If `f(x)=[{:(mx+1,if x le (pi)/(2)),(sinx+n,ifxgt(pi)/(2)):}` is continuous at `x = (pi)/(2)`, then

Options

A. `m=1,n=0`
B. `m=(npi)/(2)+1`
C. `n=m((pi)/(2))`
D. `m=n=(pi)/(2)`

Answer

C

Explanation

Given function is
`f(x)={{:(mx+1",",xle(pi)/(2)),(sinx+n",",xgt(pi)/(2)):}`
As given this function is continuous at `x=(pi)/(2)`.
So, limit of funcotion when `xto(pi)/(2)=f((pi)/(2))`
`impliesunderset(h-0)lim(sin((pi)/(2)+h)+n)=(mpi)/(2)+1`
`impliessin""(pi)/(2)+n=(mpi)/(2)+1`
`implies1+n=(mpi)/(2)+1`
`impliesn=(mpi)/(2)`