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Question #41214

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \\begin{pmatrix} 2 & 2x & 2x^2 \\\\ 2 & 2x-2 & 2x^2-4x+2 \\\\ 2 & 2[x] & 2[x]^2 \\end{pmatrix}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $f(x) = \\sqrt{\\det(\\text{adj}(A(x)))}$ . Define the sets $S_1$ and $S_2$ in the interval $(0, 10)$ as:\n $S_1 = \\{x \\in (0, 10) \\mid f(x) \\text{ is discontinuous}\\}\$ \n $S_2 = \\{x \\in (0, 10) \\mid f(x) \\text{ is non-differentiable}\\}\$ \nIf $M$ is the maximum value of $f(x)$ on $\\mathbb{R}$ , find the value of $10M + n(S_1) + n(S_2)$ , where $n(S)$ denotes the number of elements in set $S$ .

Options

A.
29
B.
38
C.
14
D.
20

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

Adjoint-Determinant Relation|Factor Theorem in Determinants|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the synergy between the Adjoint-Determinant relation and the calculus of step functions.

A common trap is assuming that the presence of the discontinuous greatest integer function $[x]$ strictly implies the discontinuity of the determinant.

Students often incorrectly conclude $n(S_1) = 9$ , leading to the trap answer 38.

Furthermore, evaluating the determinant requires recognizing the Vandermonde structure and correctly factoring out $2^3$ from the $3 \\times 3$ matrix, where failing to cube the scalar yields the trap answer 14.

Finally, the absolute value from $\\sqrt{|A|^2}$ creates sharp corners at integers, making it continuous but non-differentiable.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Adjoint-Determinant Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free