if x = a (cos 2θ + 2θ sin 2θ) and y = a (sin 2θ − 2θ cos 2θ),
find (d ^{ 2 } y)/(dx ^{ 2 } ) at θ = π/8 .
Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
CBSE Class 12 Sample Paper for 2020 Boards
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Question 14 (OR 1st Question)
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Question 28 (OR 1st Question) Important
Question 28 (OR 2nd Question) You are here
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Question 31 (OR 1st Question) Important
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Question 35 (OR 1st Question) Important
Question 35 (OR 2nd Question) Important
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CBSE Class 12 Sample Paper for 2020 Boards
Last updated at May 29, 2023 by Teachoo
Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Question 28 (OR 2nd Question) if x = a (cos 2𝜃 + 2𝜃 sin 2𝜃) and y = a (sin 2𝜃 − 2𝜃 cos 2𝜃), find (𝑑^2 𝑦)/(𝑑𝑥^2 ) at 𝜃 = 𝜋/8 . Now, x = a (cos 2𝜃 + 2𝜃 sin 2𝜃) 𝑑𝑥/𝑑𝜃 = a (–2 sin 2𝜃 + 2 sin 2𝜃 + 2𝜃 × 2 cos 2𝜃) 𝑑𝑥/𝑑𝜃 = a (4𝜃 cos 2𝜃) And, y = a (sin 2𝜃 − 2𝜃 cos 2𝜃) 𝑑𝑦/𝑑𝜃 = a (2 cos 2𝜃 – 2 cos 2𝜃 + 2𝜃 × 2sin 2𝜃) 𝑑𝑦/𝑑𝜃 = a (4𝜃 sin 2𝜃) Therefore, Dividing (2) and (1) 𝑑𝑦/𝑑𝑥=(𝑎(4𝜃 sin2𝜃))/(𝑎(4𝜃 cos2𝜃)) 𝑑𝑦/𝑑𝑥=sin2𝜃/cos2𝜃 𝑑𝑦/𝑑𝑥=tan2𝜃 Now, differentiation again with respect to x (𝑑^2 𝑦)/(𝑑𝑥^2 )=(𝑑(tan2𝜃))/𝑑𝑥 = (𝑑(tan2𝜃))/𝑑𝑥×𝑑𝜃/𝑑𝜃 = (𝑑(tan2𝜃))/𝑑𝜃×𝑑𝜃/𝑑𝑥 = 2 sec2 2θ ×𝑑𝜃/𝑑𝑥 = 2 sec2 2θ ×1/(𝑑𝑥/𝑑𝜃) = 2 sec2 2θ ×1/(𝑎(4𝜃 cos2𝜃)) We need to find value of (𝑑^2 𝑦)/(𝑑𝑥^2 ) at 𝜃 = 𝜋/8 . Putting 𝜃 = 𝜋/8 in the equation (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 2 sec2 (2×𝜋/8) ×1/(𝑎(4 × 𝜋/8 × cos〖(2×𝜋/8)〗)) = 2 sec2 (𝜋/4) ×1/(𝑎(𝜋/2 × 〖cos 〗〖𝜋/4〗)) = 2 × 1/cos^2(𝜋/4) ×2/(𝑎𝜋 × 〖cos 〗〖𝜋/4〗 ) = 2 × 1/(1/√2)^2 ×2/(𝑎𝜋 × (1/√2) ) = 2 × 1/((1/2) ) ×2/(𝑎𝜋 × (1/√2) ) = 2 × 2 ×(2√2)/𝑎𝜋 = (𝟖√𝟐)/𝒂𝝅