Problem 97 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a \neq b \neq c\), solve for \(x\left|\begin{array}{ccc}0 & x-a & x-b \\\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0\).

Step-by-Step Solution

Answer

The solution to this exercise depends on the exact values of \(a\), \(b\), and \(c\). Solving for \(x\) in the final equation \(3x^3 + (a^2 - 4b)x - 2abx + b^2x = 0\) depends on the exact values of \(a\), \(b\), and \(c\).

Step 1: Calculate the Determinant of the Matrix

Starting with the given 3x3 matrix and its determinant formula: \[|A| = aei + bfg + cdh - ceg - bdi - afh\] where \(a, b, c\) are the elements of the first row, \(d, e, f\) are the elements of the second row, and \(g, h, i\) are the elements of the third row, we can substitute the given values into the determinant formula to get: \[0*(0- (x+c)^2) - (x-a)*(0 - (x+b)^2) + (x-b)*(2x*2x - x^2) = 0\].

Step 2: Simplify the Equation

After the substitution and simplifying, the equation should look like this: \[-(x-a)(x+b)^2 + (x-b)(2x)^2 = 0\]. This simplifies further to: \[- (x^3 - a^2x + 2abx - b^2x) + 4x^2(x - b) = 0\]. This simplifies further to: \[-x^3 + a^2x - 2abx + b^2x + 4x^3 -4bx^2 = 0\]. After combining like terms, the final equation should look like: \[3x^3 + (a^2 - 4b)x - 2abx + b^2x = 0\].

Step 3: Solve for x

To solve this cubic equation for \(x\), use methods such as factoring, using the rational root theorem, using synthetic division or using the cubic formula.

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical method used to solve systems of linear equations with as many equations as unknowns. It applies to systems where the matrix of coefficients is square and assumes the determinant of the matrix is non-zero. While Cramer's Rule isn't directly used in the problem provided, understanding it helps build foundational knowledge on why determinants are crucial in solving linear systems. Cramer's Rule utilizes the determinant of a matrix to find the values of variables, where each variable is replaced in the system one by one with the constant term's determinant.

• For example, in a system with variables x, y, and z, you would solve for each variable separately by replacing its column in the matrix with the constants to find their unique determinant.
• The solution is computed by the ratio of these new determinants to the original determinant of the matrix.

This procedure is neat - when it works - but gets cumbersome with larger systems due to the need for determinant calculations. In this exercise, understanding the determinant is the key, which is why grasping Cramer's Rule's reliance on the determinant becomes beneficial.

Matrix algebra

Matrix algebra is a vital branch of mathematics dealing with the study of matrices and their operations. Matrices are arrays or grids used to solve systems of linear equations and represent linear transformations. Here, the components are neatly organized into rows and columns, and we can perform mathematical operations such as addition, subtraction, and multiplication. One particularly significant operation is finding the determinant, which is essential for determining the invertibility of a matrix.

• To calculate a determinant, specific formulas are applied, such as the one for a 3x3 matrix involving products and sums of its elements in a determinant formula.
• The determinant also indicates unique properties like matrix invertibility; if it's zero, the matrix has no inverse.

In the given exercise, the determinant calculation is crucial for determining the relationship between the matrix components and the solutions to the equation. Matrix algebra provides a structured way to approach complex problems like this, underscoring its importance in various mathematical fields.

Cubic equations

Cubic equations are polynomials of degree three and take the general form \(ax^3 + bx^2 + cx + d = 0\).Solving these equations is an integral part of algebra, offering insights into more complex mathematical behavior than linear equations.These equations can have up to three real roots, and there are several methods for solving them.

• One common technique is to factor the equation, breaking it down into simpler parts that are easier to solve.
• You might also use the Rational Root Theorem, which helps in finding potential rational solutions.
• Another method is synthetic division, a shortcut that works well with polynomial division.

In the original problem, after simplifying the matrix determinant, we end up with a cubic equation.Understanding the possible approaches to solve cubic equations ensures efficient problem-solving and a deeper comprehension of polynomial behavior and its solutions.Exploration of cubic equations sets the stage for even more advanced studies in calculus and beyond.Mastery of these techniques can simplify the apparent complexity of higher-order polynomials.