Before You Begin

Question #40624

Difficulty: 8/10 | 🟢 Free

The Question

Let $n = \lim_{k \to \infty} \sum_{r=1}^{2k} \frac{5}{k} \left[ \frac{5r}{k} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ . Find the term independent of $x$ in the binomial expansion of $\left( x \sqrt{x} - \frac{1}{x} \right)^n$ .

Options

A.
$-\binom{45}{18}$
B.
$\binom{50}{20}$
C.
$\binom{10}{4}$
D.
$\binom{45}{18}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Term Independent of x|Integral as the Limit of a Sum|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's rigor in converting the limit of a Riemann sum into a definite integral, specifically when the summand contains a Greatest Integer Function (GIF).

The primary trap lies in ignoring the step-function discontinuities and treating the argument as continuous, which incorrectly yields $n = \int_0^2 25x \, dx = 50$ , leading directly to the trap option $\binom{50}{20}$ .

A secondary trap involves missing the $2k$ upper limit of the summation, incorrectly integrating from 0 to 1 to get $n = 10$ .

Finally, the student must carefully track the negative sign in the binomial expansion, as the odd value of $r=27$ results in a negative coefficient, distinguishing the correct answer from its positive counterpart.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Term Independent of x
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free