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Let and ( < ) be the roots of the equation x^2 - 12x + 32 = 0 . Let and be the roots of x^2 - px + q = 0 , and and be the roots of x^2 - ux + v = 0 . If , , form an arithmetic progression and , , form a geometric progression, then the value of v - q u - p is equal to

Correct answer

12

Step-by-step solution

The roots of the equation x^2 - 12x + 32 = 0 are given by: (x - 4)(x - 8) = 0 x = 4 or x = 8 Since It is given that , , are in an arithmetic progression. The common difference is - = 8 - 4 = 4 . Thus, = + 4 = 8 + 4 = 12 . It is also given that , , are in a geometric progression. The common ratio is = 12 8 = 3 2 . Thus, = 3 2 = 12 3 2 = 18 . The roots of x^2 - px + q = 0 are = 8 and = 12 . Sum of roots: p = 8 + 12 = 20 Product of roots: q = 8 12 = 96 The roots of x^2 - ux + v = 0 are = 12 and = 18 . Sum of roots: u

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