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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 .

If x = a (cos 2θ + 2θ sin 2θ) and y = a (sin 2θ − 2θ cos 2θ),  find

Question 28 (OR 2nd Question) - CBSE Class 12 Sample Paper for 2020 Boards - Part 2
Question 28 (OR 2nd Question) - CBSE Class 12 Sample Paper for 2020 Boards - Part 3 Question 28 (OR 2nd Question) - CBSE Class 12 Sample Paper for 2020 Boards - Part 4

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

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 sin⁡2𝜃))/(𝑎(4𝜃 cos⁡2𝜃)) 𝑑𝑦/𝑑𝑥=sin⁡2𝜃/cos⁡2𝜃 𝑑𝑦/𝑑𝑥=tan⁡2𝜃 Now, differentiation again with respect to x (𝑑^2 𝑦)/(𝑑𝑥^2 )=(𝑑(tan⁡2𝜃))/𝑑𝑥 = (𝑑(tan⁡2𝜃))/𝑑𝑥×𝑑𝜃/𝑑𝜃 = (𝑑(tan⁡2𝜃))/𝑑𝜃×𝑑𝜃/𝑑𝑥 = 2 sec2 2θ ×𝑑𝜃/𝑑𝑥 = 2 sec2 2θ ×1/(𝑑𝑥/𝑑𝜃) = 2 sec2 2θ ×1/(𝑎(4𝜃 cos⁡2𝜃)) 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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.