Problem 40 - A Complete Resource Book in Mathematics for JEE Main

Question

If the three-digit numbers \(A 28,3 B 9\) and \(62 \mathrm{C}\), where \(A\), \(B\) and \(C\) are integers between 0 and 9 , are divisible by a fixed integer \(k\), then the determinant \(\left|\begin{array}{ccc}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|\) is (A) divisible by \(k\) (B) divisible by \(k^{2}\) (C) divisible by \(2 k\) (D) None of these

Step-by-Step Solution

Answer

The determinant is divisible by \(k\). (Answer A)

Step 1: Conditions for Divisibility

First, we need to establish the conditions for divisibility of the numbers \( A28 \), \( 3B9 \), and \( 62C \) by a fixed integer \( k \). This means each of these numbers can be expressed as: \( A28 = kx \), \( 3B9 = ky \), and \( 62C = kz \), where \( x \), \( y \), and \( z \) are integers.

Step 2: Set Up the Determinant

The determinant given is \( \left| \begin{array}{ccc} A & 3 & 6 \ 8 & 9 & C \ 2 & B & 2 \end{array} \right| \). We will compute this determinant to determine its divisibility by \( k \).

Step 3: Compute the Determinant

Calculate the determinant: \( \left| \begin{array}{ccc} A & 3 & 6 \ 8 & 9 & C \ 2 & B & 2 \end{array} \right| = A \cdot (9 \times 2 - B \times C) - 3 \cdot (8 \times 2 - 2 \times C) + 6 \cdot (8 \times B - 9 \times 2) \).Simplify to get the expression for the determinant.

Step 4: Expand and Simplify

Continue simplifying: \( = A(18 - BC) - 3(16 - 2C) + 6(8B - 18) \).Expand each term: \( = 18A - ABC - 48 + 6C + 48B - 108 \).Combine like terms: \( = 18A + 6C + 48B - ABC - 156 \).

Step 5: Analyze Divisibility by \(k\)

Given the condition of divisibility of the initial numbers by \(k\), the determinant can be reconstructed as linear combinations of multiples of \(k\), but not \(k^2\). Therefore, the determinant is divisible by \(k\), but not necessarily by \(k^2\) or \(2k\).

Step 6: Conclusion

Based on our analysis and calculations, the determinant is divisible by \(k\) because the original numbers were structured to be divisible by \(k\). Therefore, the answer is (A) divisible by \(k\).

Key Concepts

Divisibility in Mathematics

Divisibility is a fundamental concept in mathematics that helps us understand how numbers can be divided evenly by others without leaving a remainder. This concept is crucial when solving problems involving integers and modular arithmetic.
To determine if a number is divisible by another, you can use several methods:

• Check if the division leaves a remainder of zero.
• Apply specific divisibility rules for certain numbers such as 2, 3, 5, etc. For example, a number is divisible by 2 if its last digit is even.
• Express numbers as multiples of another number using variables. In the given exercise, numbers like \( A28 \) are expressed in terms of \( k \).

Understanding divisibility is key in problems like the exercise provided, where numbers are manipulated to maintain a specific divisibility property throughout algebraic operations. By confirming that multiple numbers share a common divisor, it can simplify calculations and validate solutions.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. In the context of the exercise, we use matrices to represent systems and compute their determinants.
Determinants are scalar values that can be calculated from a square matrix. The determinant provides information about the matrix's properties such as:

• If it is invertible (a non-zero determinant indicates a matrix is invertible).
• Whether it changes volume or preserves orientation in transformation operations.
• The solvability of a system of linear equations represented by the matrix.

In the exercise, the determinant of the matrix \(\left| \begin{array}{ccc} A & 3 & 6 \ 8 & 9 & C \ 2 & B & 2 \end{array} \right|\) is calculated to analyze its divisibility properties. The computations involve expanding along one row and combining terms to find if the matrix's determinant is divisible by certain integer values.

JEE Mathematics

JEE Mathematics is known for its challenging questions and is a critical component of India's Joint Entrance Examination main test, which students must pass to enter prestigious engineering institutes.
In the context of JEE preparation, understanding concepts like determinants and divisibility is crucial. To excel in such examinations, students should focus on:

• Grasping the fundamental properties of numbers and matrices.
• Practicing a variety of computational techniques for solving algebraic expressions.
• Developing problem-solving strategies for complex and multi-step problems, as seen in the exercise.

Successful navigation of these mathematical concepts can significantly boost a student's performance in JEE math sections, allowing them to apply their knowledge flexibly and confidently in various scenarios.