Question 15 - CBSE Sample Paper Class 12 - 2017-18
Last updated at May 29, 2018 by Teachoo

If y = log (√x + 1 / √x)
^{
2
}
then prove that x (x + 1)
^{
2
}
y
_{
2
}
+ (x + 1)
^{
2
}
y
_{
1
}
= 2
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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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