Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

If \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of the determinant \(\left|\begin{array}{ccc}{[e]} & {[\pi]} & {\left[\pi^{2}-6\right]} \\ {[\pi]} & {\left[\pi^{2}-6\right]} & {[e]} \\\ {\left[\pi^{2}-6\right]} & {[e]} & {[\pi]}\end{array}\right|\), then (A) \(-8\) (B) 8 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The value of the determinant is -8, so the correct answer is (A).

Step 1: Understanding the Floor Function

First, let's understand what the notation \([x]\) means. This notation represents the greatest integer less than or equal to \(x\). It's also called the floor of \(x\). We'll apply this for \(e\), \(\pi\), and \(\pi^2 - 6\) to simplify the matrix.

Step 2: Calculate Values Using the Floor Function

Calculate \[ [e] = 2, [\pi] = 3, [\pi^2 - 6] = [9.869 - 6] = 3.\] These values were derived because \(e\) is approximately 2.718, \(\pi\) is approximately 3.141, and \(\pi^2\) is approximately 9.869.

Step 3: Substitute Values into the Determinant

Substitute the calculated values into the determinant matrix: \(\left|\begin{array}{ccc} 2 & 3 & 3 \ 3 & 3 & 2 \ 3 & 2 & 3 \end{array}\right|\).

Step 4: Calculate the Determinant

Use the formula for a 3x3 determinant: \(a(ei−fh)−b(di−fg)+c(dh−eg)\). Apply it to get: \[ 2(3 \times 3 - 2 \times 2) - 3(3 \times 3 - 3 \times 2) + 3(3 \times 2 - 3 \times 3) \]= \[ 2(9 - 4) - 3(9 - 6) + 3(6 - 9) \]= \[ 2(5) - 3(3) - 3(3) \]= \[ 10 - 9 - 9 = -8 \].

Step 5: Determine the Correct Answer

Based on the determinant calculation, the value is \(-8\). Therefore, the correct answer from the options is (A) \-8\.

Key Concepts

Greatest Integer Function

The Greatest Integer Function is a mathematical function that assigns to any real number the greatest integer less than or equal to it. This function is denoted as \[ x \], where \( x \) is any real number. This function is crucial in various mathematical applications, especially in simplifying expressions by reducing them to integer values.

For example:

• If \( x = 2.9 \), then \[ x \] = 2.
• If \( x = 3.5 \), then \[ x \] = 3.
• If \( x = -1.2 \), then \[ x \] = -2.

In our exercise, we are using this function to simplify the matrix elements involving \( e \), \( \pi \), and \( \pi^2 - 6 \). The greatest integer function gives us integer values that can be easily worked with in determinant calculations, thus making the solution process much more straightforward.

Floor Function

The Floor Function is closely related to the Greatest Integer Function, as it essentially performs the same operation. It maps a real number to the largest preceding integer. Notated as \( \lfloor x \rfloor \), this function is ubiquitous in mathematics due to its simplicity and practical use in integer approximations.

For mathematicians, mathematicians use the floor function to:

• Approximate results from continuous to discrete.
• Simplify expressions when exact integers are required.

For example, understanding the floor function helps us understand the simplifications made in the given exercise:

• \( \lfloor e \rfloor = 2 \), since \( e \approx 2.718 \).
• \( \lfloor \pi \rfloor = 3 \), since \( \pi \approx 3.141 \).
• \( \lfloor \pi^2 - 6 \rfloor = 3 \), calculated from \( \pi^2 \approx 9.869 \).

These approximations allow operations within matrices and other integer-based calculations to be conducted with precision.

Matrix Calculation

Matrices are arrays of numbers, expressions, or symbols arranged in rows and columns. They are powerful tools in linear algebra for solving systems of equations and other mathematical operations.

The determination of a matrix, especially the determinant of a 3x3 matrix, follows a specific formula and series of steps:

• Identify the elements in the matrix.
• Apply the determinant formula: \( a(ei−fh)−b(di−fg)+c(dh−eg) \).

For our specific matrix beginning with elements \( 2 \), \( 3 \), \( 3 \), ... the calculation leads to:

• Calculate each term separately.
• Multiply and sum the components carefully.
• Derive the result: in this case, \[ -8 \].

These calculations transform abstract numerical arrays into meaningful solutions, providing insight and demonstrating significant topics in algebraic theory.

Drichlet Theorem

Dirichlet's Theorem is an important mathematical concept in number theory. While it may seem unrelated to the immediate problem of finding a matrix determinant, understanding such theorems broadens a student's perspective on how different areas of mathematics interrelate.

Dirichlet's Theorem typically relates to sequences and primes, stating that in any arithmetic progression with integers and a starting point \( a \) and different integer steps \( d \), there are infinitely many prime numbers if \( a \) and \( d \) are coprime.

This context illustrates how foundational principles, like Dirichlet's Theorem, supply the necessary background and exploration paths mathematicians can follow. Although not used directly, these ideas scaffold deeper solutions and guide us toward generalized understanding.