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1. An angle 'x' is chosen at random from from the interval 0
Find the probability 'p' that the numbers sin²x, cos²x and (sin x)*(cos x) are not the lengths of the sides of a triangle.
Answer :
p = [2*(tan⁻¹2)]/π
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2. A circle of radius 1 unit is placed randomly inside a 15*36 rectangle ABCD so that it completely lies inside the rectangle. Find the probability 'p' that the circle does not touch or intersect the diagonal AC.
Answer:
p = (375)/(442)
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1. An angle 'x' is chosen at random from from the interval 0
Answer :
p = [2*(tan⁻¹2)]/π
--------------------------------------…
2. A circle of radius 1 unit is placed randomly inside a 15*36 rectangle ABCD so that it completely lies inside the rectangle. Find the probability 'p' that the circle does not touch or intersect the diagonal AC.
Answer:
p = (375)/(442)
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1.
By triangular inequality,
sin^2 x + cos^2 x > sinx cosx
=> 1 > sinx cosx is true for all values of x.
Next, sin^2 x > cos^2 x + sinx cosx
=> tan^2 x > 1 + tanx
=> tan^2 x - tanx - 1 > 0
=> (tanx - 2) (tanx + 1) > 0
=> tanx > 2 and tanx > - 1 or tanx < - 2 or tanx < 1
=> 0 < x < arctan2 or arctan < -2 (not possible)
=> rquired probability, p
= (arctan2 - 0) / (π/2 - 0)
= [2*(tan⁻¹2)]/π.
2.
Please refer to the last solved problem of the following link for the solution.
http://amc.maa.org/d-publication/d1-puba…
By triangular inequality,
sin^2 x + cos^2 x > sinx cosx
=> 1 > sinx cosx is true for all values of x.
Next, sin^2 x > cos^2 x + sinx cosx
=> tan^2 x > 1 + tanx
=> tan^2 x - tanx - 1 > 0
=> (tanx - 2) (tanx + 1) > 0
=> tanx > 2 and tanx > - 1 or tanx < - 2 or tanx < 1
=> 0 < x < arctan2 or arctan < -2 (not possible)
=> rquired probability, p
= (arctan2 - 0) / (π/2 - 0)
= [2*(tan⁻¹2)]/π.
2.
Please refer to the last solved problem of the following link for the solution.
http://amc.maa.org/d-publication/d1-puba…