Q95 / 155 · WBJEE 2025 · Mathematics

Let $a_n$ denote the term independent of $x$ in the expansion of $\left[x+\frac{\sin (1 / n)}{x^2}\right]^{3 n}$, then $\lim _{n \rightarrow \infty} \frac{\left(a_n\right) n!}{{ }^{3 n} P_n}$ equals

  • Correct answer
    A
    $0$
  • B
    $1$
  • C
    e
  • D
    $\frac{\mathrm{e}}{\sqrt{3}}$

Solution

$\left[x+\frac{\sin (1 / n)}{x^2}\right]^{3 n}$ $\begin{aligned} & T_{r+1}={ }^{3 n} C_r(x)^{3 n-r}\left(\frac{\sin (1 / n)}{x^2}\right)^r \\ & ={ }^{3 n} C_r(x)^{3 n-3 r}(\sin (1 / n))^r \\ & r=n \end{aligned}$ $\begin{aligned} & a_n={ }^{3 n} C_n(\sin (1 / n))^n \\ & \lim _{n \rightarrow \infty} \frac{{ }^{3 n} C_n(\sin (1 / n))^n \cdot n!}{{ }^{3 n} P_n}=0 \end{aligned}$

Hence, the correct answer is option: 'A'.

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