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 math problems: 1. (a) Given that K = \1, 2, 3, , 11\: (i) List the prime numbers in K: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Step 1: Identify numbers in K that fit the definition of a prime number. The prime numbers in K are 2, 3, 5, 7, 11. Prime numbers: \2, 3, 5, 7, 11\ (ii) Find the probability that a number selected at random from the set K is not a prime number. Step 1: Determine the total number of elements in K. Total number of elements in K = 11. Step 2: Determine the number of prime numbers in K. From (i), there are 5 prime numbers. Step 3: Determine the number of non-prime numbers in K. Number of non-prime numbers = Total elements - Number of prime numbers Number of non-prime numbers = 11 - 5 = 6. Step 4: Calculate the probability. P(not prime) = Number of non-prime numbersTotal number of elements P(not prime) = (6)/(11) Probability: (6)/(11) 1. (b) Factorize completely: (x+y)(2m-n) - m(x+y) Step 1: Identify the common factor in both terms. The common factor is (x+y). Step 2: Factor out the common factor. (x+y)(2m-n) - m(x+y) = (x+y)[(2m-n) - m] Step 3: Simplify the expression inside the square brackets. (x+y)[2m-n-m] = (x+y)[m-n] Factorized expression: (x+y)(m-n) 1. (c) A typist charges GH¢ 68.00 for the first 8 sheets typed, and GH¢ 11.00 for each additional sheet. Calculate the amount earned if the typist typed a total of 35 sheets. Step 1: Calculate the number of additional sheets. Additional sheets = Total sheets - First 8 sheets Additional sheets = 35 - 8 = 27 sheets. Step 2: Calculate the cost for the additional sheets. Cost for additional sheets = Number of additional sheets × Cost per additional sheet Cost for additional sheets = 27 × GH¢ 11.00 = GH¢ 297.00. Step 3: Calculate the total amount earned. Total amount earned = Cost for first 8 sheets + Cost for additional sheets Total amount earned = GH¢ 68.00 + GH¢ 297.00 = GH¢ 365.00. Amount earned: GH¢ 365.00 2. (a) The image of P(2, 5) when translated by the vector r is (-3, 8). Find: (i) r; Step 1: Let the point P = (x, y) = (2, 5) and its image P' = (x', y') = (-3, 8). Let the translation vector be r = (a, b). The translation formula is P' = P + r. (x', y') = (x+a, y+b) (-3, 8) = (2+a, 5+b) Step 2: Equate the corresponding coordinates to find a and b. For the x-coordinate: -3 = 2+a a = -3 - 2 a = -5. For the y-coordinate: 8 = 5+b b = 8 - 5 b = 3. Translation vector r: -5 \\ 3 (ii) the image Q' of Q(-4, -6) when translated by r. Step 1: Use the translation vector r = (-5, 3) found in (i) and the point Q = (-4, -6). Q' = Q + r Q' = (-4, -6) + (-5, 3) Step 2: Add the corresponding coordinates. Q' = (-4 + (-5), -6 + 3) Q' = (-4 - 5, -6 + 3) Q' = (-9, -3) Image Q': (-9, -3) 2. (b) Joyce and Richard contributed GH¢ 10,000.00 and GH¢ 5,000.00 respectively to start a business. They agreed that Richard will be paid one-third of the profit as a manager, and the rest of the profit will be shared in the ratio of their contributions. If the profit of GH¢ 9,000.00 was made, how much did: (i) Joyce receive; (ii) Richard receive in total? Step 1: Calculate Richard's manager's share. Richard's manager's share = (1)/(3) × GH¢ 9,000.00 = GH¢ 3,000.00. Step 2: Calculate the remaining profit to be shared. Remaining profit = Total profit - Richard's manager's share Remaining profit = GH¢ 9,000.00 - GH¢ 3,000.00 = GH¢ 6,000.00. Step 3: Determine the ratio of their contributions. Joyce's contribution : Richard's contribution = GH¢ 10,000.00 : GH¢ 5,000.00 Ratio = 10,000 : 5,000 = 2 : 1. Sum of ratios = 2 + 1 = 3. Step 4: Calculate Joyce's share from the remaining profit. Joyce's share = (2)/(3) × GH¢ 6,000.00 = GH¢ 4,000.00. (i) Joyce receive: GH¢ 4,000.00 Step 5: Calculate Richard's share from the remaining profit. Richard's share = (1)/(3) × GH¢ 6,000.00 = GH¢ 2,000.00. Step 6: Calculate Richard's total receipt. 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. (ii) Richard receive in total: GH¢ 5,000.00