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7.Binomial Theorem
hard
જો ${\left( {\frac{2}{x} + {x^{{{\log }_e}x}}} \right)^6}(x > 0)$ ના વિસ્તરણમાં ચોથું પદ $20\times 8^7$ હોય તો $x$ ની કિમત મેળવો.
A
$8^3$
B
$8^{-2}$
C
$8$
D
$8^2$
(JEE MAIN-2019)
Solution
${T_4} = {T_{3 + 1}} = \left( {\frac{6}{3}} \right){\left( {\frac{2}{x}} \right)^3} \cdot {\left( {{x^{{{\log }_8}x}}} \right)^3}$
$20 \times 8^{7}=\frac{160}{x^{3}} \cdot x^{3000} x$
$8^{6}=x^{\log _{2} x}-3$
$2^{18}=x^{\log _{2} x-3}$
$\Rightarrow 18=\left(\log _{2} x-3\right)\left(\log _{2} x\right)$
Let $\log _{2} x=t$
$\Rightarrow t^{2}-3 t-18=0$
$\Rightarrow(t-6)(t+3)=0$
$\Rightarrow \mathrm{t}=6,-3$
$\log _{2} x=6$
$ \Rightarrow x=2^{6}=8^{2}$
$\log _{2} x=-3$
$\Rightarrow x=2^{-3}=8^{-1}$
Standard 11
Mathematics
