a) Illustrate the information provided in a Venn Diagram.

Here's the solution for question 11: a) Illustrate the information provided in a Venn Diagram. Let Y represent people who speak Yoruba, H for Hausa, and I for Igbo. Let the number of people in each region of the Venn diagram be as follows: • a: speak Yoruba only • b: speak Hausa only • c: speak Igbo only • d: speak Yoruba and Hausa only • e: speak Yoruba and Igbo only • f: speak Hausa and Igbo only • g: speak all three languages (Yoruba, Hausa, and Igbo) • h: speak none of the languages Given information: Total people = 50 g = N(Y H I) = 4 h = N(none) = 6 Number of people who speak exactly two languages = d+e+f = 14 e = (1)/(2)d f = d - 1 Step 1: Find the values for d, e, f. Substitute the expressions for e and f into d+e+f=14: d + (1)/(2)d + (d-1) = 14 2.5d - 1 = 14 2.5d = 15 d = (15)/(2.5) = 6 Now find e and f: e = (1)/(2)(6) = 3 f = 6 - 1 = 5 So, d=6, e=3, f=5. Step 2: Set up equations for a, b, c using the total number of people. The sum of all regions must equal the total number of people: a+b+c+d+e+f+g+h = 50 Substitute the known values: a+b+c+6+3+5+4+6 = 50 a+b+c+24 = 50 a+b+c = 26 This means the number of people who speak one language only is 26. Step 3: Express a, b, c in terms of m. Given: N(Y) = m^2+3 N(H) = 5m-3 N(I) = 3m+1 From the Venn diagram regions: N(Y) = a+d+e+g = a+6+3+4 = a+13 N(H) = b+d+f+g = b+6+5+4 = b+15 N(I) = c+e+f+g = c+3+5+4 = c+12 Equating these: a+13 = m^2+3 a = m^2-10 b+15 = 5m-3 b = 5m-18 c+12 = 3m+1 c = 3m-11 Step 4: Illustrate the Venn Diagram with the calculated values. The values for the regions are: • Yoruba and Hausa and Igbo (g): 4 • Yoruba and Hausa only (d): 6 • Yoruba and Igbo only (e): 3 • Hausa and Igbo only (f): 5 • None of the languages (h): 6 The values for a, b, c depend on m, which will be found in part (b)(i). For the diagram, we will use the final values of a, b, c after finding m. b) Find the: (i) value of m; (ii) number of people who speak one language only. (i) Value of m. Step 1: Use the equation a+b+c=26 and the expressions for a, b, c in terms of m. (m^2-10) + (5m-18) + (3m-11) = 26 m^2 + 8m - 39 = 26 m^2 + 8m - 65 = 0 Step 2: Solve the quadratic equation for m. Using the quadratic formula m = -B ± sqrt(B^2 - 4AC)2A: m = -8 ± sqrt(8^2 - 4(1)(-65))2(1) m = -8 ± sqrt(64 + 260)2 m = -8 ± sqrt(324)2 m = (-8 ± 18)/(2) Two possible values for m: m_1 = (-8 + 18)/(2) = (10)/(2) = 5 m_2 = (-8 - 18)/(2) = (-26)/(2) = -13 Step 3: Check for valid values of m. The number of people in any region cannot be negative. If m = 5: a = 5^2 - 10 = 25 - 10 = 15 b = 5(5) - 18 = 25 - 18 = 7 c = 3(5) - 11 = 15 - 11 = 4 All these values are positive, so m=5 is a valid solution. If $