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Question #40439

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be the set of all positive integers which are multiples of $2$ or $3$ . Consider the function $S(\theta) = \sum_{n \in A} (-1)^n \sin^n \theta$ for $\theta \in \left( \frac{\pi}{2}, \pi \right)$ . If $I = \int_{2\pi/3}^{5\pi/6} \sqrt{ S(\theta) + \frac{\sin^3 \theta}{1-\sin^6 \theta} } \, d\theta$ then the value of $e^I$ is

Options

A.
$\sqrt{3}$
B.
$\frac{1}{\sqrt{3}}$
C.
$3$
D.
$\frac{1}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Conditional Identity Constraints|Binomial Theorem for Negative/Fractional Indices|Sign Flipping Rule|Inclusion-Exclusion Principle

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Inclusion-Exclusion Principle within an infinite geometric series expansion, coupled with a classic quadrant-based sign flipping trap.

Students must first correctly apply IEP to separate the alternating series into convergent sub-series $S_2 + S_3 - S_6$ .

The critical trap occurs when simplifying $\sqrt{\tan^2 \theta}$ .

Students frequently assume $\sqrt{x^2} = x$ and write $\tan \theta$ , ignoring that in the second quadrant ( $\theta \in [2\pi/3, 5\pi/6]$ ), $\tan \theta$ is negative.

This omission leads to integrating $\tan \theta$ instead of $-\tan \theta$ , resulting in the trap answer $1/\sqrt{3}$ instead of the correct $\sqrt{3}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Conditional Identity Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free