JEE Advanced 2025 Paper 1 · Mathematics · Probability · hard

A number a is chosen at random (uniformly) from the interval [0,3] and independently a number b is chosen at random (uniformly) from [0,2]. Find the…

Numerical answer

A number

a

is chosen at random (uniformly) from the interval

[0,3]

and independently a number

b

is chosen at random (uniformly) from

[0,2]

. Find the probability that the equation

t^{2}+at+b=0

has real roots. Give the answer correct to two decimal places.

Show answer and solution

Answer: 0.37

Real roots require

a^2\ge4b

, i.e.

b\le a^2/4

. The sample space is the rectangle

0\le a\le3

0\le b\le2

of area

6

. The curve

b=a^2/4

reaches

b=2

a=2\sqrt2\approx2.828<3

. For

0\le a\le2\sqrt2

the favourable

b

range has length

a^2/4

; for

2\sqrt2\le a\le3

the whole range of length

2

is favourable. Favourable area

=\displaystyle\int_0^{2\sqrt2}\frac{a^2}{4}da+2\left(3-2\sqrt2\right)=\frac{(2\sqrt2)^3}{12}+6-4\sqrt2=\frac{4\sqrt2}{3}+6-4\sqrt2=6-\frac{8\sqrt2}{3}\approx2.2288

. Probability

=\dfrac{2.2288}{6}=1-\dfrac{4\sqrt2}{9}\approx0.3715

, which rounds to

0.37

. A direct numerical integration agrees with

0.37146

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