Before You Begin

Question #41783

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha$ and $\beta$ ( $\alpha > \beta$ ) be the roots of the quadratic equation $x^2 - 3x + 1 = 0$ . A sequence is defined as $T_n = \alpha^n - \beta^n$ for $n \in \mathbb{N}$ . Consider the linear programming problem: Maximize $Z = T_2 x + T_1 y$ subject to the constraints: $3x + y \le 15$ $x + 2y \le 10$ $x \ge 0, y \ge 0$ If $S$ is the set of all points $(x,y)$ in the feasible region at which $Z$ attains its maximum value, find the value of $\int_S (x+y) \, ds$ , where $ds$ is the arc length element along $S$ .

Options

A.
$6\sqrt{10}$
B.
$5\sqrt{10}$
C.
$15\sqrt{5}$
D.
$3\sqrt{10}$

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|Multiple Optimal Solutions|Quadratic Roots in Sequences

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of sequence evaluation involving irrational roots and the Corner Point Method in Linear Programming.

The primary trap lies in the non-integer coefficients of the objective function ( $3\sqrt{5}$ and $\sqrt{5}$ ), which often causes students to doubt their calculations or fail to recognize that the slope of the objective function exactly matches the active constraint $3x+y \le 15$ .

Consequently, they might only identify a single optimal point instead of the entire line segment (Multiple Optimal Solutions).

The final integration step over the solution set $S$ rigorously verifies if the student has correctly identified $S$ as a continuous segment rather than discrete points, punishing those who merely find the maximum value of $Z$ .

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