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Question #40920

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x)$ be a $3 \times 3$ matrix whose $(i,j)^{\text{th}}$ entry is given by $m_{ij} = \sin(ijx)$ for $1 \le i, j \le 3$ . Let $P_x(\lambda) = \lambda^3 - c_1(x)\lambda^2 + c_2(x)\lambda - c_3(x) = 0$ be the characteristic equation of $M(x)$ . Evaluate the limit: $\lim_{x \to 0} \frac{c_3(x)}{x^9}$

Options

A.
$-720$
B.
$720$
C.
$-120$
D.
$0$

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

Definition of Limit|Characteristic Equation|Determinant Scaling Property|Linear Dependency of Rows

Why This Question Is Tricky

Common Mistakes

The trap is set when students encounter a 0/0 limit involving a determinant and attempt raw expansion or blind application of L'Hôpital's Rule.

Because the determinant vanishes at x=0 due to linear dependency of rows (all rows approach 0), L'Hôpital's requires 9 successive derivatives, which is an insurmountable algebraic pitfall.

The question tests the student's ability to recognize that the matrix elements can be decoupled using Taylor series and matrix multiplication (or multilinearity).

By factoring the matrix into a product of a constant Vandermonde-derived matrix and a scaled transpose, the determinant scaling property elegantly extracts the x^9 factor, bypassing the trap entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Definition of Limit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free