If 2 + 3i is one of the roots of the equation 2x^3 - 9x^2 + kx - 13 = 0, k R, then the real root of this equation (where i^2 = -1):

JEE Main 2015 Mathematics question

If is one of the roots of the equation , , then the real root of this equation (where ):

(1)Exists and is equal to Correct
(2)Does not exist
(3)Exists and is equal to 1
(4)Exists and is equal to

Answer: (1)

Solution

The coefficients are real, so complex roots occur in conjugate pairs: is also a root. Let the third root be , which must be real since a cubic with real coefficients has at least one real root. By Vieta's formula the sum of the roots is , so $(2+3i) + (2-3i) + α =

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Question ID VK-115285