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GATE | GATE CS 2019 | Question 56

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Suppose Y is distributed uniformly in the open interval (1, 6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _________.

Note: This was Numerical Type question.
(A) 0.80
(B) 0.17
(C) 0.20
(D) 1


Answer: (A)

Explanation: For a quadratic polynomial ax2 + bx + c = 0. There are three condition:

b2 - 4ac > 0   {real and distinct root, i.e., two real roots}
b2 - 4ac = 0     {real and equal roots, i.e., only one real root}
b2 - 4ac < 0    {imaginary roots} 

Polynomial 3x2 + 6xY + 3Y + 6 has only real roots,

⇒ b2 – 4ax ≥ 0
⇒ (6Y)2 – 4(3) (3Y+ 6) ≥ 0
⇒ Y2 – Y + 2 ≥ 0

Y ∈ (–∞, – 1] ∩ [2, ∞)
⇒ Y ∈ [2, 6) 

Since y is uniformly distributed in (1, 6).

Probability distributed function,

 f(Y) = (1/5), 1 < y > 6

Hence,

So, answer is 0.8.

Quiz of this Question


Last Updated : 19 Feb, 2019
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