InterviewSolution
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InterviewSolution
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Let Delta(x)=|{:(cos^(2)x,cosxsinx,-sinx,),(cosx sinx,sin^(2)x,cosx,),(sinx,-cosx,0,):}|"then" int_(0)^(pi//2)[Delta(x)+Delta'(x)]dx equals |
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Answer» `(pi)/(3)` Let `f(n)=K,""kinN` `implies k-(1)/(2)ltsqrt(n)LTK+(1)/(2)impliesk^(2)-k+(1)/(4)ltnltk^(2)+k+(1)/(4)impliesk^(2)-k+1lenlek^(2)+k""(therefore nin N)` `therefore` Number of integers for which `f(n)=k" is "(k^(2)+k)-(k^(2)-k)=2k` `{:(therefore,,f(n)=1" for",,n=1.2),(,,f(n)=2" for",,n=3.4.5.6),(,,f(n)=3" for",,n=7.8.9.10.11.12):}` `therefore underset(2)ubrace((1)/(f(1))+(1)/(f(2)))+underset(2)ubrace((1)/(f(3))+(1)/(f(4))+(1)/(f(5))+(1)/(f(6)))+.......+(1)/(f(2015))+(1)/(f(2016))` Sum of first 2 terms = 2 Sum of next 4 terms =2 Sum of next 6 terms = 2 |
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