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QJEEM21Mac938: JEE Main 2021 Mathematics - Class 11 JEEM21Mac938 Solution: If , the number of solutions of the given equation when is…

If 3(cos⁡2x)=(3−1)cos⁡x+1\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1 , the number of solutions of the given equation when x∈[0,π2]x \in \left[ {0,{\pi \over 2}} \right] is __________.

Step-by-Step Verified Solution

3(cos⁡2x)=(3−1)cos⁡x+1\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1

⇒\Rightarrow 3cos⁡2x−3cos⁡x+cos⁡x−1=0\sqrt 3 {\cos ^2}x - \sqrt 3 \cos x + \cos x - 1 = 0

⇒3cos⁡x(cos⁡x−1)+(cos⁡x−1)=0\Rightarrow \sqrt 3 \cos x(\cos x - 1) + (\cos x - 1) = 0

⇒(cos⁡x−1)(3cos⁡x+1)=0\Rightarrow (\cos x - 1)(\sqrt 3 \cos x + 1) = 0

cos⁡x=1\cos x = 1

⇒x=0\Rightarrow x = 0 [asx∈[0,π2][as x \in \left[ {0,{\pi \over 2}} \right] ]

and cos⁡x=−13\cos x = - {1 \over {\sqrt 3 }} (not possible in x∈[0,π2]x \in \left[ {0,{\pi \over 2}} \right] ]

∴\therefore Number of solution = 1

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