Problem 88 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} 4 x & 6 x+2 & 8 x+1 \\ 6 x+2 & 9 x+3 & 12 x \\ 8 x+1 & 12 x & 16 x+2 \end{array}\right|=0 $$

Step-by-Step Solution

Answer

The given equation is: \(\det(A) = 0\), where \(A =\begin{bmatrix} 4x & 6x+2 & 8x+1 \\ 6x+2 & 9x+3 & 12x \\ 8x+1 & 12x & 16x+2 \end{bmatrix}\) Computing the determinant, expanding, and simplifying, we obtain: \(-100x^2+67x+5 = 0\) Solving the quadratic equation with the quadratic formula, we find two real solutions: \(x \approx 0.103\) and \(x \approx -0.482\).

Step 1: Compute the determinant of the 3x3 matrix

To compute the determinant of the given 3x3 matrix, we will use the following formula for a 3x3 matrix A with elements a_{ij}: $$\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})$$ Applying this formula for the given matrix, we get: $$\det(A) = 4x[(9x+3)(16x+2) - (12x)(12x)] - (6x+2)[(6x+2)(16x+2) - (8x+1)(12x)] + (8x+1)[(6x+2)(12x) - (9x+3)(8x+1)]$$

Step 2: Expand and simplify the determinant

Expanding and simplifying each term in the determinant, we get: $$\det(A) = 4x[(144x^2+30x+6) - 144x^2] - (6x+2)[(96x^2+28x+4) - (96x^2+14x)] + (8x+1)[(72x^2+24x) - (72x^2+17x+3)]$$ Now, we can combine like terms: $$\det(A) = 4x[30x+6] - (6x+2)(14x+4) + (8x+1)(-17x+3)$$

Step 3: Set the determinant to zero and solve the equation

Now that the determinant is simplified, set it equal to zero and solve for x: $$0 = 4x(30x+6) - (6x+2)(14x+4) + (8x+1)(-17x+3)$$ Expanding the equation and combining like terms: $$0 = 120x^2+24x - 84x^2-32x+12x+8 - 136x^2+40x+17x-3$$ Combining terms: $$0 = -100x^2+67x+5$$ To solve this quadratic equation, we'll find the discriminant, and then if it has real solutions, we'll use the quadratic formula: Discriminant: $$D = b^2 - 4ac = 67^2 - 4(-100)(5) = 4489 + 2000 = 6489$$ Since the discriminant is positive, the equation has two real solutions for x: $$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-67 \pm \sqrt{6489}}{-200}$$ So, the values of x are approximately: $$x \approx 0.103 \text{ and } x \approx -0.482$$

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in mathematics and appear in various fields such as physics, engineering, and economics. A standard quadratic equation is written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The solutions or "roots" of a quadratic equation can be real or complex numbers.

Quadratic equations can be solved using several methods:

• Factoring: If the quadratic can be factored easily, this method can be quick. It involves writing the quadratic as a product of two binomials.
• Completing the square: This method manipulates the equation into a perfect square trinomial, then takes the square root of both sides.
• Quadratic formula: The most universal method, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which always finds the roots if they exist.

In the example problem, after simplifying the matrix determinant, we arrived at a quadratic equation \( -100x^2 + 67x + 5 = 0 \). This equation helps us find the values of \( x \) that nullify the determinant expression.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and operations on them. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The size of a matrix is defined by its number of rows and columns, such as 3x3 for the exercise.

Matrices can be manipulated in various ways:

• Addition and Subtraction: Matrices of the same dimension can be added or subtracted by adding or subtracting corresponding elements.
• Scalar Multiplication: Every element of a matrix can be multiplied by a scalar (a constant number).
• Matrix Multiplication: A more complex operation where elements are multiplied and summed in a specific pattern. It is only possible if the number of columns in the first matrix matches the number of rows in the second.
• Determinant: A special number that can be calculated from a square matrix. It provides insights into the properties of the matrix, such as whether it is invertible.

The primary focus in the exercise was calculating the determinant of a 3x3 matrix, which required a specific formula to solve. Understanding matrix algebra is essential for dealing with more complex mathematical problems and systems.

Discriminant

The discriminant in the context of quadratic equations indicates the nature of the roots. It is a part of the quadratic formula \( D = b^2 - 4ac \). The value of the discriminant determines the types of solutions we can expect:

• Positive Discriminant: If \( D > 0 \), the quadratic equation has two distinct real roots.
• Zero Discriminant: If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
• Negative Discriminant: If \( D < 0 \), the roots are complex and appear as conjugate pairs.

In the provided solution, the discriminant was calculated to be 6489, a positive number, indicating the existence of two real solutions for \( x \). The discriminant is crucial as it helps predict how the solutions manifest even before solving the equation completely.