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CBSE Class 12 Math 2020 Delhi Set 1 Solved Paper

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Question : 23 of 36
Marks: +1, -0
If x=acos⁡θ;y=bsin⁡θx = a \cos \theta ; y = b \sin \theta, then find   d2ydx2\;\frac{d^{2} y}{d x^{2}}.
OR
Find the differential of sin⁡2x\sin^{2} x w.r.t. ecos⁡xe^{\cos x}.
Solution:     👈: Video Solution
OR
Let P=sin⁡xP = \sin x
Q=ecos⁡xQ = e^{\cos x}
On differentiating eq.(1) w.r.t. we get
  dpdx=ddx(sin⁡2x)=2sin⁡xcos⁡x\;\frac{d p}{d x} = \frac{d}{d x} (\sin^{2} x) = 2 \sin x \cos x
On differentiating eq.(2) w.r.t we get
  dθdx=ddx(ecos⁡x)=−ecos⁡xsin⁡x.\;\frac{d\theta}{d x} = \frac{d}{dx} (e^{\cos x}) = - e^{\cos x} \sin x.
dividing eq.(3) and (4)
  dpdxdθdx=2sin⁡xcos⁡x−ecos⁡xsin⁡x=−2cos⁡xecos⁡x\;\frac{ \frac{d p}{d x} }{ \frac{d \theta}{d x} } = \frac{2 \sin x \cos x}{-e^{\cos x} \sin x} = \frac{-2 \cos x}{e^{\cos x}}
[  dpdxdθdx=−2cos⁡xecos⁡x]\left[\;\frac{ \frac{d p}{d x} }{ \frac{d \theta}{d x} } = \frac{-2 \cos x}{e^{\cos x}}\right]
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