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Question #40447

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be the real-valued continuous function defined on the largest possible interval $I$ containing $\frac{\pi}{4}$ such that $f\left(\frac{\pi}{4}\right) = 0$ and $f'(x) = \frac{\sin 2x}{\cos x \cos 3x}$ . Let $S$ be the set of all $x \in I$ satisfying $f(x) = \frac{1}{3} \ln 2$ . Find the value of $\sum_{x \in S} \sec^2 x$ .

Options

A.
$\frac{5}{3}$
B.
$\frac{8}{3}$
C.
1
D.
$\frac{10}{3}$

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

Extraneous Solution Rejection|Logarithmic Integration Modulus|Factor-Product Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle logarithmic integration of trigonometric functions and the subsequent extraneous solutions generated by exponentiating log-modulus expressions.

The student must convert a product into a sum ( $\tan 3x - \tan x$ ) to integrate, which necessitates absolute values to evaluate the constant of integration at $x = \pi/4$ .

When solving the resulting equation, dropping the absolute value or failing to verify the roots against the domain $I = (\pi/6, \pi/2)$ leads to the 'ghost' solution $\cos^2 x = 1$ .

The trap specifically targets students who mechanically apply algebraic operations without respecting the domain restrictions imposed by the singularities of the original integrand.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Extraneous Solution Rejection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free