GATE Computer Science (CS) 2019 Solved Papers

Question : 57 of 65

Marks: +1, -0

Suppose

Y

is distributed uniformly in the open interval

(1,6)

. The probability that the polynomial

3 {x}^{2}+6 {xy}+3 {Y}+6

has only real roots is (rounded off to 1 decimal place) _________.

Solution:

It is given that, Polynomial

3 x^{2}+6 x Y+

3 {Y}+6

has only real roots

b^{2}-4 a x ≥ 0

(6 Y)^{2}-4(3)(3 Y+6) ≥ 0

{Y}^{2}- {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

P(2 < y < 6)=∫↙{2}↖{6} f(Y) d y

=\;{1}/{5}[Y]_{2}^{6}=\;{4}/{5}=0.8

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