If [ sin x ] + [ 2 π x ] + [ 5 π 2 x ] = 10 π 9 x when the number of solutions in the interval ( 30 , 40 ) is (where [.] is GIF)

Question

If [ sin x ] + [ 2 π x ​ ] + [ 5 π 2 x ​ ] = 10 π 9 x ​ when the number of solutions in the interval ( 30 , 40 ) is (where [.] is GIF)

TestHub · Accepted Answer

The equation is [ sin x ] = 10 π 9 x ​ − [ 2 π x ​ ] − [ 5 π 2 x ​ ] . Since [ sin x ] is an integer, 10 π 9 x ​ − [ 2 π x ​ ] − [ 5 π 2 x ​ ] must also be an integer. In the interval ( 30 , 40 ) , we have 4.77 < 2 π x ​ < 6.37 and 9.55 < 5 π 2 x ​ < 12.73 . Thus, [ 2 π x ​ ] can be 4, 5, or 6, and [ 5 π 2 x ​ ] can be 9, 10, 11, or 12. Since − 1 ≤ sin x ≤ 1 , [ sin x ] can only be -1, 0, or 1. The number of solutions is 1. The concept used is the greatest integer function (GIF).

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