Verified Solution • 2025–26 Session moderate

QJEEM23Maec02: JEE Main 2023 Mathematics - Class 11 JEEM23Maec02 Solution: Let denote the greatest integer . If the constant term in the…

Let [t][t] denote the greatest integer ≤t\leq t . If the constant term in the expansion of (3x2−12x5)7\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7} is α\alpha , then [α][\alpha] is equal to ___________.

Step-by-Step Verified Solution

Let Tr+1\mathrm{T}_{r+1} be the constant term.

Tr+1=7Cr(3x2)7−r(−12x5)r\mathrm{T}_{r+1}={ }^7 \mathrm{C}_r\left(3 x^2\right)^{7-r}\left(\frac{-1}{2 x^5}\right)^r

For constant term, power of xx should be zero.

amp; i.e., 14−2r−5r=0amp;⇒14=7r⇒r=2\begin{aligned}& \text { i.e., } 14-2 r-5 r=0 \\\\& \Rightarrow 14=7 r \Rightarrow r=2\end{aligned}

Now, constant term =α=\alpha

amp;⇒7C2(3)5(−12)2=αamp;⇒21×243×14=αamp;⇒[α]=[1275.75]=1275\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}

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