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Question #41123

Difficulty: 8/10 | 🟢 Free

The Question

Let $U_n = \{z \in \mathbb{C} : z^n = 1\}$ . For each integer $n \ge 3$ , define the mapping $f_n: U_n \to U_n$ by $f_n(z) = z^5 \exp\left( \frac{1}{2} (\text{Log}(z) + \text{Log}(\bar{z})) \right)$ where $\text{Log}(z)$ denotes the principal branch of the complex logarithm, with its imaginary part in $(-\pi, \pi]$ . Let $S$ be the set of all integers $n \in \{10, 11, \dots, 20\}$ such that $f_n$ is a bijective mapping. Find the sum of the elements of $S$ .

Options

A.
60
B.
120
C.
75
D.
135

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

Surjective Mapping (Onto)|Roots of Unity Properties|Argument and Principal Value

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the principal branch cut of the complex logarithm and its effect on mapping properties.

A common mistake is assuming $\text{Log}(\bar{z}) = -\text{Log}(z)$ for all $z$ , which ignores the discontinuity at the negative real axis where $\text{Log}(-1) = i\pi$ .

Students falling for this trap will incorrectly conclude $f_n(z) = z^5$ for all $z$ , leading them to include even values of $n$ (like 12, 14, 16, 18) where $\gcd(5, n) = 1$ , yielding the trap answer 120.

Recognizing the branch cut reveals a 'lethal collision' $f_n(1) = f_n(-1) = 1$ , destroying bijectivity for all even $n$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Surjective Mapping (Onto)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free