les parties réelle et imaginaire de \( z^{\prime} \) 0,5pt 5) Résoudre dans \( \mathbb{C} \), l'équation \( (E) \). \( 1 p t \) 0,75pt

2) a) Determine the whether the following expression is tautology, contradiction or contingency. i) \( \quad p \rightarrow((q \vee r) \wedge(p \wedge q)) \). ii) \( \quad p \vee q \leftrightarrow(r \wedge q) \).

Part II. The following problems have exactly one best answer. Chose the correct answer. 1. Which of the following compound propositions is necessarily true? A. \( p \wedge \neg p \) C. \( p \Leftrightarrow \neg p \) B. \( p \vee \neg p \) D. . \( (p \vee \neg p) \Rightarrow p \) 2. Which of the following represents a set? A. The Collection of good students in a class. C. \( \{a, b\}, c \). B. The collection of cows with 6 legs. D. The collection of interesting birds. 3. Which of the following is a proposition? A. What is your name? C. May God bless you! B. \( 2 x-3=0 \) D. 123 is a prime number. 4. Let A be any set and \( \mathrm{P}(\mathrm{A}) \) be the power set of A , which of the following is true? A. \( \{\varnothing\} \subseteq A \) B. \( \emptyset \in A \) C. \( \emptyset \in P(A) \) D. \( A \subseteq P(A) \) 5. Which of the following statement is true? A. \( (\forall x \in R)(\exists y \in R)(2 x+y=-4) \) C. \( (\forall x \in R)(\exists y \in R)(x y=1) \) B. \( (\exists x \in R)(\exists y \in R)(|x y|=-3) \) D. \( (\exists x \in R)(\forall y \in R)(2 x+y=-4 \) 6. Let A be any set with certain number of elements, then which of the following is false? A. A may contain 5 proper subsets C. A may contain 16 subsets B. A may contain only one proper subset D. A may contain only one subset 7. Let p and q be propositions which of the following implies that. \( p \wedge q \) is true. A. . \( p \vee q \) is true B. . \( \neg p \wedge q \) is false C. \( p \Rightarrow q \) is true D. \( p \Rightarrow \neg q \) is false