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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Basics of Mathematics

JEE Advanced · Mathematics · Basics of Mathematics

Let x and y are positive real number such that _9 x+ _ 27 y=(7)/(2)...

Let xx and yy are positive real number such that log⁡9x+log⁡27y=72\log _9 x+\log _{27} y=\frac{7}{2} and log⁡27x+log⁡9y=23\log _{27} x+\log _9 y=\frac{2}{3}, then:

Options

  1. Axy=243x y=243
  2. Bxy=729x y=729
  3. Cxy=316\frac{x}{y}=3^{16}
  4. Dxy=317\frac{x}{y}=3^{17}

Correct answer

D. xy=317\frac{x}{y}=3^{17}

Step-by-step solution

Let log⁡3x=A\log _3 x=A \quad and log⁡3y=B\quad \log _3 y=B∴ and A2+B3=72 and ⇒ and 3A+2B=21 and  and A3+B2=23⇒2A+3B=4∴A=log⁡3x=11⇒x=311 and B=log⁡3y=−6⇒y=3−6\begin{array}{llll}\therefore \text{ and } \frac{A}{2}+\frac{B}{3}=\frac{7}{2} \text{ and } \Rightarrow \text{ and } 3 A+2 B=21 \\ \text { and } \text{ and } \frac{A}{3}+\frac{B}{2}=\frac{2}{3}& \Rightarrow & 2 A+3 B=4 \\ \therefore & A=\log _3 x=11 & \Rightarrow & x=3^{11} \\ \text { and }& B=\log _3 y=-6 & \Rightarrow & y=3^{-6}\end{array}$$
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