Before You Begin

Question #61719

Difficulty: 5/10 | 🟢 Free

The Question

Let $M(x) = \begin{bmatrix} \sin^{-1}(x) & \cos^{-1}(x) & \frac{\pi}{2} \\ x & 2x & 3x \\ 2x^2 & 2x & \frac{3x}{2}(2x+1) \end{bmatrix}$ and $A(x) = x M(x)$ . If $F(x) = \det(A(x))$ , which of the following statements is INCORRECT?

Options

A.
$F'\left(\frac{1}{2}\right) = 0$
B.
$\text{adj}\left(M\left(\frac{1}{2}\right)\right)$ is a non-zero matrix
C.
$F''\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{8}$
D.
$F(x)$ attains a local minimum at $x = \frac{1}{2}$

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

Derivative of Implicit Inverse Forms|Differentiation of Determinants|Determinant Scaling Property|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Common Mistakes

This question tests the intersection of determinant scaling properties and the derivative of determinants via the adjugate matrix.

Students often fall into the trap of attempting brute-force row-wise differentiation of the complex inverse trigonometric matrix.

The 'Lethal Collision' occurs because scaling the matrix by $x$ requires applying $\det(xM) = x^3\det(M)$ , and evaluating the derivative at $x=1/2$ relies on recognizing that $M(1/2)$ has a rank of 1.

A rank 1 matrix has a zero adjugate, which instantly collapses the derivative of its determinant to zero (via Jacobi's formula $\frac{d}{dx}\det(M) = \text{tr}(\text{adj}(M)M')$ ), bypassing the need for tedious expansion while simultaneously proving that the adjugate is a zero matrix.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Derivative of Implicit Inverse Forms
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free