Before You Begin

Question #41510

Difficulty: 8/10 | 🟢 Free

The Question

Let $\hat{v} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector in $\mathbb{R}^3$ such that $n = 0$ and $m \ge 0$ . Let $\theta$ be the angle made by $\hat{v}$ with the positive $x$ -axis. Let $S$ be the set of all possible values of $\theta$ satisfying the inequality $\frac{m - l}{m + l} \ge 0$ . Define the integral $I$ as: $I = \int_{S} \frac{\cos^2(2\theta)}{1 + \left(\frac{62}{333}\right)^l} d\theta$ Then the value of $I$ is:

Options

A.
$\frac{\pi}{8}$
B.
$\frac{\pi}{4}$
C.
$\frac{\pi}{16}$
D.
$\frac{3\pi}{8}$

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

Definite Integral Properties (King's Property)|Axis of Integration Selection|Cross Multiplication Trap|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests the 'Axis of Integration Selection' by requiring a shift from 3D direction cosines to a 2D polar angle $\theta$ .

The core trap is the 'Cross Multiplication Trap' in solving the rational trigonometric inequality $\frac{\tan \theta - 1}{\tan \theta + 1} \ge 0$ .

Students often erroneously cross-multiply to get $\tan \theta \ge 1$ , yielding the incomplete set $[\pi/4, \pi/2)$ , missing the interval $(\pi/2, 3\pi/4)$ where both numerator and denominator are negative.

This mistake leads to incorrect bounds and an analytically unsolvable integral.

The correct symmetric bounds allow elegant evaluation via King's Property, while the constant 62/333 acts as a psychological distractor typical of JEE Advanced.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Definite Integral Properties (King's Property)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free