This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 2. (a) Construct a multiplication table for modulo 9 on the set \2, 4, 6, 8\. Step 1: Create a table and calculate each product a × b 9 for a, b \2, 4, 6, 8\. For example, 2 2 = (2 × 2) 9 = 4 9 = 4. 2 6 = (2 × 6) 9 = 12 9 = 3. 6 8 = (6 × 8) 9 = 48 9 = 3. 8 8 = (8 × 8) 9 = 64 9 = 1. The completed multiplication table is: |c|c|c|c|c| & 2 & 4 & 6 & 8 \\ 2 & 4 & 8 & 3 & 7 \\ 4 & 8 & 7 & 6 & 5 \\ 6 & 3 & 6 & 0 & 3 \\ 8 & 7 & 5 & 3 & 1 \\ 2. (b) i. Using the table, find the truth set of n n = n - 6. Step 1: Test each value of n from the set \2, 4, 6, 8\ in the equation n n = n - 6. For n=2: 2 2 = 4. 2 - 6 = -4. 4 ≠ -4. For n=4: 4 4 = 7. 4 - 6 = -2. 7 ≠ -2. For n=6: 6 6 = 0. 6 - 6 = 0. 0 = 0. This is true. For n=8: 8 8 = 1. 8 - 6 = 2. 1 ≠ 2. Step 2: Identify the values of n for which the equation holds true. The only value for which the equation is true is n=6. The truth set is \6\. 2. (b) ii. Evaluate (6 4) (4 2). Step 1: Evaluate the expression inside the first parenthesis, 6 4, using the table. 6 4 = 6 Step 2: Evaluate the expression inside the second parenthesis, 4 2, using the table. 4 2 = 8 Step 3: Substitute these results back into the original expression and evaluate. (6 4) (4 2) = 6 8 From the table, 6 8 = 3. The evaluation is 3. 3. (a) The scores obtained by 10 students in a test have the following statistics: x = 290 and x^2 = 8469. Calculate: i. The mean ii. The variance Given: Number of students N = 10, x = 290, x^2 = 8469. 3. (a) i. The mean Step 1: Use the formula for the mean, x = ( x)/(N). x = (290)/(10) Step 2: Calculate the mean. x = 29 The mean is 29. 3. (a) ii. The variance Step 1: Use the formula for variance, ^2 = ( x^2)/(N) - (x)^2. ^2 = (8469)/(10) - (29)^2 Step 2: Perform the division and squaring. ^2 = 846.9 - 841 Step 3: Calculate the variance. ^2 = 5.9 The variance is 5.9. 3. (b) The mean of 14 numbers is 19. If each of the numbers is increased by 4. What is the new mean? Step 1: Understand the effect of adding a constant to each number in a set. If each number in a set is increased by a constant value, the mean of the set also increases by that same constant value. Step 2: Add the increase to the original mean. New Mean = Original Mean + Increase New Mean = 19 + 4 Step 3: Calculate the new mean. New Mean = 23 The new mean is 23. That's 2 down. 3 left today — send the next one.