Before You Begin

Question #41788

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all positive real numbers $\alpha$ such that the numerically greatest term in the binomial expansion of $(3 - 2\alpha)^{15}$ is the $7^{\text{th}}$ term. Consider the Linear Programming Problem (LPP) for a fixed $\alpha \in S$ :

Maximize $Z = 5x + 7y$ subject to the constraints: $x + y \le \frac{\alpha}{5 \cdot 3^{13}} \sum_{k=1}^{15} k \binom{15}{k} 2^{k-1}$ $x \ge 5$ $y \ge 6$

Which of the following statements is true regarding the optimal solution of the LPP?

Options

A.
The maximum value of $Z$ is $67$
B.
The maximum value of $Z$ is $72$
C.
The LPP has an unbounded solution
D.
The LPP has no feasible solution

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

Inconsistency Detection|Binomial Coefficient Summation Properties|Numerically Greatest Term

Why This Question Is Tricky

Common Mistakes

This question integrates the concept of the Numerically Greatest Term (NGT) in binomial expansions with Linear Programming feasibility.

The primary trap lies in the student's assumption that a complex-looking LPP will naturally have a bounded optimal solution at one of its vertices.

By tying the LPP's upper bound constraint to the domain of $\alpha$ derived from the NGT condition, the system is deliberately engineered to be inconsistent.

Students who blindly apply vertex-solving algorithms without analyzing the structural bounds of $\alpha$ will fail to recognize the empty feasible region.

Quick Facts About This Question

Question Information

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