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Question #41690

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_1: x^2 - a x + 8 = 0$ and $E_2: x^2 - b x + 32 = 0$ be two quadratic equations with real coefficients having a positive common root $r$ . Let the non-common roots of $E_1$ and $E_2$ be $\alpha$ and $\beta$ respectively. It is given that $\alpha, r, \beta$ form a geometric progression in that order.

Consider the linear programming problem: Maximize $Z = (a+b)x + (ab)y$ subject to the constraints: $\alpha x + r y \le a$ $r x + \beta y \le b$ $x \ge 0, y \ge 0$

Find the maximum value of $Z$ .

Options

A.
54
B.
72
C.
108
D.
216

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

Redundant Constraints|Condition for Common Roots|Quadratic Roots in Sequences

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the condition for common roots in quadratics with Linear Programming.

The primary trap lies in the constraints: students often blindly attempt to find an intersection point for the two linear inequalities using Cramer's rule or substitution.

However, the coefficients derived from the geometric progression of the roots force the two constraints to be parallel and identical (redundant).

Failing to simplify and recognize this redundancy leads to algebraic confusion or incorrect feasible regions.

The problem strictly requires establishing the discrete parameters first to unlock the LPP.

Quick Facts About This Question

Question Information

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