Before You Begin

Question #40600

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of ways to select 4 items from a collection comprising 3 identical apples, 3 identical bananas, and 3 identical cherries. Let $I = \int_{1/e}^{e^{N/6}} \frac{|\ln x|}{x^2} dx$ . Which of the following inequalities holds true?

Options

A.
$\frac{N}{12} < I < \frac{N}{8}$
B.
$\frac{N}{8} < I < \frac{N}{6}$
C.
$\frac{N}{6} < I < \frac{N}{4}$
D.
$\frac{N}{4} < I < \frac{N}{3}$

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

Selection from Identical Items|Integration by Parts (LIATE Rule)|Logarithmic Integration Modulus|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of combinatorics (selection from identical items with constraints) and definite integration.

The primary trap lies in the 'Logarithmic Integration Modulus', where students often forget to split the integral at $x=1$ , leading to incorrect sign evaluations and a collapsed integral value.

Furthermore, the 'Inequalities in Definite Integrals' phase forces candidates to rigorously bound $e^2$ to verify the final range, penalizing those who rely on superficial approximations or miscalculate the combinatorial bound $N$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Selection from Identical Items
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free