Let f(x)= `sin(picosx)` and g(x)=`cos(1pisinx)` e two function defined for `x gt 0 ` define the following sets whose elements are written in increasing order.
`X={x:f(x)=0},Y={x:f'(x)=0}`
`Z={x:g(x)=0},W={x:g'(x)=0}`
`{:("List I","List II"),((I)" "X,(P)" "supe{(pi)/(2),(3pi)/(2),4pi,7pi}),((II)" "Y,(Q)" an arithmetic progression"),((III)" "Z,(R)" not an arithmetic progression"),((IV)" W",(S)" "supe{(pi)/(6),(7pi)/(6),(13pi)/(6)}),(,(T)" "supe{(pi)/(3),(2pi)/(3),pi}),(,(U)" "supe{(pi)/(6),(3pi)/(4)}):}`
Q. Which of th following is the only correct combination

Options

A. IV-(P),(R),(S)
B. III-(R),(U)
C. III-(P),(Q),(U)
D. IV-(Q),(T)

Answer

A

Explanation

`f(x)=0impliessin(picosx)=0`
`impliespicosx=npiimpliescosx=n`
`impliescosx=-1,0,1impliesX={npi,(2n+1)(pi)/(2)}={n(pi)/(2),n in I}`
`f'(x)=0impliescos(picosx)(-pisinx)=0`
`impliespicosx=(2n+1)(pi)/(2)` or `x=npi`
`impliescosx=n+(1)/(2)` or `x=npi`
`impliescosx=+-(1)/(2)` or `x=npi`
`impliesY={2npi+-(pi)/(3),2npi+-(2pi)/(3),n pi, n in I}`
`g(x)=0impliescos(2pisinx)=0implies2pisinx=(2n+1)(pi)/(2)`
`impliessinx=(2n+1)/(4)=+-(1)/(4),+-(3)/(4)`
`impliesZ={npi+-sin^(-1)((1)/(4)),n pi +-sin^(-1)((3)/(4)), n in I}`
`g'(x)=0implies-sin(2pisinx)(2picosx)=0`
`implies2pisinx=n pi` or `x=(2n+1)(pi)/(2)`
`impliesinx=(n)/(2)=0,+-(1)/(2),+-1` or `x=(2n+1)(pi)/(2)`
`impliesW={npi,(2n+1)(pi)/(2),n pi +-(pi)/(6), n in I}`