Q57 / 160 · TS EAMCET 2018 (05 May Shift 2) · Mathematics
If the coefficients of $(2 \alpha+4)$ th and $(\alpha-2)$ th terms in the expansion $(1+x)^{2018}$ are equal, then $\alpha=$
- Correct answerA673
- B674
- C675
- D676
Solution
General term in the expansion of $(1+x)^{2019}$ is
$$
\mathrm{T}_{p+1}={ }^{2019} \mathrm{C}_p \chi^p
$$
$\therefore$ Coefficient of $(2 \alpha+4)$ th term $={ }^{2019} C_{2 \alpha+3}$
and coefficient of $(\alpha-2)$ th term $={ }^{2019} C_{\alpha-3}$
According to given condition, we have
$$
\begin{aligned}
& { }^{2019} C_{2 \alpha+3}={ }^{2019} C_{\alpha-3} \\
& \Rightarrow 2 \alpha+3=\alpha-3 \text { or } 2 \alpha+3+\alpha-3=2019 \\
& {\left[{ }^4 \cdot{ }^n C_x={ }^n C_y=x=y \text { or } x+y=n\right]} \\
& \Rightarrow \quad \alpha=-6 \text { or } 3 \alpha=2019 \Rightarrow \alpha=673
\end{aligned}
$$
$[\because \alpha$ can not be negative $]$
Hence, the correct answer is option: 'A'.