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Question #41382

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(t) = \begin{pmatrix} t-3 & 1 & 1 \\ 1 & t-3 & 1 \\ 1 & 1 & t-3 \end{pmatrix}$ and $f(x) = \int_0^x (\det(A(t)))^{1/3} dt$ . If $S$ is the sum of the abscissae of all the points of inflection of the curve $y = f(x)$ , find the value of $S$ .

Options

A.
2
B.
5
C.
6
D.
7

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

Point of Inflection Conditions|Factor Theorem in Determinants|Singular vs Non-Singular Matrices|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous definition of inflection points combined with determinant properties.

The 'Lethal Collision' occurs because students often assume inflection points only exist where $f''(x) = 0$ (yielding only $x=2$ ).

They fail to account for points where the matrix becomes singular, causing $f''(x)$ to be undefined (at $x=1$ and $x=4$ ).

Furthermore, among the undefined points, they must correctly identify that $x=4$ is an inflection point due to a sign change in the fractional power, while $x=1$ is not because the squared term prevents a sign change.

This perfectly filters students who rely on rote differentiation without analyzing the domain and sign changes of the second derivative.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Point of Inflection Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free