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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You're on a roll — Here are the solutions to the questions: 1. a) i) List the elements in set 'A' and 'B'. Step 1: Define the universal set U. U = \Odd numbers less than 19\ U = \1, 3, 5, 7, 9, 11, 13, 15, 17\ Step 2: Define set A. A = \Factors of 15\ The factors of 15 are the numbers that divide 15 exactly. A = \1, 3, 5, 15\ Step 3: Define set B. B = \Prime numbers\ (within U) Prime numbers in U are numbers greater than 1 that are only divisible by 1 and themselves. B = \3, 5, 7, 11, 13, 17\ The elements in set A are \1, 3, 5, 15\ and in set B are \3, 5, 7, 11, 13, 17\. ii) Find the set of A B and A B. Step 1: Find the intersection A B. A B contains elements common to both A and B. A = \1, 3, 5, 15\ B = \3, 5, 7, 11, 13, 17\ A B = \3, 5\ Step 2: Find the union A B. A B contains all elements in A or B (or both). A B = \1, 3, 5, 7, 11, 13, 15, 17\ The set A B is \3, 5\ and A B is \1, 3, 5, 7, 11, 13, 15, 17\. iii) Illustrate U, A and B on a Venn diagram. Step 1: Draw a rectangle for U and two overlapping circles for A and B. Step 2: Place elements: A B = \3, 5\. A only = \1, 15\. B only = \7, 11, 13, 17\. Elements in U but not in A B = \9\. ` U +---------------------------------+ | +-------+ +-------+ | | | A | | B | | | | 1,15| | 7,11,| | | | +--+---+--+ | | | | |3,5| | | | | | +--+---+--+ | | | | | |13,17| | | +-------+ +-------+ | | | | 9 | +---------------------------------+ ` iv) Shade the region for prime factors of 15 on the Venn diagram. The prime factors of 15 are 3 and 5. This corresponds to the region A B. Shade the overlapping region of circles A and B. 1. b) Find the value of x if 3^x-1 = 27. Step 1: Express 27 as a power of 3. 27 = 3^3 Step 2: Rewrite the equation with the same base. 3^x-1 = 3^3 Step 3: Equate the exponents. x - 1 = 3 Step 4: Solve for x. x = 3 + 1 x = 4 The value of x is 4. 2. a) Determine the combination of GH¢20, GH¢50, GH¢100, GH¢200 and GH¢500 notes to make GH¢1,000,000. Here is one possible combination: Step 1: Use GH¢500 notes. Number of GH¢500 notes = GH¢500,000GH¢500 = 1,000 notes. Remaining amount = GH¢1,000,000 - GH¢500,000 = GH¢500,000. Step 2: Use GH¢200 notes. Number of GH¢200 notes = GH¢200,000GH¢200 = 1,000 notes. Remaining amount = GH¢500,000 - GH¢200,000 = GH¢300,000. Step 3: Use GH¢100 notes. Number of GH¢100 notes = GH¢200,000GH¢100 = 2,000 notes. Remaining amount = GH¢300,000 - GH¢200,000 = GH¢100,000. Step 4: Use GH¢50 notes. Number of GH¢50 notes = GH¢50,000GH¢50 = 1,000 notes. Remaining amount = GH¢100,000 - GH¢50,000 = GH¢50,000. Step 5: Use GH¢20 notes. Number of GH¢20 notes = GH¢50,000GH¢20 = 2,500 notes. Remaining amount = GH¢50,000 - GH¢50,000 = GH¢0. The combination is: • 1,000 notes of GH¢500 • 1,000 notes of GH¢200 • 2,000 notes of GH¢100 • 1,000 notes of GH¢50 • 2,500 notes of GH¢20 2. b) A rectangular sheet of metal has length 44cm and breadth 20cm. It is folded to form a cylinder with its breadth becoming the height. Calculate i) the radius of the cylinder formed. Step 1: Identify the dimensions of the cylinder. The length of the rectangular sheet becomes the circumference of the base of the cylinder. Circumference (C) = 44 cm The breadth of the rectangular sheet becomes the height of the cylinder. Height (h) = 20 cm Step 2: Calculate the radius (r) using the circumference formula. C = 2 r 44 = 2 × (22)/(7) × r 44 = (44)/(7) × r r = (44 × 7)/(44) r = 7 cm The radius of the cylinder formed is 7 cm. ii) the volume of the cylinder [Take = (22)/(7)] Step 1: Use the formula for the volume of a cylinder. V = r^2 h Step 2: Substitute the values for r and h. V = (22)/(7) × (7 cm)^2 × 20 cm V = (22)/(7) × 49 cm^2 × 20 cm V = 22 × 7 cm^2 × 20 cm V = 154 cm^2 × 20 cm V = 3080 cm^3 The volume of the cylinder is 3080 cm^3. 2. c) Given that m = 7 \\ 2x and n = 5y - 3 \\ 8 . If m = n, find the value of x and y. Step 1: Equate the corresponding components of the vectors. For the first component: 7 = 5y - 3 For the second component: 2x = 8 Step 2: Solve the equation for y. 7 = 5y - 3 7 + 3 = 5y 10 = 5y y = (10)/(5) y = 2 Step 3: Solve the equation for x. 2x = 8 x = (8)/(2) x = 4 The value of x is 4 and the value of y is 2. 3. a) Given that x = -3 and y = (3)/(4). Find the value of i) x^2(y - 1) Step 1: Substitute the values of x and y into the expression. x^2(y - 1) = (-3)^2 ((3)/(4) - 1) Step 2: Simplify the terms. (-3)^2 = 9 (3)/(4) - 1 = (3)/(4) - (4)/(4) = -(1)/(4) Step 3: Multiply the simplified terms. 9 × (-(1)/(4)) = -(9)/(4) The value of x^2(y - 1) is -(9)/(4). ii) y^2 - (3)/(x) Step 1: Substitute the values of x and y into the expression. y^2 - (3)/(x) = ((3)/(4))^2 - (3)/(-3) Step 2: Simplify the terms. ((3)/(4))^2 = (3^2)/(4^2) = (9)/(16) (3)/(-3) = -1 Step 3: Perform the subtraction. (9)/(16) - (-1) = (9)/(16) + 1 (9)/(16) + (16)/(16) = (25)/(16) The value of $y^