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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2023 · Mathematics · Binomial Theorem

If the sixth term in the binomial expansion of (x^ -(8)/(3) +x^2 _...

If the sixth term in the binomial expansion of (x−83+x2log⁡10x)8\left(x^{-\frac{8}{3}}+x^2 \log _{10} x\right)^8 is 5600 , then what is the value of xx ?

Options

  1. A66
  2. B88
  3. C99
  4. D1010

Correct answer

D. 1010

Step-by-step solution

In the expansion of (x−8/3+x2log⁡10x)8T6=8C5(x−8/3)3⋅(x2log⁡10x)5⇒5600=56⋅x−8⋅x10⋅log⁡10x⇒56⋅x2⋅log⁡10x=5600⇒56⋅x2⋅log⁡10x=5600∴x2log⁡10x=100=(10)2⋅log⁡1010∴x=10\quad\left(x^{-8 / 3}+x^2 \log _{10} x\right)^8T_6={ }^8 C_5\left(x^{-8 / 3}\right)^3 \cdot\left(x^2 \log _{10} x\right)^5\Rightarrow 5600=56 \cdot x^{-8} \cdot x^{10} \cdot \log _{10} x\Rightarrow 56 \cdot x^2 \cdot \log _{10} x=5600\Rightarrow 56 \cdot x^2 \cdot \log _{10} x=5600\therefore \quad x^2 \log _{10} x=100=(10)^2 \cdot \log _{10} 10\therefore \quad x=10.
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