Exercises 1. Which of the following sentences are propositions? For those that are, indicate value. a. 123 is a prime number. b. 0 is an even number. c. \( x^{2}-4=0 \). d. Multiply \( 5 x+2 \) by 3 . e. What an impossible question! 2. State the negation of each of the following statements. a. \( \sqrt{2} \) is a rational number. b. 0 is not a negative integer. c. 111 is a prime number.

Explain the difference between exponential growth and exponential decay.

Q. 15 Find the eigenvalues and eigenvectors of the matrix \( \left[\begin{array}{ll}6 & 5 \\ 3 & 4\end{array}\right] \).

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)

20. Show that \( \sqrt{3} \) is an irrational number. 21. Show that \( ((p \Rightarrow q) \wedge(q \Rightarrow r)) \Rightarrow(p \Rightarrow r) \) is tautology or not. (2 pts) 22. Show that \( p \Rightarrow(q \vee r) \) and \( (p \Rightarrow q) \vee(p \Rightarrow r) \) are equivalent. ( 2 pts)

1. नीचे दी गई श्रेणी में अगला \( 4,16,8,64,12,256,16,10 \) (a) 4096 (b) 24 (c) 320 (d) 4100