Draw the truth table for the compound statement (p q) and write A B and (A B) in simple English.

Here is the marking guide for the paper: 3. c) Hence or otherwise, determine the roots of the equation f(x)=0 The function f(x) is not provided in the image. Therefore, the roots of the equation f(x)=0 cannot be determined. 4. a) Draw the truth table for the compound statement (p q) Step 1: Create columns for p, q, p q, and (p q). Step 2: Fill in all possible truth values for p and q. Step 3: Calculate the truth values for p q. The disjunction p q is true if at least one of p or q is true. Step 4: Calculate the truth values for (p q). This is the negation of p q. | p | q | p q | (p q) | | :---- | :---- | :--------- | :--------------- | | T | T | T | F | | T | F | T | F | | F | T | T | F | | F | F | F | T | 4. b) Consider the statements A: If the rain is falling B: I will use an umbrella Write in simple English i) A B This statement means "A AND B". The rain is falling AND I will use an umbrella. ii) (A B) This statement means "NOT (A AND B)". Using De Morgan's laws, (A B) is equivalent to A B. The rain is NOT falling OR I will NOT use an umbrella. 5. a) Given the matrix Q = 2 & 3 \\ x-2 & x . Find i) the transpose, Q^T. Step 1: To find the transpose of a matrix, swap its rows and columns. Q^T = 2 & x-2 \\ 3 & x Q^T = 2 & x-2 \\ 3 & x ii) the value of x for which Q = Q^T. Step 1: For Q to be equal to Q^T, their corresponding elements must be equal. 2 & 3 \\ x-2 & x = 2 & x-2 \\ 3 & x Step 2: Equate the off-diagonal elements. 3 = x-2 Step 3: Solve for x. x = 3+2 x = 5 (Also, x-2 = 3 gives x=5, and the diagonal elements 2=2 and x=x are always equal). x=5 5. b) Consider P = 4 & 3 \\ 1 & 2 . Calculate i) P^2 Step 1: P^2 means P × P. P^2 = 4 & 3 \\ 1 & 2 4 & 3 \\ 1 & 2 Step 2: Perform matrix multiplication. The element in row i, column j of the product matrix is the dot product of row i of the first matrix and column j of the second matrix. P^2 = (4)(4) + (3)(1) & (4)(3) + (3)(2) \\ (1)(4) + (2)(1) & (1)(3) + (2)(2) P^2 = 16 + 3 & 12 + 6 \\ 4 + 2 & 3 + 4 P^2 = 19 & 18 \\ 6 & 7 P^2 = 19 & 18 \\ 6 & 7