Aspire Faculty ID #12161 · Topic: JEE Main 2 September 2020 (Evening) · Just now
JEE Main 2 September 2020 (Evening)

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :

Solution

There are \(n\) stations on a circle. Each pair is connected by a straight track.

Blue lines connect nearest neighbours, so the number of blue lines is 
Blue $ = n. $ 
Total lines is $ \binom{n}{2} = \frac{n(n-1)}{2}. $ 

Hence red lines are $ \text{Red} = \binom{n}{2} - n. $ 

Given Red =$ 99 \times \text{Blue} $ 
$ \binom{n}{2} - n = 99n $
$ \;\;\Longrightarrow\;\; \frac{n(n-1)}{2} - n = 99n$
$ \;\;\Longrightarrow\;\; \frac{n(n-1)}{2} = 100n$
$ \;\;\Longrightarrow\;\; n-1 = 200$
$ \;\;\Longrightarrow\;\; n = 201.$
 Final Answer: \(\boxed{201}\)
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