A) \[-3\sqrt{1-{{y}^{2}}}\]
B) \[9y\]
C) \[-9y\]
D) \[3\sqrt{1-{{y}^{2}}}\]
Correct Answer: C
Solution :
Given, \[y={{\cos }^{2}}\frac{3x}{2}-{{\sin }^{2}}\frac{3x}{2}\] \[\Rightarrow \] \[y={{\cos }^{2}}\frac{3x}{2}-\left( 1-{{\cos }^{2}}\frac{3x}{2} \right)\] \[\Rightarrow \] \[y=2{{\cos }^{2}}\frac{3x}{2}-1\] \[\Rightarrow \] \[y=2{{\cos }^{2}}\frac{3x}{2}-1\] \[\Rightarrow \] \[\frac{dy}{dx}=2-2\cos \frac{3x}{2}\left( -\sin \frac{3x}{2} \right)\left( \frac{3}{2} \right)\] \[\Rightarrow \] \[\frac{dy}{dx}=-6\cos \frac{3x}{2}\sin \frac{3x}{2}\] \[\Rightarrow \] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-6\left[ \cos \frac{3x}{2}\left( \cos \,\frac{3x}{2} \right).\frac{3}{2}-\sin \frac{3x}{2} \right.\] \[\left. \sin \,\frac{3x}{2}.\frac{3}{2} \right]\] \[\Rightarrow \] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-9\left[ {{\cos }^{2}}\frac{3x}{2}-{{\sin }^{2}}\frac{3x}{2} \right]\] \[\Rightarrow \] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-9y\]
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