If x= ∫ 0^y dt/√(1 + 9t^2) and (d^2 y) / (dx^2 ) = ay, then a is equal to

(A) 3Β 

(B) 6Β 

(C) 9Β 

(D) 1

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  1. Chapter 7 Class 12 Integrals (Term 2)
  2. Serial order wise

Transcript

Question 6 If π‘₯= ∫1_0^𝑦▒〖 𝑑𝑑/√(1 + 9𝑑^2 )γ€— and (𝑑^2 𝑦)/(𝑑π‘₯^2 )=π‘Žπ‘¦, then π‘Ž is equal to 3 (B) 6 (C) 9 (D) 1 Since π‘₯= ∫1_0^𝑦▒〖 𝑑𝑑/√(1 + 9𝑑^2 )γ€— Changing variable from t to y π‘₯= ∫1_0^𝑦▒〖 𝑑𝑦/√(1 + 9𝑦^2 )γ€— It means x is a function of y So, we can write 𝒅𝒙/π’…π’š=𝟏/√(𝟏 + πŸ—π’š^𝟐 ) And, ππ’š/𝐝𝐱=√(1 + 9𝑦^2 ) Differentiating again w.r.t x (𝐝^𝟐 π’š)/(𝐝𝐱^𝟐 )=𝟏/(𝟐√(1 + 9𝑦^2 )) ×𝒅(πŸ—π’š^𝟐 )/𝒅𝒙 (d^2 𝑦)/(dx^2 )=1/(2√(1 + 9𝑦^2 )) Γ— 9 Γ— 2𝑦 Γ—π’…π’š/𝒅𝒙 Putting value of 𝑑𝑦/𝑑π‘₯ (𝐝^𝟐 π’š)/(𝐝𝐱^𝟐 )=𝟏/(𝟐√(1 + 9𝑦^2 )) Γ— πŸ— Γ— πŸπ’š Γ—βˆš(1 + 9𝑦^2 ) (𝐝^𝟐 π’š)/(𝐝𝐱^𝟐 )=πŸ—π’š Thus, a = 9 So, the correct answer is (c)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.