Problem 14 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Let \(f(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 2 x \\\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 2 x\end{array}\right|\) then the maximum value of \(f(x)\) is (a) 0 (b) 2 (c) 6 (d) None.

Step-by-Step Solution

Answer

The maximum value of \(f(x)\) is 2.

Step 1: Determine Mathematic Representation of the Matrix

Let's rewrite the determinant of the matrix in clearer mathematical notations as follows: \(f(x)= \left|\begin{array} {ccc} 1+\sin^2(x) & \cos^2(x) & 4 \sin 2x \ \sin^2(x) & 1 + \cos^2(x)& 4 \sin 2x \ \sin^2(x) & \cos^2(x) & 1+ 4 \sin 2x \end{array} \right|\)

Step 2: Calculate the Determinant

Calculating the determinant of the provided matrix:\n\(f(x)= (1+\sin^2(x))\cdot [(1+\cos^2(x))\cdot (1+4\sin 2x)- (4\sin 2x \cos^2(x))] - (\cos^2(x)) \cdot[\sin^2(x) \cdot (1+4\sin 2x) - 4\sin 2x \sin^2(x)] + (4\sin 2x)\cdot [\sin^2(x) \cos^2(x) - \sin^2(x)\cdot \sin^2(x)]=2.\)

Step 3: Take the Absolute Value

The determinant \(f(x)\) is already a positive number, thus the absolute value \(\left|f(x)\right|\) remains 2.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and cosine are essential tools in mathematics, particularly in analyzing periodic phenomena.

• Sine, denoted as \(\sin\), measures the vertical component of an angle on the unit circle.
• Cosine, denoted as \(\cos\), measures the horizontal component.
• Double angle identity for sine: \( \sin(2x) = 2 \sin(x) \cos(x) \). This is useful when dealing with expressions like \(\sin(2x)\).

In the context of matrix determinants, these functions are used to express the components of the matrix. This can complicate computations, but it also allows for more complex relationships within the matrix structure. Understanding these functions' properties is essential for solving determinant-related problems.

Matrix Algebra

Matrix algebra involves operations on matrices, including addition, multiplication, and finding determinants. A matrix is a rectangular array of numbers arranged in rows and columns.

• The determinant is a special number that can be calculated from a square matrix. It is instrumental in solving systems of linear equations.
• The provided exercise involves calculating the determinant of a \(3\times3\) matrix.
• Determinants for a \(3\times3\) matrix can be found using a specific formula involving the elements of the matrix.

Determinant calculation involves expanding along a row or a column using co-factors, as shown in the solution. Knowing how to compute a determinant is vital for various fields, including engineering and physics. It helps in finding eigenvalues and in understanding transformations.

Mathematical Optimization

Mathematical optimization seeks to find the best solution from a set of possible solutions. This can involve maximizing or minimizing a function subject to constraints. In the context of this exercise:- The objective was to find the maximum value of the determinant function \(f(x)\) derived from the matrix involving trigonometric functions.- The pre-calculated value of the determinant indicates that the determinant reaches a value of 2.- Optimization might involve exploring the behavior of the function and determining which values of \(x\) yield critical points, although the step-by-step solution simplifies this by direct calculation.Understanding how to optimize functions is crucial in various applications, from designing efficient systems to financial markets where maximum return or minimum risk is desired.