Q174 / 270 · NDA 2023 (Phase 1) · Mathematics
If the sixth term in the binomial expansion of
$\left(x^{-\frac{8}{3}}+x^2 \log _{10} x\right)^8$ is 5600 , then what is the value of $x$ ?
- A$6$
- B$8$
- C$9$
- Correct answerD$10$
Solution
In the expansion of $\quad\left(x^{-8 / 3}+x^2 \log _{10} x\right)^8$
$T_6={ }^8 C_5\left(x^{-8 / 3}\right)^3 \cdot\left(x^2 \log _{10} x\right)^5$
$\Rightarrow 5600=56 \cdot x^{-8} \cdot x^{10} \cdot \log _{10} x$
$\Rightarrow 56 \cdot x^2 \cdot \log _{10} x=5600$
$\Rightarrow 56 \cdot x^2 \cdot \log _{10} x=5600$
$\therefore \quad x^2 \log _{10} x=100=(10)^2 \cdot \log _{10} 10$
$\therefore \quad x=10$.
Hence, the correct answer is option: 'D'.