Before You Begin

Question #40485

Difficulty: 8/10 | 🟢 Free

The Question

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$ . A point $w = u + iv$ in the complex plane is defined by $u = \arccos x$ and $v = \arccos y$ . Let $S$ be the locus of $w$ as $x$ and $y$ vary, and let $A$ be the area of the region enclosed by $S$ . Let $k$ be the number of points on $S$ where the implicit derivative $\frac{dv}{du}$ , obtained directly from the equation $\cos^2 u + \cos^2 v = 1$ , yields the indeterminate form $\frac{0}{0}$ . Find the value of $\frac{k A}{\pi^2}$ .

Options

A.
1
B.
2
C.
4
D.
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

Locus Problems via Transformation|Algebraic-Trigonometric Graphical Mapping|Implicit Differentiation Implicit Domain

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of inverse trigonometric transformations and implicit differentiation.

The 'Lethal Collision' occurs when students apply implicit differentiation to the algebraic relation $\cos^2 u + \cos^2 v = 1$ to find the slope of the locus.

A common trap is assuming that setting the numerator to zero yields horizontal tangents.

However, due to the domain restrictions of the principal branches of $\arccos$ , the boundary points force both the numerator and denominator to zero simultaneously, yielding a $0/0$ indeterminate form.

This masks the fact that these points are actually non-differentiable corners of the square locus, forcing students to distinguish between blind algebraic manipulation and domain-valid geometric reality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Locus Problems via Transformation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free