Let Jn,m = ∫(0 to 1/2) (xⁿ / (xᵐ - 1)) dx, ∀n m and n, m ∈ N. Consider a matrix A = [aij]3×3 where

Q: Is it really necessary to simplify the integral before solving it?

Absolutely! In JEE problems, complexity is often a mask for simplicity. By factoring the numerator, you transform a daunting integral into a simple power rule problem, saving precious time and reducing the chance of error.

Q: How do I know for sure that $A$ is upper triangular?

The definition $a_{ij} = 0$ for $i > j$ is the formal definition of an upper triangular matrix. Recognizing this structure is a key skill, as it allows you to compute the determinant simply by multiplying the diagonal elements.

Q: Is there a shortcut for $|\text{adj } A^{-1}|$?

Yes! Instead of calculating the adjoint and then the inverse, use the property $|\text{adj } M| = |M|^{n-1}$ and $|M^{-1}| = 1/|M|$. This turns a matrix-level problem into a simple scalar calculation.

Question

Let $J_{n,m} = \int_{0}^{1/2} \frac{x^n}{x^m-1} dx, \forall n > m$ and $n, m \in \mathbb{N}$. Consider a matrix $A = [a_{ij}]_{3 	imes 3}$ where $a_{ij} = \begin{cases} J_{6+i,3} - J_{i+3,3}, & i \le j \ 0, & i > j \end{cases}$. Then $|	ext{adj } A^{-1}|$ is :

Options:

(A) $(15)^2 	imes 2^{42}$

(B) $(15)^2 	imes 2^{34}$

(C) $(105)^2 	imes 2^{38}$

(D) $(105)^2 	imes 2^{36}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given the integral definition $J_{n,m} = \int_{0}^{1/2} \frac{x^n}{x^m-1} dx$ and a matrix $A$ whose elements are defined by $a_{ij} = J_{6+i,3} - J_{i+3,3}$. Our objective is to determine the value of $|	ext{adj } A^{-1}|$.

### The Art of Simplification

When evaluating $a_{ij} = \int_0^{1/2} \frac{x^{6+i}}{x^3-1} dx - \int_0^{1/2} \frac{x^{i+3}}{x^3-1} dx$, we combine the integrands due to the identical limits of integration. This yields:

$$a_{ij} = \int_0^{1/2} \frac{x^{6+i} - x^{i+3}}{x^3-1} dx$$

By factoring the numerator, we observe that $x^{6+i} - x^{i+3} = x^{i+3}(x^3 - 1)$. This allows the denominator to cancel out entirely, simplifying the expression to:

$$a_{ij} = \int_0^{1/2} x^{i+3} dx$$

Applying the power rule, we find the general term for the matrix elements:

$$a_{ij} = \left[ \frac{x^{i+4}}{i+4} ight]_0^{1/2} = \frac{1}{(i+4)2^{i+4}}$$

### The Matrix Structure

The problem specifies that $a_{ij} = 0$ for $i > j$, confirming that $A$ is an **upper triangular matrix**. The determinant of an upper triangular matrix is the product of its diagonal elements, $|A| = a_{11} \cdot a_{22} \cdot a_{33}$.

Calculating the diagonal elements for $i=1, 2, 3$:

$$a_{11} = \frac{1}{5 \cdot 2^5}, \quad a_{22} = \frac{1}{6 \cdot 2^6}, \quad a_{33} = \frac{1}{7 \cdot 2^7}$$

Multiplying these values, we obtain the determinant of $A$:

$$|A| = \frac{1}{(5 \cdot 6 \cdot 7) \cdot 2^{5+6+7}} = \frac{1}{210 \cdot 2^{18}}$$

### Final Calculation

We seek the value of $|	ext{adj } A^{-1}|$. Using the property $|	ext{adj } M| = |M|^{n-1}$ for an $n 	imes n$ matrix, and noting that $n=3$, we have:

$$|	ext{adj } A^{-1}| = |A^{-1}|^2 = \left( \frac{1}{|A|} ight)^2 = \frac{1}{|A|^2}$$

Substituting our calculated value for $|A|$:

$$\frac{1}{|A|^2} = (210 \cdot 2^{18})^2 = 210^2 \cdot 2^{36}$$

Since $210 = 105 \cdot 2$, we can simplify the expression as $(105 \cdot 2)^2 \cdot 2^{36} = 105^2 \cdot 2^2 \cdot 2^{36}$. The final result is:

**$105^2 \cdot 2^{38}$**

Select Answer:

Visualized Solution

The Integral and the Matrix

Defining $a_{ij}$

Combining Integrals

Algebraic Simplification

Definite Integration

Upper Triangular Matrix

Computing $a_{ii}$

Product of Diagonal Elements

Matrix Properties

Finding the Final Value

The Sigma Insight: Adjoint and Inverse of a Matrix

Analyzing the Setup

The Art of Simplification

The Matrix Structure

Final Calculation

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