KVPY SB SX Exam 2011 Question Paper

Question : 4 of 120

Marks: +1, -0

Consider the cubic equation

x^3 + ax^2+ bx + c = 0

, where a, b, c are real numbers. Which of the following statements is correct?

Solution:

f(x) = x^3 + ax^2 + bx + c

f ' (x) = 3x^2 + 2ax + b

D = 4a^2 - 4 . 3 . b

= 4 (a^2 - 3b)

a^2 < 2b

⇒ a^2 < 3b

⇒ D < 0

⇒ f '(x) = 0 has non real roots

Hence f(x) = 0 has 1 real and two imaginary roots

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