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Question #66444

Difficulty: 5/10 | 🟢 Free

The Question

Let the system of linear equations $x + y + z = 3$ $2x + 3y + z = 6$ $3x + 4y + \alpha z = \beta$ have infinitely many solutions. Let $S$ be the set of all solutions $(x, y, z)$ of this system that also satisfy the inequalities $x \ge 0$ , $y \ge 0$ , and $z \ge 0$ . Let $M$ be the maximum value of the objective function $Z = 2x + 4y + 5z$ on $S$ , $L$ be the square of the length of the line segment represented by $S$ , and $N$ be the number of points in $S$ where all coordinates are integers. Which of the following is NOT a correct value?

Options

A.
$\alpha + \beta = 11$
B.
$M = \frac{27}{2}$
C.
$L = \frac{27}{2}$
D.
N = 3

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

System of Linear Equations (Cramer/Inverse)|Multiple Optimal Solutions|System of Inequalities

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Common Mistakes

This question tests the intersection of linear algebra (Cramer's Rule/consistency) with linear programming (feasible regions).

The primary trap lies in students mechanically finding the condition for infinite solutions (determinant equals zero) but failing to parameterize the dependent system to find the actual geometric locus of the solutions.

By introducing inequality constraints, the problem forces a cognitive shift from unbounded algebraic solutions to a bounded geometric line segment.

Students who do not correctly intersect the parameterized line with the first octant will incorrectly evaluate the bounds for the objective function or miscount the integer coordinate points, specifically falling for the trap of assuming unboundedness or misidentifying the segment's endpoints.

The negative phrasing ('NOT a correct value') further penalizes partial analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	System of Linear Equations (Cramer/Inverse)
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free