question_answer

Three points \[P,Q,R\] are selected at random from the circumference of a circle. The probability that the points lie on a semicircle is

A) \[\frac{1}{2}\]

B) \[\frac{2}{3}\]

C) \[\frac{3}{4}\]  

D) \[\frac{\pi }{2}\]

Correct Answer: C

Solution :

Let the length  of circumference in \[2l.\]Let \[x\] denotes the clockwise are length of \[PQ\] and let \[y\] denotes the clockwise are length of \[PR.\] then \[0<x<2l\,\operatorname{and}\,<y<2l\]

Thus, sample space is the set of points inside a square of edge length \[2l.\]

Now \[P,Q,R\]lie on a semicircle under following conditions:

(1)  \[x<l,y<l\]

(2) \[x<l\,\operatorname{and}\,y-x>l\]

(3) \[x>l,y>l\]

(4) \[y<l,\,and\,x-y>l\]

Thus, required probability \[=\frac{Area\,of\,shaded\,region\,}{Area\,of\,square\,S}\]\[=\frac{{{l}^{2}}+\frac{1}{2}{{l}^{2}}+{{l}^{2}}+\frac{1}{2}{{l}^{2}}}{{{\left( 2l \right)}^{2}}}=\frac{3}{4}\]