(iii) x(10910x)2-31 (log10x)4-31091...
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(iii) x(10910x)2-31 (log10x)4-310910 X+1 > 1000

JEE/Engineering Exams
Maths
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[ text { Given } x^{left(log _{10} xright)^{2}-3 log _{14} x+1}>100 ] Consider ( log _{10} x=a quad ) Then, ( quad x=10^{a} ) ( left(10^{2} Rightarrowleft(10^{2}right)^{left(a^{2}-3 a+1right)}>10^{3}right. ) ( Rightarrow quad 10^{a^{3}-3 a^{2}+a} quad>10^{3} ) ( Rightarrow quad a^{3}-3 a^{2}+a quad>3 quad ) since bases are same) ( Rightarrow quad a^{3}-3 a^{2}+a-3 quad>0 ) ( Rightarrow quad a^{2}(a-3)+1(a-3)>0 ) ( Rightarrow quadleft(a^{2}+1right)(a-3) quad>0 ) ( a^{2}+1>0 ) and ( a-3>0 ) ( a^{2}>-1 ) and ( a>3 ) Since ( a in R quad, quad a^{2} quad ) is always positive for all real values Jhus, ( quad a>3 ) When ( a=3, x=10^{3}=1000 ) Jhus when ( a>3, quad x>1000 ) Jtence, ( quad[x in(1000, infty)] ) For any queries ( +operatorname{sen} 2 k 1 ) andicom Don't frel hesitated to ask doubt thorough this mail : id.
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