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QJEEM16Ma59e7: JEE Main 2016 Mathematics - Class 11 JEEM16Ma59e7 Solution: If , then the number of real values of , which satisfy the…

If 0 \le x < 2\pi , then the number of real values of xx , which satisfy the equation  cos⁡x+cos⁡2x+cos⁡3x+cos⁡4x=0\,\cos x + \cos 2x + \cos 3x + \cos 4x = 0 is:

Multiple Choice Options

  • A 7
  • B 9
  • C 3
  • D 5

Step-by-Step Verified Solution

cos⁡x+cos⁡2x+cos⁡3x+cos⁡4x=0\cos x + \cos 2x + \cos 3x + \cos 4x = 0

⇒\Rightarrow (cos⁡x+cos⁡3x)(\cos x + \cos 3x) + (cos⁡2x+cos⁡4x)(\cos 2x + \cos 4x) = 0

⇒2cos⁡2xcos⁡x+2cos⁡3xcos⁡x=0\Rightarrow 2\cos 2x\cos x + 2\cos 3x\cos x = 0

⇒2cos⁡x(2cos⁡5x2cos⁡x2)=0\Rightarrow 2\cos x\left( {2\cos {{5x} \over 2}\cos {x \over 2}} \right) = 0

cos⁡x=0,cos⁡5x2=0,cos⁡x2=0\cos x = 0,\cos {{5x} \over 2} = 0,\cos {x \over 2} = 0

x=π,π2,3π2,π5,3π5,7π5,9π5x = \pi ,{\pi \over 2},{{3\pi } \over 2},{\pi \over 5},{{3\pi } \over 5},{{7\pi } \over 5},{{9\pi } \over 5}

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