If y = log (√x + 1 / √x) then prove that x (x + 1) 2 y 2 + (x + 1) 2 y 1 = 2

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

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If y = log (root x + 1/root)^2, then prove x (x+1)^2 y2 + (x+1)^2 y1 =

Question 15 - CBSE Class 12 Sample Paper for 2018 Boards - Part 2
Question 15 - CBSE Class 12 Sample Paper for 2018 Boards - Part 3 Question 15 - CBSE Class 12 Sample Paper for 2018 Boards - Part 4 Question 15 - CBSE Class 12 Sample Paper for 2018 Boards - Part 5 Question 15 - CBSE Class 12 Sample Paper for 2018 Boards - Part 6

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Transcript

Question 15 If y = log ( +1/ )^2then prove that x (x + 1)2 y2 + (x + 1)2 y1 = 2 Given y = log ( +1/ )^2 Using log a2 = 2 log a y = 2 log ( +1/ ) y = 2 log (( + 1)/ ) y = 2 log (( + 1)/ ) Using log / = log a log b y = 2 log ( +1) 2 log y = 2 log ( +1) 2 log ( )^(1/2) Using log a2 = 2 log a y = 2 log ( +1) 2 1/2 log y = 2 log ( +1) log Now finding y1 and y2 From (1) y = 2 log ( +1) log Differentiating w.r.t x / = 2 (log ( + 1) )/ (log )/ / = 2 1/( + 1) 1/ / = 2/( + 1) 1/ / = (2 ( +1))/( ( + 1)) / = (( 1))/( ( + 1)) y1 = (( 1))/( ( + 1)) Now, from (2) y1 = (( 1))/( ( + 1)) Differentiating w.r.t x y2 = ((( 1))/ ( + 1) )^ = x (x + 1)2 ( ^2 +2 + 1)/( ^2 ( + 1)^2 ) + (x + 1)2 (( 1))/( ( + 1)) = ( ^2 +2 + 1)/ + (x + 1) (( 1))/ = ( ^2 +2 + 1)/ + (( + 1)( 1))/ = ( ^2 +2 + 1)/ + ( ^2 1)/ = ( ^2 +2 + 1 + ^2 1)/ = 2 / = 2 = RHS Hence proved

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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.