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.
ScanSolve AI Answer
Step 1: Solve 1(a)(i) - List prime numbers in K. The set K = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. The prime numbers in K are 2, 3, 5, 7, 11. Prime numbers in K = \2, 3, 5, 7, 11\ Step 2: Solve 1(a)(ii) - Find the probability that a number selected from K is not prime. The total number of elements in K is 11. The number of prime numbers in K is 5. The number of non-prime numbers in K is 11 - 5 = 6. P(not prime) = Number of non-prime numbersTotal number of elements = (6)/(11) The probability is (6)/(11). Step 3: Solve 1(b) - Factorize completely (x+y)(2m-n) - m(x+y). Factor out the common term (x+y): (x+y)(2m-n) - m(x+y) = (x+y)[(2m-n) - m] Simplify the expression inside the square brackets: (x+y)(2m-n-m) = (x+y)(m-n) The factorized expression is (x+y)(m-n). Step 4: Solve 1(c) - Calculate the total amount earned by the typist. Total sheets typed = 35. Cost for the first 8 sheets = GH¢ 68.00. Number of additional sheets = 35 - 8 = 27 sheets. Cost per additional sheet = GH¢ 11.00. Cost for additional sheets = 27 × GH¢ 11.00 = GH¢ 297.00. Total amount earned = Cost for first 8 sheets + Cost for additional sheets Total amount = GH¢ 68.00 + GH¢ 297.00 = GH¢ 365.00 The total amount earned is GH¢ 365.00. Step 5: Solve 2(a)(i) - Find the image of P(2, 5) translated by r = (-3, 8). To find the image P'(x', y'), add the components of the translation vector to the coordinates of P. P'(x', y') = (2 + (-3), 5 + 8) = (2 - 3, 5 + 8) = (-1, 13) The image of P(2, 5) is (-1, 13). Step 6: Solve 2(a)(ii) - Find the image Q' of Q(-4, -6) translated by r = (-3, 8). Similarly, add the components of the translation vector to the coordinates of Q. Q'(x', y') = (-4 + (-3), -6 + 8) = (-4 - 3, -6 + 8) = (-7, 2) The image of Q(-4, -6) is (-7, 2). Step 7: Solve 2(b) - Calculate Joyce's and Richard's receipts. Total profit = GH¢ 9,000.00. Richard's manager's share = (1)/(3) × GH¢ 9,000.00 = GH¢ 3,000.00. Remaining profit for distribution = GH¢ 9,000.00 - GH¢ 3,000.00 = GH¢ 6,000.00. Joyce's contribution = GH¢ 10,000.00. Richard's contribution = GH¢ 5,000.00. Ratio of contributions (Joyce : Richard) = 10,000 : 5,000 = 2 : 1. Total ratio parts = 2 + 1 = 3. 2(b)(i) Joyce's receipt: Joyce's share from remaining profit = (2)/(3) × GH¢ 6,000.00 = GH¢ 4,000.00. Joyce received GH¢ 4,000.00. 2(b)(ii) Richard's total receipt: Richard's share from remaining profit = (1)/(3) × GH¢ 6,000.00 = GH¢ 2,000.00. Richard's total receipt = Richard's manager's share + Richard's share from remaining profit Richard's total receipt = GH¢ 3,000.00 + GH¢ 2,000.00 = GH¢ 5,000.00 Richard received in total GH¢ 5,000.00. Step 8: Solve 3(a) - Calculate Asana's total payment. Original daily wage = GH¢ 25.00. Increase in wage = 10\% of GH¢ 25.00 = 0.10 × GH¢ 25.00 = GH¢ 2.50. New daily wage = GH¢ 25.00 + GH¢ 2.50 = GH¢ 27.50. Number of days worked = 30 days. Total payment = New daily wage × Number of days worked Total payment = GH¢ 27.50 × 30 = GH¢ 825.00 Asana will be paid GH¢ 825.00. Step 9: Solve 3(b)(i) - Find the perimeter of the room. The dimensions of the L-shaped room are: Outer top length = 8 m Outer left length = 5 m Outer right length = 3 m Inner horizontal length = 8 m - 5 m = 3 m Inner vertical length = 5 m - 3 m = 2 m The perimeter is the sum of all outer side lengths: Perimeter = 8 m + 5 m + 3 m + 2 m + 5 m + 3 m = 26 m The perimeter is 26 m. Step 10: Solve 3(b)(ii) - Find the area of the room. Divide the L-shaped room into two rectangles. Rectangle 1 (top part): 8 m × 3 m. Area_1 = 8 × 3 = 24 m^2. Rectangle 2 (bottom-left part): $5