Problem 89 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\left|\begin{array}{ccc}\cos (\theta+\alpha) & -\sin (\theta+\alpha) & \cos 2 \alpha \\ \sin \theta & \cos \theta & \sin \alpha \\ -\cos \theta & \sin \theta & \lambda \cos \alpha\end{array}\right|\) is (A) independent of \(\theta\) for all \(\lambda \in \mathrm{R}\) (B) independent of \(\theta\) and \(\alpha\) when \(\lambda=1\) (C) independent of \(\theta\) and \(\alpha\) when \(\lambda=-1\) (D) None of these

Step-by-Step Solution

Answer

(B) independent of \(\theta\) and \(\alpha\) when \(\lambda=1\).

Step 1: Understand the Matrix Structure

We are given a 3x3 matrix and need to find its determinant. The structure of the matrix is: \[\left|\begin{array}{ccc} \cos (\theta+\alpha) &-\sin (\theta+\alpha) & \cos 2 \alpha \\sin \theta & \cos \theta & \sin \alpha \-\cos \theta & \sin \theta & \lambda \cos \alpha\end{array}\right|\]Knowing the rows and columns helps us plan the determinant calculation.

Step 2: Expand the Determinant Using Cofactors

To find the determinant of the matrix, expand it along the first row: \[d = \cos(\theta+\alpha) \cdot \left( \cos\theta \cdot \lambda \cos \alpha - \sin\theta \cdot \sin\alpha \right) + \sin(\theta+\alpha) \cdot \left( \sin\theta \cdot \lambda \cos \alpha + \cos\theta \cdot \sin\alpha \right) + \cos 2\alpha \cdot \left( \sin\theta \cdot \sin\theta + \cos\theta \cdot \cos\theta \right)\]

Step 3: Simplify With Trigonometric Identities

Using the identity \(\sin^2x + \cos^2x = 1\), the third term simplifies: - \( \cos 2\alpha \cdot 1 = \cos 2\alpha \)So, the determinant expression is: \[ d = \lambda \cos \alpha \cdot ( \cos(\theta+\alpha)\cos\theta + \sin(\theta+\alpha)\sin\theta ) - \sin\alpha ( \cos(\theta+\alpha) \sin\theta - \sin(\theta+\alpha) \cos\theta ) + \cos 2\alpha \]

Step 4: Apply Angle Sum Identities

Apply the angle sum identities: - \(\cos(\theta+\alpha) = \cos\theta \cos\alpha - \sin\theta \sin\alpha\)- \(\sin(\theta+\alpha) = \sin\theta \cos\alpha + \cos\theta \sin\alpha\)Substitute back to see the simplification in each term, particularly noticing complete trigonometric identities that may emerge.

Step 5: Evaluate for Specific \(\lambda\) Values

With full substitution, check for simplifications, especially when \(\lambda = 1\) and \(\lambda = -1\). Simplify the determinant: - For \(\lambda = 1\), the determinant becomes \( \cos 2\alpha \), which is independent of \(\theta\) and \(\alpha\).- For \(\lambda = -1\), no simplification leading to independence from \(\theta\) and \(\alpha\) appears.

Step 6: Conclude the Independence

Upon testing \(\lambda = 1\), we find the determinant simplifies to depend only on \(\cos 2\alpha\), achieving independence from both \(\theta\) and \(\alpha\). This matches option (B).

Key Concepts

Trigonometric identities

Trigonometric identities are fundamental tools in mathematics, especially when dealing with angles and periodic functions. They allow us to simplify expressions and solve equations involving trigonometric functions. The two primary identities used here are the angle sum identities:

• \( \cos(\theta + \alpha) = \cos \theta \cos \alpha - \sin \theta \sin \alpha \)
• \( \sin(\theta + \alpha) = \sin \theta \cos \alpha + \cos \theta \sin \alpha \)

These identities help express complex trigonometric expressions in a simpler form. In this exercise, applying these identities is essential for breaking down the determinant into manageable parts, revealing the relationships between the variables \( \theta \) and \( \alpha \). Further, using the identity \( \sin^2 x + \cos^2 x = 1 \) is crucial to simplify certain terms that appear in the determinant expression. These identities allow us to see the inherent symmetries or dependencies present in these expressions, particularly when checking independence from specific variables.

Matrix expansion

Matrix expansion is a technique used to compute the determinant of a matrix, especially when dealing with higher-order matrices like the 3x3 matrix in our exercise. The determinant gives a scalar value that can reveal important properties of the matrix such as invertibility. The method usually starts by selecting a row or column and finding the sum of products of each element with its corresponding cofactor.

In this exercise, we expanded along the first row of the matrix. This involves multiplying each element in the first row by its cofactor — which is itself the determinant of the minor left when that row and column are removed — and summing up these products. It helps to unravel complex expressions into simpler components and evaluates the unique value of the determinant. Understanding this method is essential for breaking down and calculating matrices in determinant form.

Cofactor method

The cofactor method is pivotal for calculating determinants of square matrices. For a 3x3 matrix, each element's cofactor is determined by the determinant of the 2x2 matrix left when the element's row and column are removed. Multiplying this cofactor by the corresponding matrix element and summing them across the row (or column) provides the overall determinant.

To perform the cofactor expansion effectively, it helps to:

• Identify each minor matrix: leaving out the row and column of each element.
• Calculate the determinant of each 2x2 minor matrix.
• Apply the sign pattern (+, -, + for the first row) for each cofactor.

Not only does the cofactor method help in finding a determinant, but it also strengthens the understanding of linear dependency and matrix properties. In our problem, dissecting the matrix into these cofactors, and subsequently applying trigonometric identities, highlights the dependencies of variables \( \theta \) and \( \alpha \) on the determinant value.