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Question #40619

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $5 \times 5$ matrix where $a_{ij}$ is the coefficient of $x^2$ in the binomial expansion of $(1+x)^{i+j+2}$ for $1 \le i, j \le 5$ . If $B = [b_{ij}]$ is the adjoint matrix of $A$ , then the value of $\sum_{i=1}^5 \sum_{j=1}^5 (b_{ij}^2 + 2^{b_{ij}})$ is equal to

Options

A.
0
B.
25
C.
50
D.
125

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

General Term Formulation|Adjoint-Determinant Relation|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

This question tests the 'Singular vs.

Non-Singular' threshold using binomial coefficients.

Students often recognize that the determinant is zero but incorrectly assume the adjoint matrix has rank 1 (which happens when rank(A) = n-1).

The trap lies in the polynomial nature of the binomial expansion $\binom{i+j+2}{2}$ , which restricts the rank of the $5 \times 5$ matrix to at most 3.

Because the rank is less than or equal to $n-2$ , all $4 \times 4$ cofactors vanish, making the adjoint matrix exactly the zero matrix.

Failing to identify this forces students into impossible cofactor calculations.

Quick Facts About This Question

Question Information

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