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Question #41694

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable uniformly distributed over the set $\{3, 7, 11, 15\}$ . Let $\mu$ and $\sigma^2$ denote the mean and variance of $X$ , respectively.

Consider the following linear programming problem: Maximize $Z = \sigma^2 x + \mu y$ subject to the constraints: $x + y \le 34$ $2x + y \le 39$ $5x + 2y \le 88$ $9x + 2y \le 144$ $x \ge 0, y \ge 0$

The maximum value of $Z$ is:

Options

A.
361
B.
369
C.
371
D.
381

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

Corner Point Method Precision|Random Variable Mapping|Variance of Discrete Variable|Equiprobable Outcomes

Why This Question Is Tricky

Common Mistakes

This question cross-pollinates Probability with Linear Programming, a hallmark of JEE Advanced.

The primary trap lies in the 'Corner Point Method Precision'.

The feasible region is tightly constrained such that the boundary contains adjacent integer coordinates (e.g., $(9, 21)$ yielding $Z=369$ ).

If a student relies on a rough graphical sketch instead of algebraically solving for the exact intersection $(10, 19)$ , they will fall for the 369 trap.

Additionally, it tests the fundamental variance formula $\sigma^2 = E[X^2] - (E[X])^2$ ; forgetting to subtract the squared mean leads to an incorrect objective function gradient, completely shifting the optimal vertex.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Corner Point Method Precision
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free