If x = a sec πœƒ, y = b tan πœƒ, then (d 2 y)/(dx 2 ) at θ = π/6 is :

(a) (-3√3 b)/a 2  (b) (-2√3 b)/a
(c) (-3√3 b)/a  (d) (-b)/(3√3 a 2 )

This question is inspired from Question 31 (Choice 2) - CBSE Class 12 Sample Paper for 2021 Boards

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Transcript

Question 22 If x = a sec πœƒ, y = b tan πœƒ, then (𝑑^2 𝑦)/(𝑑π‘₯^2 ) at πœƒ = πœ‹/6 is : (a) (βˆ’3√3 𝑏)/π‘Ž^2 (b) (βˆ’2√3 𝑏)/π‘Ž (c) (βˆ’3√3 𝑏)/π‘Ž (d) (βˆ’π‘)/(3√3 π‘Ž^2 ) Given π‘₯=π‘Ž sec β‘πœƒ, 𝑦=𝑏 tanβ‘πœƒ 𝑑𝑦/𝑑π‘₯ = 𝑑𝑦/𝑑π‘₯ Γ— π‘‘πœƒ/π‘‘πœƒ 𝑑𝑦/𝑑π‘₯ = 𝑑𝑦/π‘‘πœƒ Γ— π‘‘πœƒ/𝑑π‘₯ π’…π’š/𝒅𝒙 = (π’…π’š/π’…πœ½)/(𝒅𝒙/π’…πœ½) Calculating π’…π’š/π’…πœ½ 𝑦 = 𝑏 tanβ‘πœƒ 𝑑𝑦/π‘‘πœƒ = 𝑑(𝑏 tanβ‘πœƒ )/π‘‘πœƒ 𝑑𝑦/π‘‘πœƒ = 𝑏 𝑑(tanβ‘πœƒ )/π‘‘πœƒ π’…π’š/π’…πœ½ = 𝑏 .sec^2β‘πœƒ Calculating 𝒅𝒙/π’…πœ½ π‘₯=π‘Ž sec β‘πœƒ 𝑑π‘₯/π‘‘πœƒ = 𝑑(π‘Ž sec β‘πœƒ)/π‘‘πœƒ 𝑑π‘₯/π‘‘πœƒ = π‘Ž 𝑑(sec β‘πœƒ)/π‘‘πœƒ 𝒅𝒙/π’…πœ½ = π‘Ž (secβ‘πœƒ.tanβ‘πœƒ ) 𝑑𝑦/𝑑π‘₯ = (𝑏 secβ‘πœƒ)/(π‘Ž tanβ‘πœƒ ) 𝑑𝑦/𝑑π‘₯ = (𝑏 . 1/cosβ‘πœƒ )/(π‘Ž (sinβ‘πœƒ/cosβ‘πœƒ ) ) 𝑑𝑦/𝑑π‘₯ = 𝑏 Γ— 1/cosβ‘πœƒ Γ— cosβ‘πœƒ/γ€–a sinγ€—β‘πœƒ 𝑑𝑦/𝑑π‘₯ = 𝑏/π‘Ž (1/sinβ‘πœƒ ) π’…π’š/𝒅𝒙 = 𝒃/𝒂 𝒄𝒐𝒔𝒆𝒄 𝜽 Now, (𝒅^𝟐 π’š)/(𝒅𝒙^𝟐 ) = 𝒃/𝒂 (𝐝(𝒄𝒐𝒔𝒆𝒄 𝜽))/𝒅𝒙 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑏/π‘Ž (d(π‘π‘œπ‘ π‘’π‘ πœƒ))/𝑑θ ×𝑑θ/𝑑π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑏/π‘Ž (d(π‘π‘œπ‘ π‘’π‘ πœƒ))/𝑑θ Γ—πŸ/(𝒅𝒙/π’…πœ½) (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑏/π‘Ž (βˆ’π‘π‘œπ‘ π‘’π‘ ΞΈ cot⁑θ) Γ—πŸ/(𝒂 𝒔𝒆𝒄 ΞΈ 𝒕𝒂𝒏 ΞΈ) (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (βˆ’π‘)/π‘Ž^2 1/sin⁑〖θ γ€— cot⁑θ Γ— 𝒄𝒐𝒔 ΞΈ 𝒄𝒐𝒕 ΞΈ (𝒅^𝟐 π’š)/(𝒅𝒙^𝟐 ) = (βˆ’π’ƒ)/𝒂^𝟐 〖𝒄𝒐𝒕〗^πŸ‘β‘πœ½ We need to find (𝑑^2 𝑦)/(𝑑π‘₯^2 ) π‘Žπ‘‘ π‘₯ = πœ‹/6 Putting π‘₯ = 𝝅/πŸ” (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (βˆ’π‘)/π‘Ž^2 γ€–π‘π‘œπ‘‘γ€—^3β‘γ€–πœ‹/6γ€— (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (βˆ’π‘)/π‘Ž^2 (√3)^3 (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = (βˆ’π‘)/π‘Ž^2 3√3 (𝒅^𝟐 π’š)/(𝒅𝒙^𝟐 ) = (βˆ’πŸ‘βˆšπŸ‘ 𝒃)/𝒂^𝟐 So, the correct answer is (A)

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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.