Before You Begin

Question #41777

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the region in the complex plane defined by the system of inequalities $\text{Re}\left(z e^{-i \frac{(2k-1)\pi}{12}}\right) \le \cos\frac{\pi}{12}$ for $k=1, 2, \dots, 12$ . Let $f(z) = \text{Im}((1+i)z)$ . If $M$ is the maximum value of $f(z)$ for $z \in R$ , and $L$ is the length of the line segment representing the set of all points in $R$ where this maximum is attained, then the value of $M \cdot L$ is:

Options

A.
$\frac{1}{\sqrt{2}}$
B.
$\frac{\sqrt{3}}{2}$
C.
1
D.
$\frac{\sqrt{3}-1}{2}$

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

Multiple Optimal Solutions|De Moivre’s Theorem Applications|Geometry of Complex Inequalities

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of complex inequalities as a feasible region in a Linear Programming Problem (LPP).

The primary trap lies in the objective function $f(z) = x+y$ .

Students often incorrectly assume the maximum occurs at $z = e^{i\pi/4}$ on the unit circle, failing to realize this point lies outside the dodecagon.

Furthermore, the problem tests the advanced LPP concept of multiple optimal solutions; the gradient of the objective function is parallel to the boundary edge connecting $e^{i\pi/6}$ and $e^{i\pi/3}$ , causing the optimal solution space to be a line segment rather than a single vertex.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Multiple Optimal Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free