For x     ϵ     0 ,   π , the equation sin ⁡ x + 2 sin ⁡ 2 x…

For x  ϵ  0, π, the equation sin⁡x+2sin⁡2x-sin⁡3x=3 has
  1. Infinitely solution
  2. Three solutions
  3. One solution
  4. No solution

Solution

sin⁡x+2sin⁡2x-sin⁡3x=3
sin⁡x+4 sin⁡xcos⁡x-3sin⁡x+4sin3⁡x=3  
sin⁡x -2+4cos⁡x+41-cos2⁡x=3
sin⁡x 2-4cos2⁡x-4cos⁡x+1+1=3
sin⁡x 3-2cos⁡x-12 =3
⇒  sin⁡x=1 and 2cos⁡x-1=0
⇒   x=π2  and x=π3
Which is not possible at same time
Hence, no solution

Asked in: JEE Advanced 2014 (Paper 2)

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