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Question #41242

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x) = x^3 - 4x^2 + 3x + 2$ . Let $r_1, r_2, r_3$ be the roots of $P(x)=0$ such that $r_1$ and $r_2$ are irrational and $r_3$ is rational. Consider the system of linear equations in $x, y, z$ : $x + y + z = 1$ $r_1 x + r_2 y + r_3 z = 2$ $r_1^2 x + r_2^2 y + [\lambda] z = 3$ where $[\cdot]$ denotes the greatest integer function. Let $S$ be the set of all real values of $\lambda$ for which the system has no solution. Then $S$ is equal to:

Options

A.
$[4, 5)$
B.
$[5, 6)$
C.
$[6, 7)$
D.
$\emptyset$

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

System of Linear Equations (Cramer/Inverse)|Conjugate Root Theorem|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Conjugate Root Theorem and Cramer's Rule, specifically exploiting the discontinuity of the greatest integer function.

Students often mechanically calculate determinants without recognizing that the coefficients, derived from a polynomial, possess symmetric properties (sum and product of conjugate roots) that drastically simplify the Vandermonde-like determinant.

The trap lies in assuming the system always has a unique solution or failing to verify consistency ( $\Delta_x \neq 0$ ) when the main determinant vanishes.

By linking the determinant to a step function, the problem forces a transition from a unique solution space to an inconsistent one precisely at the integer threshold, a classic JEE Advanced maneuver to test boundary condition analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	System of Linear Equations (Cramer/Inverse)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free