Verified Solution • 2025–26 Session moderate

QJEEM21Ma8624: JEE Main 2021 Mathematics - Class 11 JEEM21Ma8624 Solution: If the fourth term in the expansion of is 4480, then the value…

If the fourth term in the expansion of (x+xlog⁡2x)7{(x + {x^{{{\log }_2}x}})^7} is 4480, then the value of x where x ∈\in N is equal to :

Multiple Choice Options

  • A 3
  • B 1
  • C 4
  • D 2

Step-by-Step Verified Solution

T4 = 7C3x4(xlog⁡2x)3=4480{}^7{C_3}{x^4}{({x^{{{\log }_2}x}})^3} = 4480

⇒35x4(xlog⁡2x)3=4480\Rightarrow 35{x^4}{({x^{{{\log }_2}x}})^3} = 4480

⇒x4(xlog⁡2x)3=128\Rightarrow {x^4}{({x^{{{\log }_2}x}})^3} = 128

take log w.r.t. base 2 we get,

4log⁡2x+3log⁡2(xlog⁡2x)=log⁡21284{\log _2}x + 3{\log _2}({x^{{{\log }_2}x}}) = {\log _2}128

Let log⁡2x=y{\log _2}x = y

4y+3y2=74y + 3{y^2} = 7

⇒y=1,−73\Rightarrow y = 1,{{ - 7} \over 3}

⇒log⁡2x=1,−73\Rightarrow {\log _2}x = 1,{{ - 7} \over 3}

⇒\Rightarrow x=2,x=2−7/3x = 2,x = {2^{ - 7/3}}

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