15. Vector \( \mathbf{q} \) has a magnitude of 7 and is parallel to vector \( \mathbf{p} \). Given that \( \mathbf{p}=3 \mathbf{i}-\mathbf{j}+1 \frac{1}{2} \mathbf{k} \), Express vector \( \mathbf{q} \) in terms of \( \mathbf{i}, \mathbf{j} \) and \( \mathbf{k} \).

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 \( \begin{array}{llll}\text { A. } 8 & \text { B. } 7 & \text { C. } 3 & \text { D. } 4\end{array} \)

Explain the difference between exponential growth and exponential decay.

Use ajdoint matrix to evaluate the inverse of matrix \( A=\left(\begin{array}{cccc}3 & -3 & 2 & 4 \\ -4 & 4 & -2 & -2 \\ 2 & -3 & -3 & -3 \\ -1 & 2 & -4 & -3\end{array}\right) \)

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