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

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

The determinant is independent of \(\theta\) and \(\alpha\) when \(\lambda = -1\) (Option C).

Step 1: Evaluating the Determinant

Given the matrix \[\begin{bmatrix}\cos (\theta+\alpha) & -\sin (\theta+\alpha) & \cos 2\alpha \\sin \theta & \cos \theta & \sin \alpha \-\cos \theta & \sin \theta & \lambda \cos \alpha\end{bmatrix}\],we need to substitute it into the formula for the determinant of a 3x3 matrix, denoted as \[a_{11}(b_{22}c_{33} - b_{23}c_{32}) - a_{12}(b_{21}c_{33} - b_{23}c_{31}) + a_{13}(b_{21}c_{32} - b_{22}c_{31})\].

Step 2: Simplify 3x3 Determinant

Substitute the elements of the given matrix into the determinant formula:\[\left|\begin{array}{ccc}\cos (\theta+\alpha) & -\sin (\theta+\alpha) & \cos 2\alpha \\sin \theta & \cos \theta & \sin \alpha \-\cos \theta & \sin \theta & \lambda \cos \alpha\end{array}\right|= \cos(\theta+\alpha)\left(\cos\theta \cdot \lambda\cos\alpha - \sin\theta \cdot \sin \alpha\right)+ \sin(\theta+\alpha)\left(\sin\theta \cdot \lambda\cos\alpha + \cos\theta \cdot \sin \alpha\right)+ \cos 2\alpha \cdot \left(\sin\theta \cdot \sin\theta + \cos\theta \cdot \cos\theta\right)\].

Step 3: Evaluate Each Component

First, simplify components involving \(\cos 2\alpha\):\[\cos 2\alpha(\sin^2\theta + \cos^2\theta) = \cos 2\alpha \cdot 1 = \cos 2\alpha\].Next, expand remaining components:\[\cos(\theta+\alpha)\left(\lambda\cos(\alpha)\cos(\theta) - \sin(\theta)\sin(\alpha)\right) + \sin(\theta+\alpha)\left(\lambda\cos(\alpha)\sin(\theta) + \cos(\theta)\sin(\alpha)\right)\].

Step 4: Use Trigonometric Identities

Using trigonometric identities:\[\cos(\theta+\alpha)\sin\alpha + \sin(\theta+\alpha)\cos\alpha = \sin(\theta+\alpha+\alpha)\].The expression does not simplify to become independent of both \(\theta\) and \(\alpha\) when \(\lambda eq -1\). Also note that collective term can be 0 independent of \(\theta\) and \(\alpha\) when \(\lambda = -1\).

Step 5: Conclusion

By computing the determinant and analyzing factors, it simplifies when \(\lambda = -1\) to be independent of \(\theta\) and \(\alpha\) only. C: Independent of \(\theta\) and \(\alpha\) when \(\lambda = -1\).

Key Concepts

3x3 matrix

A 3x3 matrix is a square array consisting of nine elements arranged in three rows and three columns. This is a key concept used throughout various fields of mathematics, including linear algebra and calculus. In this context, it is particularly important for calculating determinants.

The determinant of a 3x3 matrix is a special number that is calculated from its elements and can indicate various properties of the matrix, such as whether it is invertible. The formula for computing the determinant involves a combination of the elements across the matrix. For instance, if you have a 3x3 matrix \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \\end{bmatrix}\]you calculate its determinant using the following pattern: \[a(ei − fh) − b(di − fg) + c(dh − eg)\]Here, each term is a combination of products of two elements in the matrix, some of which are subtracted from each other. This operation can be broken down further to effectively carry out the computation within the specific context of the problem at hand.

Understanding this structure is crucial for solving more complex problems, such as those involving trigonometric elements or specific properties of the determinant.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities serve as a powerful tool in simplifying expressions and solving trigonometric equations.

Some crucial identities include:

• Pythagorean identities, such as \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Angle sum and difference identities, like \(\cos(\theta + \alpha) = \cos \theta \cos \alpha - \sin \theta \sin \alpha\)
• Double angle formulas, for example, \(\cos 2\alpha = 2 \cos^2 \alpha - 1\) or \(\cos 2\alpha = 1 - 2 \sin^2 \alpha\)

Using these identities allows mathematicians and students to transform complex trigonometric expressions into simpler forms. In the exercise, recognizing the identity \(\cos(\theta+eta)\cos \alpha + \sin(\theta+eta)\sin \alpha = \cos (\theta+eta-eta)\) was essential for solving the determinant problem.

These simplifications can reveal hidden properties of equations and significantly reduce the complexity of calculations.

Properties of determinants

Determinants have several fascinating properties that can simplify computations and reveal more about the nature of matrices. Understanding these properties can be crucial when tackling problems involving complex variable expressions.

Some important properties include:

• A square matrix is invertible if and only if its determinant is not zero. This means the matrix has a unique solution to the associated system of equations.
• Switching two rows or columns of a matrix multiplies its determinant by -1. This is a useful property during row operations and can check for consistency in the system.
• The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements, simplifying calculations in specific configurations.
• A determinant becomes zero if two rows or columns are identical, indicating that the matrix does not have full rank.

In the given exercise, the independence of the determinant from variables \(\theta\) and \(\alpha\) by altering the value of \(\lambda\) highlights how properties can be explored further through strategic manipulation. Specifically, using row operations and recognizing patterns aids in understanding how different configurations and values affect the determinant's outcome.

These insights are valuable for a range of applications, from solving linear systems to finding eigenvalues.