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Question #41279

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \min(\sin 2x, x^2+x)$ . Let $a = \lim_{x \to 0^+} \frac{f(x)}{x}$ and $b = \lim_{x \to 0^-} \frac{f(x)}{x}$ . Consider the system of linear equations: $x + ay + z = 1$ $x + 2y + bz = 2$ $2x + 3y + 3z = \lambda$ Let $S$ be the set of all real values of $\lambda$ for which the system has infinitely many solutions. Find the sum of the elements of $S$ .

Options

A.
3
B.
2
C.
1
D.
0

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

Cramer's Rule Constraints|Indeterminate Forms|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question bridges Calculus and Linear Algebra by using one-sided limits of a non-differentiable minimum function to define the coefficients of a linear system.

The core trap lies in the evaluation of $f(x) = \min(\sin 2x, x^2+x)$ near $x=0$ .

Students often incorrectly assume differentiability or blindly apply L'Hopital's rule without checking the one-sided dominance of the functions, leading to incorrect coefficients (e.g., $a=2, b=2$ or $a=1, b=1$ ).

This incorrect evaluation forces the system's determinant to be non-zero, resulting in a unique solution for all $\lambda$ and an empty set $S$ , trapping the student into answering 0.

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