Let $[t]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7}$ is $\alpha$, then $[\alpha]$ is equal to ___________.
Answer (integer)
1275
Solution
Let $\mathrm{T}_{r+1}$ be the constant term.
<br/><br/>$$
\mathrm{T}_{r+1}={ }^7 \mathrm{C}_r\left(3 x^2\right)^{7-r}\left(\frac{-1}{2 x^5}\right)^r
$$
<br/><br/>For constant term, power of $x$ should be zero.
<br/><br/>$$
\begin{aligned}
& \text { i.e., } 14-2 r-5 r=0 \\\\
& \Rightarrow 14=7 r \Rightarrow r=2
\end{aligned}
$$
<br/><br/>Now, constant term $=\alpha$
<br/><br/>$$
\begin{aligned}
& \Rightarrow{ }^7 C_2(3)^5\left(\frac{-1}{2}\right)^2=\alpha \\\\
& \Rightarrow 21 \times 243 \times \frac{1}{4}=\alpha \\\\
& \Rightarrow[\alpha]=[1275.75]=1275
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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