Let the given function is differentiable at x = 1.
`f(x)={{:(lim_(nrarroo) (ax(x-1)(cot.(pix)/(4))^(n)+(px^(2)+2))/((cot,(pix)/(4))^(n)+1)",",x in(0,1)uu(1,2)),(0",",x=1):}`
Then the value of `|a+q|` is

Options

A. 4
B. 6
C. 8
D. 10

Answer

B

Explanation

`f(x)={{:(underset(nrarroo)(lim)(ax(x-1)(cot.(pix)/(4))^(n)+(px^(2)+2))/((cot.(pix)/(4))^(n)+1)),(0", x=1"):}`
`x in (0,1)uu(1,2)`
`therefore" "f(x)={{:(ax(x-1)",",0ltxlt1),(0",",x=1),(px^(2)+2",",1ltxlt2):}`
`therefore` f(x) must be continuous at x = 1
`therefore" L.H.L. = R.H.L"=f(1)`
`rArr" "p+2=0 rArr p=-2`
Also, `f'(x)={{:(a(2x-1)",",0ltxlt1),(-4x",",xgt1):}`
`because" f(x) is differentiable at x = 1, so"`
L.H.D. = R.H.D. `rArr a=-4`
Hence `|a+p|=6`