A) \[{{2}^{20}}(\cos 2x-{{2}^{20}}\cos 4x)\]
B) \[{{2}^{20}}(\cos 2x+{{2}^{20}}\cos 4x)\]
C) \[{{2}^{20}}(\sin 2x+{{2}^{20}}\sin 4x)\]
D) \[{{2}^{20}}(\sin 2x-{{2}^{20}}\sin 4x)\]
Correct Answer: B
Solution :
\[y=2\cos x\cos 3x=\cos 4x+\cos 2x\] \[\frac{dy}{dx}=-4\sin 4x+(-2)\sin 2x\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{(-4)}^{2}}\cos 4x+{{(-2)}^{2}}\cos 2x\] Similarly, \[\frac{{{d}^{20}}y}{d{{x}^{20}}}={{(-4)}^{20}}\cos 4x+{{(-2)}^{20}}\cos 2x\] \[={{4}^{20}}\cos 4x+{{2}^{20}}\cos 2x\] \[={{2}^{20}}[{{2}^{20}}\cos 4x+\cos 2x]\]
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