NIMCET 2015 Question Paper with solutions

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Question : 31 of 120

Marks: +1, -0

The value of

\sin^{-1} 1/√2

+

\sin^{-1} {√2-√1}/√6

+

\sin^{-1} {√3-√2}/√{12}

+ ... to infinity is equal to;

π
π/3
π/2
π/4

Solution:

We have,

\sin^{-1}1/√2+\sin^{-1}{√2-√1}/√6+\sin^{-1}{√3-√2}/√12+..∞

⇒Σ↖{∞}↙{n=1}\sin^{-1}({√x-√{x-1}}/√{x(x+1)})

=Σ↖{∞}↙{n=1}\tan^{-1}√x-\tan^{-1}√{x-1}

⇒Σ↖{∞}↙{n=1}\tan^{-1}√x-\tan^{-1}√{x-1}=\tan^{-1}∞=π/2

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