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 Question 1(a)(i). The set K = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime numbers in K are 2, 3, 5, 7, 11. Prime numbers in K = \2, 3, 5, 7, 11\ Step 2: Solve Question 1(a)(ii). Total number of elements in K is 11. Number of prime numbers in K is 5. Number of non-prime numbers in K = Total numbers - Prime numbers = 11 - 5 = 6. The probability that a number selected at random from K is not a prime number is: P(not prime) = Number of non-prime numbersTotal number of numbers = (6)/(11) P(not prime) = (6)/(11) Step 3: Solve Question 1(b). To factorize (x+y)(2m-n) - m(x+y), we identify the common factor (x+y). (x+y)(2m-n) - m(x+y) = (x+y)[(2m-n) - m] = (x+y)(2m-n-m) = (x+y)(m-n) Factorized expression = (x+y)(m-n) Step 4: Solve Question 1(c). Cost for the first 8 sheets = GH¢ 68.00. Total sheets typed = 35 sheets. Number of additional sheets = 35 - 8 = 27 sheets. Cost for additional sheets = 27 × GH¢ 11.00 = GH¢ 297.00. Total amount earned = Cost for first 8 sheets + Cost for additional sheets = GH¢ 68.00 + GH¢ 297.00 = GH¢ 365.00 Total amount earned = GH¢ 365.00 Step 5: Solve Question 2(a). The translation vector is r = (-3, 8). (i) To find the image P' of P(2, 5) translated by r: P' = (2 + (-3), 5 + 8) = (2 - 3, 5 + 8) = (-1, 13) P' = (-1, 13) (ii) To find the image Q' of Q(-4, -6) translated by r: Q' = (-4 + (-3), -6 + 8) = (-4 - 3, -6 + 8) = (-7, 2) Q' = (-7, 2) Step 6: Solve Question 2(b). Total profit = GH¢ 9,000.00. Richard's manager's share = (1)/(3) × GH¢ 9,000.00 = GH¢ 3,000.00. Remaining profit = GH¢ 9,000.00 - GH¢ 3,000.00 = GH¢ 6,000.00. Contribution ratio (Joyce : Richard) = 10,000 : 5,000 = 2 : 1. Total ratio parts = 2 + 1 = 3. (i) Joyce's share from remaining profit = (2)/(3) × GH¢ 6,000.00 = GH¢ 4,000.00. Joyce receives: Joyce's total = GH¢ 4,000.00 (ii) 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 = GH¢ 3,000.00 + GH¢ 2,000.00 = GH¢ 5,000.00 Richard's total = GH¢ 5,000.00 Step 7: Solve Question 3(a). Original daily wage = GH¢ 25.00. Increase in daily 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 amount paid = New daily wage × Number of days worked = GH¢ 27.50 × 30 = GH¢ 825.00 Total amount paid = GH¢ 825.00 Step 8: Solve Question 3(b). The diagram shows an L-shaped room. Let's determine the lengths of all sides: The top horizontal side is 8 m. The right vertical side is 3 m. The bottom horizontal side is 5 m. The left vertical side is 5 m. The missing horizontal segment (part of the top side of the "cut-out") is 8 m - 5 m = 3 m. The missing vertical segment (part of the left side of the "cut-out") is 5 m - 3 m = 2 m. (i) Perimeter: Sum of all outer sides. Perimeter = 8 m + 3 m + 5 m + 2 m + 3 m + 5 m = 26 m Perimeter = 26 m (ii) Area: Divide the L-shape into two rectangles. We can consider it as a large rectangle of 8 m × 5 m with a smaller rectangle of 3 m × 2 m removed from the top right corner. Area of large rectangle = 8 m × 5 m = 40 m^2. Area of removed rectangle = (8-5) m × (5-3) m = 3 m × 2 m = 6 m^2. Area = 40 m^2 - 6 m^2 = 34 m^2 Area = 34 m^2 (iii) Cost of carpeting the floor: Cost per square metre = GH¢ 20.00. Total area = 34 m^2. Total cost = Area × Cost per square metre = 34 m^2 × GH¢ 20.00/m^2 = GH¢ 680.00 Cost of carpeting = GH¢ 680.00 Drop the next question. 📸