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Question #41191

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all complex numbers $z = x + iy$ where $x, y \in \mathbb{Z}$ , satisfying the following two conditions:

(i) $\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$

(ii) $\cos^{-1}\left(\frac{x^2+y^2 - 5}{5}\right) \cdot (x^2+y^2 - 8) \le 0$

Find the number of elements in $S$ .

Options

A.
17
B.
21
C.
25
D.
33

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

The Sum-Difference Formula Constraints|Locus Problems via Transformation|The Zero-Factor Cancellation Trap|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the student's rigor in handling inverse trigonometric identities and inequalities.

The primary trap is a 'Zero-Factor Cancellation' hazard.

In Condition (ii), students often divide by $\cos^{-1}\left(\frac{x^2+y^2-5}{5}\right)$ assuming it is strictly positive, thereby solving only $x^2+y^2 \le 8$ and losing the discrete solution set $x^2+y^2 = 10$ where the inverse cosine evaluates to zero.

Additionally, Condition (i) requires knowledge of the sum formula constraint $xy < 1$ , which restricts the locus to specific quadrants.

Failing to account for the zero-factor hides valid points in the Wavy Curve solution, leading to the trap answer 17.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Sum-Difference Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free