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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Hey cafu, good to see you again. Here are the solutions to the questions: 1. (a) Given 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. The prime numbers in K are 2, 3, 5, 7, 11. The prime numbers 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 K. There are 11 numbers in K. 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 numbers - 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 numbers P(not prime) = (6)/(11) The probability is (6)/(11). 1. (b) Factorize completely: (x+y)(2m-n) - m(x+y) Step 1: Identify the common factor. 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) 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. 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 = 27 sheets × GH¢ 11.00/sheet = 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. The amount earned is GH¢ 365.00. 2. (a) The image of P(2, 5) when translated by the vector r = (-3, 8). A translation by vector (a, b) means adding a to the x-coordinate and b to the y-coordinate. P'(x', y') = (x+a, y+b). (i) Find the image of P(2, 5): P' = (2 + (-3), 5 + 8) = (2 - 3, 5 + 8) = (-1, 13) The image of P(2, 5) is (-1, 13). (ii) Find the image Q' of Q(-4, -6) when translated by r: Q' = (-4 + (-3), -6 + 8) = (-4 - 3, -6 + 8) = (-7, 2) The image of Q(-4, -6) is (-7, 2). 2. (b) Joyce and Richard contributed GH¢ 10,000.00 and GH¢ 5,000.00 respectively. Richard is paid one-third of the profit as a manager. The rest of the profit is shared in the ratio of their contributions. Total profit is GH¢ 9,000.00. 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 = 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 = 2 : 1. Total ratio parts = 2 + 1 = 3. (i) How much did Joyce receive? Joyce's share from remaining profit = (2)/(3) × GH¢ 6,000.00 = GH¢ 4,000.00. Joyce received GH¢ 4,000.00. (ii) How much did Richard receive in total? Richard's share from remaining profit = (1)/(3) × GH¢ 6,000.00 = GH¢ 2,000.00. Richard's total receive = Richard's manager's share + Richard's share from remaining profit Richard's total receive = GH¢ 3,000.00 + GH¢ 2,000.00 = GH¢ 5,000.00. Richard received in total GH¢ 5,000.00. 3. (a) Asana's original daily wage was GH¢ 25.00. Her daily wage increased by 10%. She worked for 30 days. Step 1: Calculate the increase in daily wage. Increase = 10\% of GH¢ 25.00 = 0.10 × GH¢ 25.00 = GH¢ 2.50. Step 2: Calculate the new daily wage. New daily wage = GH¢ 25.00 + GH¢ 2.50 = GH¢ 27.50. Step 3: Calculate the total amount she will be paid for 30 days. Total paid = New daily wage × Number of days worked Total paid = GH¢ 27.50 × 30 = GH¢ 825.00. She will be paid GH¢ 825.00. 3. (b) The diagram shows the floor of a room with its dimensions. Let's determine the missing dimensions first: The total width is 8 m. The bottom segment is 5 m. So, the top inner horizontal segment is 8 m - 5 m = 3 m. The total height on the left is 5 m. The right vertical segment is 3 m. So, the bottom inner vertical segment is 5 m - 3 m = 2 m. (i) Find the perimeter: The perimeter is the sum of all the outer sides. Perimeter = 8 m + 3 m + (8-5) m + (5-3) m + 5 m + 5 m Perimeter = 8 m + 3 m + 3 m + 2 m + 5 m + 5 m Perimeter = 26 m. The perimeter is 26 m. (ii) Find the area: We can divide the L-shaped floor into two rectangles. Method 1: Divide into a top rectangle and a bottom-left rectangle. Rectangle 1 (top): Length = 8 m, Width = 3 m. Area_1 = 8 m × 3 m = 24 m^2. Rectangle 2 (bottom-left): Length = 5 m, Width = (5-3) m = 2 m. Area_2 = 5 m × 2 m = 10 m^2. Total Area = Area_1 + Area_2 = 24 m^2 + 10 m^2 = 34 m^2. Method 2: Divide into a large left rectangle and a small top-right rectangle. Rectangle 1 (left): Length $=