The equation x2 – px – 60 = 0, has two roots ‘a’ and ‘b’ such that (a – b) = 17 and p > 0. If a series starts with ‘p’ such that the consecutive terms are 4 more than the preceding term is formed, then find the product of 2nd and 4th terms of such series.
Given, x2 + px – 60 = 0 Since, sum of roots = -(-p)/1 So, a + b = p Since, product of roots of the equation = -(60/1) Therefore, ab = -60……. (1) Or, b = (-60/a) And, a – b = 17……. (2) Putting the value of ‘b’ in equation (2), we get a2 – 17a + 60 = 0 => a2 – 12a – 5a + 60 = 0 => a(a – 12) – 5(a – 12) = 0 => (a – 12)(a – 5) = 0 => a = 12, 5 When a = 12, then b = -5(-60/12) And, when a = 5, then b = -12(-72/5) Therefore, a + b = 12 + (-5) = 7 = p Therefore, series will be 7, 11, 15, 19 required product = 11 × 19 = 209
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