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Question #40560

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of distinct arrangements of 3 identical red balls and 2 identical blue balls in a straight line such that the two blue balls are not adjacent. Let $f(x) = \lim_{t \to 0} \left( \frac{\sin(Ntx)}{Ntx} \right)^{\frac{1}{t^2}}$ . If $g(x) = f(\tan^{-1}(\ln x))$ , then the value of $g'(e)$ is

Options

A.
$-\frac{3\pi}{2e} e^{-\frac{3\pi^2}{8}}$
B.
$-\frac{3\pi}{e} e^{-\frac{3\pi^2}{8}}$
C.
$-\frac{3\pi}{2} e^{-\frac{3\pi^2}{8}}$
D.
$-\frac{\pi}{2e} e^{-\frac{3\pi^2}{8}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Fundamental Principle of Counting (FPC)|Gap Method (Non-adjacency)|Indeterminate Forms|Chain Rule Complexity

Why This Question Is Tricky

Common Mistakes

This question fuses Discrete Mathematics with Calculus, creating a lethal collision of concepts.

The student must first correctly apply the Gap Method to find the combinatorial constant $N$ .

A common trap is treating identical items as distinct or forgetting the non-adjacency constraint.

This $N$ is then embedded in a $1^\infty$ indeterminate limit, requiring L'Hopital's rule or Taylor expansion.

Finally, the resulting function is subjected to a multi-layered chain rule.

An error in $N$ propagates exponentially into the exponent, and forgetting internal derivatives during the chain rule (such as the derivative of the inverse tangent or the natural logarithm) leads directly to the carefully crafted distractor options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Fundamental Principle of Counting (FPC)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free