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Question #41263

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The Question

For any integer $m \ge 1$ , let $A_{m,1}, A_{m,2}, \dots, A_{m,m}$ be $m$ arithmetic means inserted between $-1$ and $2m+1$ . Let $S_m = \sum_{k=1}^m A_{m,k}$ and $f_m(r) = \sum_{k=1}^m A_{m,k} r^{k-1}$ . Let $d_m$ be the common difference of the arithmetic progression formed by inserting $m$ arithmetic means between $0$ and $m+1$ . Define the matrix $M_n(r) = \begin{pmatrix} f_n(r) & f_{n+1}(r) & f_{n+2}(r) \\ S_n & S_{n+1} & S_{n+2} \\ d_n & d_{n+1} & d_{n+2} \end{pmatrix}$ If $\lim_{r \to 1} \frac{\det(M_n(r))}{1-r} = 399$ , then the value of $n$ is:

Options

A.
8
B.
9
C.
10
D.
11

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

Singular vs. Non-Singular Threshold|Insertion of Means|Arithmetic-Geometric Progression (AGP)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle the singular threshold of a matrix whose elements are defined by Arithmetic-Geometric Progressions (AGP).

The primary trap lies in attempting to evaluate the AGP sum $f_m(r)$ using standard summation formulas before taking the limit $r \to 1$ , which leads to a $0/0$ indeterminate form and complex derivatives if L'Hopital's rule is applied directly to the determinant.

By recognizing that the matrix elements are generated by mean-insertion identities, students are guided to use determinant column operations ( $C_3 \to C_3 - C_2$ , $C_2 \to C_2 - C_1$ ) first.

This elegantly collapses the AGP series into single terms, factoring out $(1-r)$ naturally and bypassing the singularity trap entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Singular vs. Non-Singular Threshold
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free