NIMCET 2008 Question Paper with solutions

Question : 14 of 120

Marks: +1, -0

For

a>0, a ≠ 1,

the number of values of

x

satisfying the equation

2 {\log}_{x}(a)+\log _{a x}(a)+3 \log _{x_{2}}(a)=0

Solution:

Given equation,

2 \log _{x}(a)+\log _{i x}(a)+3 \log _{a^{2} x}(a)=0

⇒ \log a[\;{2}/{\log x}+\;{1}/{\log a+\log x}+\;{3}/{2 \log a+\log x}]=0

Let

\;\; \log x=y

and

\log a=c

⇒ \;\; \;{2}/{Y}+\;{1}/{Y+c}+\;{3}/{y+2 c}=0

⇒ \;\; 2(y+c)(y+2 c)+y(y+2 c)+3 y(y+c)=0

⇒ \;\; 6 y^{2}+11 c y+4 c^{2}=0

⇒ \;\; y=\;{-11 c ± √{121 c^{2}-96 c^{2}}}/{12}

=\;{-11 c ± 5 c}/{12}=\;{-4 c}/{3},-\;{c}/{2}

⇒ \;\; \log x=\;{-4}/{3} \log a

and

\log x=-\;{1}/{2} \log a

⇒ \;\; x=a^{-4 \/ 3} ; a^{-1{/} 2}

i.e., two solutions are there.

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