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Question #40737

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two rays originating from the origin in the first quadrant, making angles $\theta_1 = \tan^{-1}(2)$ and $\theta_2 = \tan^{-1}(5)$ with the positive $x$ -axis, respectively. A vertical line $x = r$ ( $r > 0$ ) intersects $L_1$ and $L_2$ at points $P$ and $Q$ , respectively. Let $y_1(r)$ and $y_2(r)$ be the ordinates of $P$ and $Q$ . A function $f: (0, \infty) \to \mathbb{R}$ is defined as $f(r) = \int_{y_1(r)}^{y_2(r)} \left( \arcsin(\sin t) + \arccos(\cos t) \right) dt$ Find the value of $f'\left(\frac{3\pi}{5}\right)$ .

Options

A.
$\frac{19\pi}{5}$
B.
$\frac{126\pi}{5}$
C.
$3\pi$
D.
$\frac{2\pi}{5}$

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

Parametric Form Manipulation|Newton-Leibniz Formula with variable limits|Principal Value Branch Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz rule combined with the restricted principal value branches of inverse trigonometric functions.

A common trap is blindly assuming $\arcsin(\sin x) = x$ and $\arccos(\cos x) = x$ for all $x$ .

When the parameter $r = \frac{3\pi}{5}$ , the upper limit evaluates to $3\pi$ and the lower limit to $\frac{6\pi}{5}$ , both of which fall outside the standard principal domains $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $[0, \pi]$ .

Students must correctly reduce these angles using periodic properties to avoid extrinsic solutions like $\frac{126\pi}{5}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Parametric Form Manipulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free