List the prime numbers in K = 1, 2, 3, ..., 11. (b) Factorize completely: (x+y)(2m-n) - m(x+y).

Alright Don — let's do this. 1. (a) (i) List the prime numbers in K = \1, 2, 3, , 11\. Step 1: Recall the definition of a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Step 2: Identify prime numbers within the set K. The prime numbers in K are 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 set K. The set K = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\, so there are 11 elements. Step 2: Determine the number of prime numbers in K. From part (i), the prime numbers are \2, 3, 5, 7, 11\, which is 5 prime numbers. Step 3: Determine the number of non-prime numbers in K. Number of non-prime numbers = Total numbers - Number of prime numbers = 11 - 5 = 6 Step 4: Calculate the probability. P(not prime) = Number of non-prime numbersTotal number of numbers = (6)/(11) The probability is (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 term. (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) The completely factorized expression is (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. Total sheets typed = 35 sheets. Sheets covered by the initial charge = 8 sheets. Additional sheets = 35 - 8 = 27 sheets. Step 2: Calculate the cost for the additional sheets. Cost per additional sheet = GH¢ 11.00. Cost for additional sheets = 27 × GH¢ 11.00 = GH¢ 297.00. Step 3: Calculate the total amount earned. Cost for first 8 sheets = GH¢ 68.00. Total amount earned = Cost for first 8 sheets + Cost for additional sheets = GH¢ 68.00 + GH¢ 297.00 = GH¢ 365.00 The amount earned is 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 original point be P(x_1, y_1) = (2, 5) and the image be P'(x_2, y_2) = (-3, 8). Let the translation vector be r = a \\ b . Step 2: Use the translation formula: P' = P + r. -3 \\ 8 = 2 \\ 5 + a \\ b Step 3: Solve for a and b. a = -3 - 2 = -5 b = 8 - 5 = 3 The translation vector r is -5 \\ 3 . (ii) the image Q' of Q(-4, -6) when translated by r. Step 1: Let the original point be Q(x_Q, y_Q) = (-4, -6) and the translation vector be r = -5 \\ 3 from part (i). Step 2: Use the translation formula: Q' = Q + r. Q' = -4 \\ -6 + -5 \\ 3 Step 3: Calculate the coordinates of Q'. Q' = -4 + (-5) \\ -6 + 3 = -9 \\ -3 The image Q' is (-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. Total profit = GH¢ 9,000.00. 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 = GH¢ 9,000.00 - GH¢ 3,000.00 = GH¢ 6,000.00 Step 3: Determine the ratio of contributions. Joyce's contribution : Richard's contribution = GH¢ 10,000.00 : GH¢ 5,000.00 = 10,000 : 5,000 = 2 : 1 The sum of the ratio parts is 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. Joyce received 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 earnings. Richard's total = Richard's manager's share + Richard's share from remaining profit = GH¢ 3,000.00 + GH¢ 2,000.00 = GH¢ 5,000.00 Richard received in total GH¢ 5,000.00. 3. (a) Asana was engaged to collect cashew-fruits on a farm and was paid a wage of GH¢ 25.00 a day. If the daily wage increased by 10% and she worked for 30 days, how much will she be paid? Step 1: Calculate the increase in daily wage. Original daily wage = GH¢ 25.00. Increase = 10% of GH¢ 25.00 = 0.10 × GH¢ 25.00 = GH¢ 2.50. Step 2: Calculate the new daily wage. New daily wage = Original daily wage + Increase = GH¢ 25.00 + GH¢ 2.50 = GH¢ 27.50. Step 3: Calculate the total amount paid for 30 days. Total payment = New daily wage × Number of days worked