Question 3 - NCERT Exemplar - MCQs - Chapter 6 Class 12 Application of Derivatives

Last updated at April 8, 2024 by

The tangent to the curve given by x = e t . cost, y = e t . sint at t =

π/4 makes with x-axis an angle:

(A)0 

(B) π/4 

(C) π/3 

(D) π/2

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Transcript

Question 3 The tangent to the curve given by x = et . cost, y = et . sint at t = 𝜋/4 makes with x-axis an angle: 0 (B) 𝜋/4 (C) 𝜋/3 (D) 𝜋/2 Given curve 𝒙=𝒆^𝒕.𝒄𝒐𝒔𝒕 and 𝒚=𝒆^𝒕.𝒔𝒊𝒏𝒕 Finding slope of tangent 𝒅𝒚/𝒅𝒙=(𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) 𝒙=𝒆^𝒕.𝒄𝒐𝒔𝒕 Differentiating 𝑥 𝑤.𝑟.𝑡 𝑡 𝑑𝑥/𝑑𝑡=𝑒^𝑡 𝑐𝑜𝑠𝑡−𝑒^𝑡 𝑠𝑖𝑛𝑡 𝒅𝒙/𝒅𝒕=𝑒^𝑡 (𝑐𝑜𝑠𝑡−𝑠𝑖𝑛𝑡) 𝒚=𝒆^𝒕.𝒔𝒊𝒏𝒕 Differentiating y 𝑤.𝑟.𝑡 𝑑𝑦/𝑑𝑡=𝑒^𝑡 𝑠𝑖𝑛𝑡+𝑒^𝑡 𝑐𝑜𝑠𝑡 𝒅𝒚/𝒅𝒕=𝑒^𝑡 (𝑠𝑖𝑛𝑡\+𝑐𝑜𝑠𝑡) Now, 𝒅𝒚/𝒅𝒙=(𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) 𝑑𝑦/𝑑𝑥=(𝑒^𝑡 (𝑠𝑖𝑛𝑡\+𝑐𝑜𝑠𝑡)" " )/(𝑒^𝑡 (𝑐𝑜𝑠𝑡−𝑠𝑖𝑛𝑡)" " ) 𝑑𝑦/𝑑𝑥=( (𝒔𝒊𝒏𝒕\+𝒄𝒐𝒔𝒕)" " )/( (𝒄𝒐𝒔𝒕−𝒔𝒊𝒏𝒕)" " ) Putting 𝐭=𝛑/𝟒 ├ 𝒅𝒚/𝒅𝒙┤|_(𝒕=𝝅/𝟒)=( (𝑠𝑖𝑛 𝜋/4 \+ 𝑐𝑜𝑠 𝜋/4)" " )/( (𝑐𝑜𝑠 𝜋/4 − 𝑠𝑖𝑛 𝜋/4)" " ) = (1/√2+1/√2)/(1/√2−1/√2) = (2/(√2))/0 = ∞ Let 𝜽 be the angle made by tangent with the 𝑥- axis. Slope = 𝐭𝐚𝐧⁡𝜽 ∞ = tan⁡𝜃 tan⁡〖𝜃=∞〗 ∴ 𝜽= 𝝅/𝟐 So, the correct answer is (D)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.