Let f(x) = sin(πcos x) and g(x) = cos(2πsin x) be two functions defined for x > 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 contains sets X, Y, Z, and W. List II contains some information regarding these sets. List I | List II A. X | 1. {π/2, 3π/2, 4π, 7π} B. Y | 2. an arithmetic progression C. Z | 3. NOT an arithmetic progression D. W | 4. {π/6, 7π/6, 13π/6} | 5. {π/3, 2π/3, π} | 6. {π/6, 3π/4} Which of the following is only CORRECT combination?

Let f(x) = sin(πcos x) and g(x) = cos(2πsin x) be two functions defined for x > 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 contains sets X, Y, Z, and W. List II contains some information regarding these sets.

List I | List II
A. X | 1. {π/2, 3π/2, 4π, 7π}
B. Y | 2. an arithmetic progression
C. Z | 3. NOT an arithmetic progression
D. W | 4. {π/6, 7π/6, 13π/6}
| 5. {π/3, 2π/3, π}
| 6. {π/6, 3π/4}

Which of the following is only CORRECT combination?

I−(Q),(U)

I−(P),(R)

II−(Q),(T)

II−(R),(S)

Solution:

Correct option is C.II−(Q),(T)f(x)=0⇒sin⁡(πcos⁡x)=0=πcos⁡x=nπ⇒cos⁡x=n=cos⁡x=−1;,0,1⇒X=nπ,(2n+1)π2=nπ2,n∈If′(x)=0⇒cos⁡(πcos⁡x)(−πsin⁡x)=0πcos⁡x=(2n+)π2orx=nπ⇒cos⁡x=n+12orx=nπ⇒cos⁡x=±12orx=nπ⇒Y=.−2;π3,π3,0,π3,2π3,π,4π3,which is an arithmetic progressiong(x)=0⇒(2πsin⁡x)=0⇒2πsin⁡x=(2n+1)π2⇒sin⁡x=2n+14=±14,±34⇒sin⁡x=2n+14=±14,±34⇒z=nπ±sin−1;⁡14,nπ±sin−1;⁡34,n∈Ig′(x)=0⇒−sin⁡(2πsin⁡x)(2πcos⁡x)=0⇒2πsin⁡x=nπorx=(2n+)π2⇒sin⁡x=n2=0,±12,±1orx=(2n+1)π2W=nπ,(2n+1)π2,nπ±π5,n∈I

Eduprobe

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