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Question #41266

Difficulty: 8/10 | 🟢 Free

The Question

Consider the system of linear equations: $x + y + z = \lambda$ $x + 2y + 3z = \mu$ $14x + 22y + 30z = \lambda^2 + \mu^2 - 11$ If the system has infinitely many solutions, then the maximum value of $3\lambda + 4\mu$ is:

Options

A.
45
B.
55
C.
65
D.
75

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

Linear Dependency of Rows|System of Linear Equations (Cramer/Inverse)|Redundant Constraints

Why This Question Is Tricky

Common Mistakes

This question tests the 'Redundant Constraints' trap within a system of linear equations.

Students typically default to Cramer's Rule, calculating the determinants $\Delta_x, \Delta_y, \Delta_z$ and setting them to zero.

Given the large coefficients in the third row, this algebraic approach leads to severe time loss and a high probability of calculation errors (algebraic fatigue).

The intended JEE Advanced pivot is to bypass determinants entirely by identifying the linear dependency of the rows ( $R_3 = 6R_1 + 8R_2$ ) and applying this exact scalar relationship to the constant column, instantly reducing the problem to a standard coordinate geometry optimization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Linear Dependency of Rows
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free