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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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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 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.