QJEEM19Maf86d: JEE Main 2019 Mathematics - Class 11 JEEM19Maf86d Solution: If the fourth term in the binomial expansion of (x > 0) is 20 × …

Question

Verified Solution • 2025–26 Session moderate

If the fourth term in the binomial expansion of

{\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}

(x > 0) is 20 × 8⁷, then a value of x is :

Accepted Answer

{\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}

Given T₄ = 20 × 8⁷

\Rightarrow

{}^6{C_3}{\left( {{2 \over x}} \right)^3}{\left( {{x^{{{\log }_8}x}}} \right)^3}

= 20 × 8⁷

\Rightarrow

20

\times

{8 \over {{x^3}}} \times {x^{3{{\log }_8}x}}

= 20 × 8⁷

\Rightarrow

{x^{3{{\log }_8}x - 3}}

= 8⁶

Taking

{{{\log }_8}}

both side

\Rightarrow

(

{3{{\log }_8}x - 3}

)

\times

{{{\log }_8}x}

= 6

\Rightarrow

3{\left( {{{\log }_8}x} \right)^2}

- 3

{{{\log }_8}x}

= 6

\Rightarrow

{\left( {{{\log }_8}x} \right)^2}

-

{{{\log }_8}x}

= 2

\Rightarrow

(

{{{\log }_8}x}

- 2)(

{{{\log }_8}x}

+ 1) = 0

\Rightarrow

{{{\log }_8}x}

= 2 or

{{{\log }_8}x}

= -1

\Rightarrow

x = 8² or x =

{1 \over 8}

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