For the functions f( ) = ^2 + ^2 , and g( ) = ^2 + ^2 , > > 0, let _ 0 < < /2 f( ) = _ 0 < < g( ). If the first term of a G.P. is ( 2 ), its common ratio is ( 2 ) and the sum of its first 10 terms is m n , (m, n) = 1, then m + n is equal to _______.
A. A
For the function f( ) = ^2 + ^2 , applying the AM-GM inequality gives: ^2 + ^2 2 ^2 ^2 = 2 Thus, _ 0 For the function g( ) = ^2 + ^2 , we can rewrite it as: g( ) = ^2 + (1 - ^2 ) = + ( - ) ^2 Since > > 0, the maximum value of g( ) occurs when ^2 is maximum, which is 1 (at = /2). Thus, _ 0 Given that f( ) = g( ), we have: 2 = Squaring both sides (since > 0), we get: 4 = ^2 = 4 For the given Geometric Progression (G.P.): First term a = 2 = 4 2 = 2 Common ratio r = 2 = 2 4 = 1 2 The sum of the first 10 terms is: S_ 10 = a 1 - r^ 10 1 - r = 2 1 - (1/2)^ 10 1 - 1/2 = 4 (1 - 1 1024 ) = 4 ( 1023 1024 ) = 1023 256 We are given S_ 10 = m n with (m, n) = 1. Since 256 = 2^8 and 1023 is odd, they are coprime. Therefore, m = 1023 and n = 256. m + n = 1023 + 256 = 1279 Answer: 1279
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