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Question

If A1 and A2 be the areas of two regular polygons having the same perimeter and number of sides be n and 2n respectively, then A1A2 is -
  1. independent of n  
  2. depends on n 
  3. is a function of cos(πn) only with no fractional powers
  4. is a function of sin(πn) only with no fractional powers.


Solution

The correct options are
B depends on n 
C is a function of cos(πn) only with no fractional powers
Area of polygon with n sides of length x is,
A=14.nx2cotπn
So, A1=14na21cotπn, A2=14(2n)a22cotπ2n
& na1=2na2
A1A2=12×4cot(π/n)cot(π/2n)=2cosπn.sinπ2nsinπn.cosπ2n
=2cosπn.sinπ2n2sinπ2ncosπ2n.cosπ2n
=2cosπn(1+cosπn)

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