Before You Begin

Question #67142

Difficulty: 5/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix whose entries are given by $a_{ij} = \text{Coefficient of } x^2 \text{ in the expansion of } (1+x)^{n+i} (1-x)^{j}$ , where $n$ is a fixed positive integer and $i, j \in \{1, 2, 3\}$ . Find the value of $\det(A)$ .

Options

A.
$1$
B.
$-1$
C.
$0$
D.
$n$

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

Minors and Cofactors Sign Convention|Coefficient of x^r in Expansion

Why This Question Is Tricky

Common Mistakes

Students often mishandle the binomial expansion of $(1-x)^j$ when finding the coefficient of $x^2$ in the product $(1+x)^{n+i}(1-x)^j$ .

They either assume the negative sign only affects odd powers (missing the negative cross-term $-(n+i)j x^2$ ) or incorrectly add the indices to get $\binom{n+i+j}{2}$ .

If they use $\binom{n+i+j}{2}$ , the row operations yield a second row of $n+1+j$ (increasing), which flips the sign of the determinant to $-1$ .

If they miss the cross-term entirely, the entries become separable as $f(i)+g(j)$ , leading to a determinant of $0$ .

The question rigorously tests both algebraic precision in binomial products and the properties of difference operators in determinants.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Minors and Cofactors Sign Convention
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free