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I.S.I. and C.M.I. Entrance

Real numbers problem – C.M.I – U.G – 2019

Understand the problem

 .Find all real numbers x for which
$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$  

Source of the problem

C.M.I (Chennai mathematical institute ) U.G-2019

Topic

Real Numbers

Difficulty Level
8 out of 10

Start with hints

Do you really need a hint? Try it first!

It is of the the form of  $\frac{a^3+b^3}{a^2b+b^2a}$ . Do you observe ?  where a=\(2^x\) b=\(3^x\)

\(\frac{a^3+b^3+a^2b+b^2a}{a^2b+b^2a}\)=\(\frac{13}{6}\) \(\Rightarrow\frac{a^2+b^2}{ab}=\frac{13}{6}\) \(\frac{a}{b}+\frac{b}{a}=\frac{13}{6}\) let x=a/b then it is a quadratic equation    

\(6x^2-13x+6=0\) on solving we get x=3/2 or 2/3 now replace x by a/b

so again  putting value of a and b we get final ans is +1 and -1 

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

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