Solve the simultaneous equations: x + 2y = 7, 2x - 2y = 2. Consider the sets P = x: x is a prime number less than 15 and Q = x: x is an even number less than 15. List elements of P and Q, find P U Q,

Here are the solutions to the questions: Question 1 a) Solve the simultaneous equations: x + 2y = 7 (1) 2x - 2y = 2 (2) Step 1: Add Equation (1) and Equation (2) to eliminate y. (x + 2y) + (2x - 2y) = 7 + 2 3x = 9 Step 2: Solve for x. x = (9)/(3) x = 3 Step 3: Substitute the value of x into Equation (1) to find y. 3 + 2y = 7 2y = 7 - 3 2y = 4 Step 4: Solve for y. y = (4)/(2) y = 2 The solution is x=3 and y=2. x=3, y=2 b) Consider the sets: P = \x: x is a prime number less than 15\ Q = \x: x is an even number less than 15\ i) List all the elements of sets P and Q. Prime numbers less than 15 are numbers greater than 1 that are only divisible by 1 and themselves. P = \2, 3, 5, 7, 11, 13\ Even numbers less than 15 are integers divisible by 2. Q = \2, 4, 6, 8, 10, 12, 14\ ii) Find P Q. The union of sets P and Q includes all elements that are in P, or in Q, or in both. P Q = \2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14\ iii) State whether P and Q are disjoint sets. Disjoint sets have no common elements (their intersection is an empty set). The intersection of P and Q is P Q = \2\. Since P Q ≠ , the sets are not disjoint. No, P and Q are not disjoint sets. Question 2 a) Solve 2(x + 4) 5x - 1 and graph the solution on a number line. Step 1: Distribute the 2 on the left side of the inequality. 2x + 8 5x - 1 Step 2: Subtract 2x from both sides. 8 5x - 2x - 1 8 3x - 1 Step 3: Add 1 to both sides. 8 + 1 3x 9 3x Step 4: Divide by 3. (9)/(3) x 3 x This can also be written as x 3. x 3 Graph on a number line: A closed circle at 3 with an arrow extending to the left, indicating all numbers less than or equal to 3. ` <-----|-----|-----|-----|-----|-----|-----|-----> -2 -1 0 1 2 3 4 [----------- ` b) Change 11101_2 to base ten numeral. Step 1: Write out the place values for each digit in base 2. 11101_2 = (1 × 2^4) + (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) Step 2: Calculate the powers of 2. 2^4 = 16 2^3 = 8 2^2 = 4 2^1 = 2 2^0 = 1 Step 3: Substitute the powers of 2 and multiply. 11101_2 = (1 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) 11101_2 = 16 + 8 + 4 + 0 + 1 Step 4: Sum the results. 11101_2 = 29 29_10 Question 3 a) Three machines produce packages at regular intervals: Machine A every 12 minutes, Machine B every 18 minutes, and Machine C every 24 minutes. Find the first time (in minutes) when all three machines will produce a package at the same time. To find when all three machines will produce a package at the same time, we need to find the Least Common Multiple (LCM) of their intervals: 12, 18, and 24 minutes. Step 1: Find the prime factorization of each number. 12 = 2 × 2 × 3 = 2^2 × 3^1 18 = 2 × 3 × 3 = 2^1 × 3^2 24 = 2 × 2 × 2 × 3 = 2^3 × 3^1 Step 2: To find the LCM, take the highest power of each prime factor present in any of the factorizations. The highest power of 2 is 2^3. The highest power of 3 is 3^2. Step 3: Multiply these highest powers together. LCM(12, 18, 24) = 2^3 × 3^2 = 8 × 9 = 72 All three machines will produce a package at the same time after 72 minutes. 72 minutes b) If an item is sold at a 20% profit for GH¢60, find its original cost price. Let the original cost price be C. The profit is 20% of the cost price, which is 0.20C. The selling price (SP) is the cost price plus the profit. SP = C + 0.20C SP = 1.20C We are given that the selling price is GH¢60. 60 = 1.20C Step 1: Solve for C. C = (60)/(1.20) C = (60)/(120)100 C = (60 × 100)/(120) C = (6000)/(120) C = 50 The original cost price is GH¢50. GH¢50 c) Arrange the numbers -20, 15, -2, 0, 56, 4 in ascending order. Ascending order means arranging numbers from the smallest to the largest. Step 1: Identify the negative numbers and arrange them: -20, -2. Step 2: Identify zero: 0. Step 3: Identify the positive numbers and arrange them: 4, 15, 56. Combining these in order: -20, -2, 0, 4, 15, 56 -20, -2, 0, 4, 15, 56