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Question

# If (2x)ln2=(3y)ln3, 3lnx=2lny and (x0, y0) is the solution of these equations, then x0 is

A
16
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B
13
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C
12
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D
6
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Solution

## The correct option is D 12Let (2x)ln2=(3y)ln3 ⇒ (i) 3lnx=2lny ⇒ (ii)(i) ⇒(ln2)(ln2x)=(ln3)(ln3y) ⇒(ln2)(lnx+ln2)=(ln3)(ln3+lny) Let lnx=a,lny=b ⇒(ln2)(a+ln2)=(ln3)(ln3+b) ⇒ (iii)(ii)⇒(lnx)(ln3)=(ln2)(lny)Substituting a,b⇒a(ln3)=b(ln2) ⇒ (iv)Substituting the value of b from (iv) in (iii), we geta(ln2)+((ln2)2)=(ln32)+a(ln3)2ln2On simplifying we get,a=ln2((ln3)2−(ln2)2)((ln2)2−(ln3)2)⇒a=−ln2⇒lnx=−ln2⇒x=12

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