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 problems: 1. a) i) List all the members of M and N. M = factors of 12 The factors of 12 are the numbers that divide 12 exactly. M = \1, 2, 3, 4, 6, 12\ N = multiples of 2 less than 14 The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, ... Multiples of 2 less than 14 are those before 14. N = \2, 4, 6, 8, 10, 12\ 1. a) ii) Find M N. The intersection M N consists of elements that are common to both set M and set N. M = \1, 2, 3, 4, 6, 12\ N = \2, 4, 6, 8, 10, 12\ The common elements are 2, 4, 6, and 12. M N = \2, 4, 6, 12\ 1. a) iii) Represent M and N on a Venn diagram. To represent M and N on a Venn diagram: Draw two overlapping circles, one for M and one for N. Place the elements of M N = 2, 4, 6, 12 in the overlapping region. Place the elements unique to M (M \ N = 1, 3) in the part of circle M that does not overlap with N. Place the elements unique to N (N \ M = 8, 10) in the part of circle N that does not overlap with M. 1. b) A farmer uses (1)/(3) of his land to plant cassava, (2)/(3) of the remaining land to plant maize and the rest vegetables. If vegetables cover an area of 10 acres. What is the total area of the farmer's land? Step 1: Let the total area of the farmer's land be A acres. Fraction of land for cassava = (1)/(3). Area for cassava = (1)/(3)A. Step 2: Calculate the remaining land after planting cassava. Remaining land = A - (1)/(3)A = (3)/(3)A - (1)/(3)A = (2)/(3)A. Step 3: Calculate the fraction of land for maize. Fraction of remaining land for maize = (2)/(3). Area for maize = (2)/(3) × (remaining land) = (2)/(3) × (2)/(3)A = (4)/(9)A. Step 4: Calculate the total fraction of land used for cassava and maize. Total fraction for cassava and maize = (1)/(3) + (4)/(9). To add these fractions, find a common denominator, which is 9. (1)/(3) = (1 × 3)/(3 × 3) = (3)/(9) Total fraction for cassava and maize = (3)/(9) + (4)/(9) = (7)/(9). Step 5: Calculate the fraction of land for vegetables. Fraction for vegetables = 1 - (fraction for cassava and maize) Fraction for vegetables = 1 - (7)/(9) = (9)/(9) - (7)/(9) = (2)/(9). Step 6: Use the given area for vegetables to find the total area A. We are given that vegetables cover an area of 10 acres. So, (2)/(9)A = 10. To find A, multiply both sides by (9)/(2): A = 10 × (9)/(2) A = 5 × 9 A = 45 acres The total area of the farmer's land is 45 acres. 45 acres 2. a) There are oranges in three different baskets labelled a, b and c. If the number of oranges in basket a is 82, b is 102 and basket c is 96. Find the total number of oranges in the three baskets using associative property. The associative property of addition states that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum. That is, (x+y)+z = x+(y+z). Number of oranges in basket a = 82 Number of oranges in basket b = 102 Number of oranges in basket c = 96 Total number of oranges = 82 + 102 + 96. Using the associative property, we can group them as follows: (82 + 102) + 96 184 + 96 280 Alternatively: 82 + (102 + 96) 82 + 198 280 The total number of oranges in the three baskets is 280. 280 2. b) Copy and complete the table of values for the relations y_1 = 2x + 5 and y_2 = 3 - 2x from -4 to 3. First, calculate the values for y_1 = 2x + 5: For x = -4: y_1 = 2(-4) + 5 = -8 + 5 = -3 For x = -3: y_1 = 2(-3) + 5 = -6 + 5 = -1 For x = -2: y_1 = 2(-2) + 5 = -4 + 5 = 1 For x = -1: y_1 = 2(-1) + 5 = -2 + 5 = 3 For x = 0: y_1 = 2(0) + 5 = 0 + 5 = 5 For x = 1: y_1 = 2(1) + 5 = 2 + 5 = 7 For x = 2: y_1 = 2(2) + 5 = 4 + 5 = 9 For x = 3: y_1 = 2(3) + 5 = 6 + 5 = 11 Next, calculate the values for y_2 = 3 - 2x: For x = -4: y_2 = 3 - 2(-4) = 3 + 8 = 11 For x = -3: y_2 = 3 - 2(-3) = 3 + 6 = 9 For x = -2: $y_2 = 3 - 2