If `f(x)={sin(pi/2)(x-[x]),x<5 5(b-1),x=5(a b^2|x^2-11 x+24|)/(x-3),x >5` is continuous at `x=5,a ,b in R` then ([.] denotes the greatest integer function) `a=(25)/(108), b=6/5` b. `a=6/(13), b=(17)/(29)` c. `a=1/2, b=(25)/(36)` d. `a=(23)/(100), b=6/5`

Options

A. `a=(25)/(108),b=(6)/(5)`
B. `a=(6)/(13),b=(17)/(29)`
C. `a=(1)/(2),b=(25)/(36)`
D. `a=(23),(100),b=(6)/(5)`

Answer

A

Explanation

`f(5^(-))=underset(xrarr5^(-))(lim)sin((pi)/(2)(x-[x]))`
`" "=underset(xrarr5^(-))(lim)sin((pi)/(2)(x-4))=1`
`f(5^(+))=underset(xrarr5^(+))(lim)(ab^(2)|x^(2)-11x+24|)/(x-3)`
`" "=underset(xrarr5^(+))(lim)(-ab^(2)(x-3)(x-8))/(x-3)`
`rArr" "5(b-1)=1=3ab^(2)`
`rArr" "b=(6)/(5),a=(25)/(108)`