Binomial Theorem
Constant Term and Floor Function
nta_pyq_2023_apr
Grade 11
Question:
Let $[t]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3x^2-\dfrac{1}{2x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to _____.
Step-by-Step Solution
Key Concept: $T_{r+1}=\binom{7}{r}(3x^2)^{7-r}\left(-\frac{1}{2x^5}\right)^r=\binom{7}{r}(-1)^r\frac{3^{7-r}}{2^r}x^{14-7r}$. Set $14-7r=0\Rightarrow r=2$.
$\alpha=21\cdot\frac{243}{4}=\frac{5103}{4}$. $[\alpha]=1275$.
Correct Answer: 1275