(a) y+xy′=0 (5%)

(b) y′′ + y = t , y(0) = 1 , y′(0) =0 (using Laplace transform method) (15%)

(a) Calculate the Laplace Transform of f(t) (10%)

(b) Calculate the Fourier Transform of f(t) (10%)Hint: The Heaviside function H(t) and shifted Heaviside function

(a) Please draw the heuristic search tree for solving the famous 8 puzzle problem. The puzzle is contained in a tray that can be thought of as a grid of nine squares. The tray contains only eight tiles, which means that one of the grids is always empty. The tiles are numbered from 1 to 8. The rule of the game is that a tile can be slid into an empty slot. The initial and goal states of an 8-puzzle are shown as the follows. Please draw the heuristic search tree for solving the puzzle.Initial stateGoal state

(b) Use a truth table to find whether the following argument is valid.{P → Q, Q → R} |− (P → R)

(a) List and define three kinds of identification witness in entity authentication.

(b) Define a certification authority (CA) and its relation to public-key cryptography.

(a) The input data to a program is made up of a combination of three integers in the range of 1000 to 1999 (inclusive). Find the number of exhaustive tests to test all combinations of these numbers.

(b) A random number generator creates a number between 0 and 0.999. How can this random number generator be used to do random testing for the system described in (a).

(a) Write a recursive algorithm in pseudocodeto find the combination of n objects taken k at a time using the following definition.\[ c(n, k) = \begin{cases} 1 & \text{if } k = 0 \text{ or } n \\ c(n-1, k) + c(n-1, k-1) & \text{if } n > k \ge 1 \end{cases} \]

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