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Question #40633

Difficulty: 8/10 | 🟢 Free

The Question

Let $M = \begin{pmatrix} \binom{2}{0} & \binom{4}{2} \\ \binom{3}{3}-1 & \binom{5}{5} \end{pmatrix}$ and $G_n$ be the greatest binomial coefficient in the expansion of $(1+x)^n$ for a positive integer $n$ . If the matrix $A = G_n M^2 - 5 \text{adj}(2M)$ is singular, then the value of $\sum_{r=0}^n \binom{n}{r}^2$ is

Options

A.
252
B.
924
C.
70
D.
3432

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Greatest Binomial Coefficient|Determinant Scaling Property|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the determinant scaling property and adjoint properties specifically for $2 \times 2$ matrices, combined with binomial coefficients.

The primary trap is the reflexive use of the $3 \times 3$ adjoint scaling rule $\text{adj}(kM) = k^2 \text{adj}(M)$ , which is heavily ingrained in JEE aspirants.

Students who incorrectly apply this will compute $\text{adj}(2M) = 4 \text{adj}(M)$ , leading to $G_n = 20$ and $n=6$ , which yields the trap answer 924.

The problem forces careful dimension checking before applying matrix properties.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Greatest Binomial Coefficient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free