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Question #41251

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(n)$ and $g(n)$ denote the exponent of the highest power of $2$ and $3$ dividing $n!$ , respectively. Let $S$ be the set of all integers $n \in [3, 15]$ for which the matrix $M_n$ is singular, where $M_n = \begin{pmatrix} \frac{1}{n-5} & \frac{1}{n-5} & \frac{1}{n-5} \\ \frac{f(n-2)}{n-7} & \frac{f(n-1)}{n-7} & \frac{f(n)}{n-7} \\ g(n-2) & g(n-1) & g(n) \end{pmatrix}$ Find the sum of all elements in $S$ .

Options

A.
64
B.
69
C.
71
D.
76

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Determinant Scaling Property|Exponent of Prime in n!|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the 'Denominator Restriction' trap within the context of determinant scaling and Legendre's formula.

Students typically apply column operations to simplify the determinant and find the roots of the resulting equation.

However, by factoring out $(n-5)$ and $(n-7)$ from the rows, they often forget that the original matrix entries must remain defined.

Consequently, they include the extraneous roots $n=5$ and $n=7$ in their solution set, leading to incorrect sums like 76, 71, or 69.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Determinant Scaling Property
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free