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JEE MAIN 2024
1-2-2024 S2
Question
Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} X^{\frac{1}{3}}+\frac{1}{2 X^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is :
Select the correct option:
A
$\frac{4}{9}$
B
$\frac{1}{9}$
C
$\frac{1}{4}$
D
$\frac{9}{4}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$\begin{aligned} & \left(\frac{x^{\frac{1}{3}}}{3}+\frac{x^{\frac{-2}{3}}}{)^{18}}\right. \\ & \mathrm{t}_7={ }^{18} \mathrm{c}_6\left(\frac{\mathrm{x}^{\frac{1}{3}}}{3}\right)^{12}\left(\frac{\mathrm{x}^{\frac{-2}{3}}}{2}\right)^6={ }^{18} \mathrm{c}_6 \frac{1}{(3)^{12}} \cdot \frac{1}{2^6} \\ & \mathrm{t}_{13}={ }^{18} \mathrm{c}_{12}\left(\frac{\mathrm{x}^{\frac{1}{3}}}{3}\right)^6\left(\frac{\mathrm{x}^{\frac{-2}{3}}}{2}\right)^{12}={ }^{18} \mathrm{c}_{12} \frac{1}{(3)^6} \cdot \frac{1}{2^{12}} \cdot \mathrm{x}^{-6} \\ & \mathrm{~m}={ }^{18} \mathrm{c}_6 \cdot 3^{-12} \cdot 2^{-6}: \mathrm{n}={ }^{18} \mathrm{c}_{12} \cdot 2^{-12} \cdot 3^{-6} \\ & \left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}=\left(\frac{2^{-12} \cdot 3^{-6}}{3^{-12} \cdot 2^{-6}}\right)^{\frac{1}{3}}=\left(\frac{3}{2}\right)^2=\frac{9}{4}\end{aligned}$
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