Problem 1 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\theta=\frac{\pi}{5}\) and \(\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] .\) If \(\mathrm{B}=\mathrm{A}+\mathrm{A}^{4}\), then det (B): \(\quad\) [Sep.06, 2020 (II)] (a) is one (b) lies in \((2,3)\) (c) is zero (d) lies in \((1,2)\)

Step-by-Step Solution

Answer

Det B is approximately 0.764, which lies in (1,2).

Step 1: Calculate A^2

Given \( A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \), we need to find \( A^2 \). Use the matrix multiplication formula: \( A^2 = A \cdot A \).\[ A^2 = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} = \begin{bmatrix} \cos^2 \theta - \sin^2 \theta & 2 \cos \theta \sin \theta \ -2 \cos \theta \sin \theta & \cos^2 \theta - \sin^2 \theta \end{bmatrix} \].

Step 2: Verify A^2 Formulas

Recall the trigonometric identities: \( \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \) and \( 2 \cos \theta \sin \theta = \sin(2\theta) \), thus,\[ A^2 = \begin{bmatrix} \cos(2\theta) & \sin(2\theta) \ -\sin(2\theta) & \cos(2\theta) \end{bmatrix} \].

Step 3: Calculate A^4

Use \( A^2 \) to calculate \( A^4 = (A^2)^2 \).\[ A^4 = \begin{bmatrix} \cos(2\theta) & \sin(2\theta) \ -\sin(2\theta) & \cos(2\theta) \end{bmatrix} \begin{bmatrix} \cos(2\theta) & \sin(2\theta) \ -\sin(2\theta) & \cos(2\theta) \end{bmatrix} = \begin{bmatrix} \cos^2(2\theta) - \sin^2(2\theta) & 2 \cos(2\theta) \sin(2\theta) \ -2 \cos(2\theta) \sin(2\theta) & \cos^2(2\theta) - \sin^2(2\theta) \end{bmatrix} \].

Step 4: Verify A^4 Formulas

Simplify \( A^4 \) using similar trigonometric identities again: \( \cos^2(2\theta) - \sin^2(2\theta) = \cos(4\theta) \) and \( 2 \cos(2\theta) \sin(2\theta) = \sin(4\theta) \). Hence,\[ A^4 = \begin{bmatrix} \cos(4\theta) & \sin(4\theta) \ -\sin(4\theta) & \cos(4\theta) \end{bmatrix} \].

Step 5: Compute B

Given \( B = A + A^4 \), substitute the matrices we found:\[ B = \begin{bmatrix} \cos \theta + \cos(4\theta) & \sin \theta + \sin(4\theta) \ -\sin \theta - \sin(4\theta) & \cos \theta + \cos(4\theta) \end{bmatrix} \].

Step 6: Calculate det(B)

Use the determinant formula for 2x2 matrices: \( \text{det}(B) = ad - bc \), where \( B = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Here:\[ a = d = \cos \theta + \cos(4\theta), \ b = \sin \theta + \sin(4\theta), \ c = -\sin \theta - \sin(4\theta) \].Thus,\[ \text{det}(B) = (\cos \theta + \cos(4\theta))^2 - (\sin \theta + \sin(4\theta))( -\sin \theta - \sin(4\theta)) \].Simplify it using trigonometric identities: \( (\cos^2 \theta + 2\cos \theta\cos(4\theta) + \cos^2(4\theta)) + (\sin^2 \theta + 2\sin \theta\sin(4\theta) + \sin^2(4\theta)) \), which gives,\[ \text{det}(B) = 2(1 + \cos{(\theta - 4\theta)}) = 2(1 + \cos{(-3\theta)}) \].

Step 7: Evaluate det(B) for \\(\theta = \frac{\pi}{5}\\)

With \( \theta = \frac{\pi}{5} \), evaluate \( 2(1 + \cos{(-3 \frac{\pi}{5})}) \).Since \( \cos(-x) = \cos x \), it becomes, \( 2(1 + \cos{\frac{-3\pi}{5}}) \).Evaluate \( \cos{\frac{3\pi}{5}} = - \cos{\frac{2\pi}{5}} \).\[ \cos{\frac{2\pi}{5}} \approx 0.618 \Rightarrow \cos{\frac{3\pi}{5}} \approx -0.618 \].Thus, \( 2(1 - 0.618) \approx 2 \times 0.382 = 0.764 \).

Step 8: Choose Correct Option

Based on calculated \( \text{det}(B) \approx 0.764 \), compare with given intervals, and find it lies in option (d).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions like sine and cosine. They are useful in simplifying expressions, solving equations, and transforming complex trigonometric equations into more manageable forms.

Some fundamental identities include:

• The Pythagorean Identity: \( \cos^2 \theta + \sin^2 \theta = 1 \)
• The Double Angle Formulas: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \) and \( \sin(2\theta) = 2 \sin \theta \cos \theta \)
• The Addition Formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

In this particular problem, these identities are used to simplify the calculations of matrices raised to a power. For example, knowing that \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \) allows for quick simplifications in matrix multiplication, especially when calculating powers of a matrix containing trigonometric functions.

This simplification is key in moving from raw trigonometric expressions to more straightforward results, thus making the full process of solving the determinant of matrix \( B \) possible.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. This operation is essential when working with linear transformations, systems of equations, or computational problems.

To perform matrix multiplication, follow these steps:

• Take each element from the rows of the first matrix.
• Multiply them by the corresponding elements in the columns of the second matrix.
• Sum up each product for each element in the resulting matrix.

For multiplying 2x2 matrices, like in this exercise, the determinant and trigonometric identities play vital roles. When calculating \( A^2 \) or \( A^4 \), each element is derived by applying the multiplication rules along with identified trigonometric identities.

For example, in calculating \( A^4 \), we use the already computed \( A^2 \) matrix and multiply it by itself. This demonstrates how matrix multiplication is not just a mechanical procedure but is enriched by mathematical theory like trigonometric identities. Understanding this process ensures a deeper understanding of how linear transformations accumulate effects.

2x2 Matrices Determinant

The determinant of a 2x2 matrix is a scalar value that can help identify the nature of the transformations represented by that matrix, including scaling and rotation. It is calculated simply using the formula:

If \( B = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then the determinant is calculated as \( \text{det}(B) = ad - bc \).

In the exercise provided, matrices \( A \) and \( A^4 \) are used to eventually find matrix \( B \). To discover the determinant of \( B \), the formula is applied to compute the products of its diagonal elements and the difference consisting of products of its other two elements.

Solving \( \text{det}(B) \) involves evaluating trigonometric expressions like \( \cos \theta + \cos(4\theta) \) and \( \sin \theta + \sin(4\theta) \). By simplifying these using trigonometric identities, one can achieve the necessary expressions to accurately calculate \( 2(1 + \cos{(-3 \theta)}) \), finally leading to the correct interval that \( \text{det}(B) \) lies in. Understanding determinants is crucial since they reveal the properties related to the geometric transformation enacted by the matrix.