Exercice 3 On note: \[ F=\left\{(x . y . z . t) \in \mathbf{R}^{4} \mid x+y+2 z+t=0\right\} \] 1. Démontrer que \( F \) est un sous-espace vectoriel de \( \mathbf{R}^{4} \). 2. Expliquer pourquoi \( \operatorname{dim}(F) \leqslant 3 \). 3. Donner une description paramétrique de \( F \). 4. Donner une base de \( F \) et déterminer sa dimension

Part-I: Write true if the statement is true or false if the statement is false. (1pt each) 1. If \( A=\{0,1,2\} \) and \( B=\{x \in R \mid 0 \leq x \leq 2\} \), then \( A=B \). 2. \( \{\emptyset\} \) is both the element and the subset of \( \{\emptyset,\{\emptyset\}\} \). 3. If the truth value of \( p \Rightarrow q \) is \( F \), then the truth values of \( (\neg p \wedge q) \Leftrightarrow(p \vee q) \) is false. 4. If \( A=\{x \in Z \mid x \) is divisible by 2\( \} \) and \( B=\{y \in Z \mid y \) is divisible by 3\( \} \), then \( A \cap B \) is \( \emptyset \).

Part-III: Choose the best answer from given choice (1pt eacil) 11. Let \( \mathrm{B}=\{\emptyset,\{\emptyset,\{\emptyset\}\}\} \). Then the number of proper subset is \( \begin{array}{llll}\text { A. } 8 & \text { B. } 7 & \text { C. } 3 & \text { D. } 4\end{array} \)

75. Questions in which only two alternatives are possible is called A. Multiple choice questions B. Dichotomous questions C. Open ended questions D. Structured questions

Find the matris \( A^{1403} \) whenever \( A=\left[\begin{array}{cc}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{array}\right. \)

Explain the difference between exponential growth and exponential decay.