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Question

Which of the following is/are correct regarding their fundamental period?

A
1|sinx|+|cosx|π/2
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B
|sinx+cosx|π
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C
|sinx+cosx|+|sinxcosx|3π/2
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D
|sinx|cosx+sinx|cosx|2π
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Solution

The correct option is D |sinx|cosx+sinx|cosx|2π
If f(x), g(x) are complementary even functions, then period of f(x)±g(x) is
12(L.C.M of period of f(x), period of g(x))

And |sinx|,|cosx| are complementary function.

(a) We know that the fundamental period of |sinx| and |cosx| is π.
Now, the period of
|sinx+cosx12(L.C.M(π,π))=π2
So, the period of 1|sinx|+|cosx|π2

(b) |sinx+cosx|=2|sin(45°+x)|
So, the period of 2|sin(45°+x)| is π,
Therefore, the period of
|sinx+cosx|π

(c) The period of |sinx±cosx| is π.
|sinx+cosx+sinxcosx| are complementary functions​.
Period of |sinx+cosx+sinxcosx
=12(L.C.M(π,π))=π2

(d) Period of |sinx|cosx=L.C.M.(π,2π)=2π
Period of sinx|cosx|=L.C.M.(2π,π)=2π
Period of |sinx|cosx+sinx|cosx|=L.C.M.(2π,2π)=2π

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