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Question #41278

Difficulty: 8/10 | 🟢 Free

The Question

Consider the system of linear equations: $x + y + z = 1$ $2x + 3y + [\alpha]z = 2$ $8x + 6y + 3|\alpha|z = \beta$ where $\alpha, \beta \in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$ . Let $A$ be the set of all values of $\alpha$ for which there exists a real number $\beta$ such that the system has infinitely many solutions. If $S = \sum_{\alpha \in A} |\alpha|$ and $\beta_0$ is the unique value of $\beta$ that makes the system consistent for these $\alpha$ , then the value of $3S + \beta_0$ is:

Options

A.
130
B.
52
C.
122
D.
138

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Cramer's Rule Constraints|Line-Plane intersection logic|Greatest Integer Function (GIF)|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze a system of linear equations where the coefficients are non-differentiable, piecewise functions (Greatest Integer Function and Modulus).

The primary trap is assuming the modulus and GIF behave uniformly, leading students to either miss the negative roots entirely or miscalculate the bounds of the GIF inequality.

By forcing $\Delta = 0$ into a piecewise Diophantine-like inequality $n \le f(n) < n+1$ , it rigorously filters out students who rely on purely symbolic determinant expansion without analyzing the domain of $\alpha$ .

The secondary trap lies in the intersection logic: realizing that $\Delta_z = 0$ uniquely determines $\beta_0$ independent of $\alpha$ , preventing a false assumption that $\beta$ varies with each root.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Cramer's Rule Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free