Two quadratic equations I and II are given below. Equation I: ax² − 38x + ab = 0 Equation II: cy² − 65y + 2cd = 0 Note: (i) Sum of the values of a and c is 7 (ii) One of the roots of equation I is the 3rd smallest composite number. (iii) One of the roots of equation II is 6. (iv) The value of a is the smallest even prime number. Find the relation between the value of x and y.

From (ii): One of the roots of equation (I) is 3rd smallest composite number. So, one of the roots of equation (I) = 8 Also, a = smallest even prime number So, a = 2 Equation I: 2x² − 38x + 2b = 0 2 X (8)² − 38 X (8) + 2b = 0 2 X 64 − 304 + 2b = 0 128 − 304 + 2b = 0 −176 + 2b = 0 2b = 176 b = 88 Now, 2x² − 38x + 176 = 0 x² − 19x + 88 = 0 x² − 8x − 11x + 88 = 0 x(x − 8) − 11(x − 8) = 0 (x − 8) (x − 11) = 0 x = 8 or 11 From (i): a + c = 7 c = 7 - 2 = 5 Equation II: cy² − 65y + 2cd = 0 5y² − 65y + 10d = 0 One of the roots of equation II is 6. So, 5 X (6)² − 65 X (6) + 10d = 0 5 X 36 − 390 + 10d = 0 180 − 390 + 10d = 0 −210 + 10d = 0 10d = 210 d = 21 Now, 5y² − 65y + 210 = 0 y² − 13y + 42 = 0 y² − 6y − 7y + 42 = 0 y(y − 6) − 7(y − 6) = 0 (y − 6) (y − 7) = 0 So, y = 6 or 7 From I: x = 8 or 11 From II: y = 6 or 7 Therefore, x > y