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Question

Let f(x)=sinx[xπ]+12, where [x] denotes greatest integer function, then f(x) is

A
odd function
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B
even function
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C
neither odd or even
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D
both odd and even function
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Solution

The correct option is B even function
f(0)=sin0[0π]+12=0
Now putting xx
f(x)=sin(x)[xπ]+12
Now we know that, when xπ is not an integer
[xπ]+[xπ]=1
Using this we can write,
f(x)=sin(x)[xπ]1+12f(x)=sinx[xπ]+12=f(x)
When xπ is an integer,
xπ=nx=nπ
So the function will be,
f(nπ)=sin(nπ)n+12=0
So, the given function is even

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