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23 Apr 2019, 13:56
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D
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Difficulty:
75%
(hard)
Question Stats:
45% (02:17) correct
55%
(02:18)
wrong
based on 29
sessions
History
Date
Time
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Not Attempted Yet
Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
(A) 1/6
(B) 1/4
(C) \(\frac{2 - \sqrt{2}}{2}\)
(D) 1/3
(E) 1/2
(A) 1/6
(B) 1/4
(C) \(\frac{2 - \sqrt{2}}{2}\)
(D) 1/3
(E) 1/2
24 Apr 2019, 01:03
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Let a random point on the outer circle A, draw 2 chords, which are tangent to the inner circle, intersect outer circle at B and C respectively. Chord from A will intersect inner circle only when other point lies on the minor arc BC.
probability that the chord joining the two points intersects the inner circle= angle BOC/360
angle BOC= 2* angle BAC
Now sin(OAC)= 1/2=0.5
Angle OAC=30
angle BAC=2*30=60
angle BOC 2*60=120
Probability= 120/360=1/3
probability that the chord joining the two points intersects the inner circle= angle BOC/360
angle BOC= 2* angle BAC
Now sin(OAC)= 1/2=0.5
Angle OAC=30
angle BAC=2*30=60
angle BOC 2*60=120
Probability= 120/360=1/3
Bunuel
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24 Apr 2019, 02:22
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Bunuel
First select any arbitrary point on the outer now we can select the second point only from that part of the outer circle such that the two selected points are just able to form a tangent to the inner circle this gives us a region from which we can never intersect the inner circle with two points selected on it,so the required region or circumference is total- that part.
ANS D
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