Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|\begin{array}{ccc}\operatorname{cosec} \alpha & 1 & 0 \\ 1 & 2 \operatorname{cosec} \alpha & 1 \\ 0 & 1 & 2 \operatorname{cosec} \alpha\end{array}\right|=\frac{1}{2}\left(z^{3}+\frac{1}{z^{3}}\right)\), then \(z\) is equal to (A) \(\sin \alpha / 2\) (B) \(\cos \alpha / 2\) (C) \(\tan \alpha / 2\) (D) None of these

Step-by-Step Solution

Answer

(B) \(\cos \frac{\alpha}{2}\)

Step 1: Label the Matrix and Write the Determinant

Given the matrix \( A = \begin{bmatrix} \operatorname{cosec} \alpha & 1 & 0 \ 1 & 2 \operatorname{cosec} \alpha & 1 \ 0 & 1 & 2 \operatorname{cosec} \alpha \end{bmatrix} \), identify the determinant \( \det(A) \). This will be computed using cofactor expansion along the first row.

Step 2: Compute the Determinant of the Matrix

Using cofactor expansion along the first row, \( \det(A) = \operatorname{cosec} \alpha \cdot \det \left( \begin{bmatrix} 2 \operatorname{cosec} \alpha & 1 \ 1 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right) - 1\cdot \det \left( \begin{bmatrix} 1 & 1 \ 0 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right) \).

Step 3: Simplify the Determinants

For \( \det \left( \begin{bmatrix} 2 \operatorname{cosec} \alpha & 1 \ 1 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right) = 2 \operatorname{cosec} \alpha \cdot 2 \operatorname{cosec} \alpha - 1 \cdot 1 = 4 \cot^2 \alpha \). For \( \det \left( \begin{bmatrix} 1 & 1 \ 0 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right) = 2 \operatorname{cosec} \alpha \).

Step 4: Substitute Back and Solve

Substitute into the determinant equation: \( \det(A) = \operatorname{cosec} \alpha \cdot (4 \cot^2 \alpha) - 2 \operatorname{cosec} \alpha = 4 \operatorname{cosec} \alpha \cot^2 \alpha - 2 \operatorname{cosec} \alpha \).

Step 5: Equate to the Given Expression

From the problem statement, \( 4 \operatorname{cosec} \alpha \cot^2 \alpha - 2 \operatorname{cosec} \alpha = \frac{1}{2} \left(z^3 + \frac{1}{z^3} \right) \). Simplify this using trigonometric identities.

Step 6: Analyze and Use Trigonometric Identities

Recall that \( \operatorname{cosec} \alpha = 1/\sin \alpha \) and \( \cot^2 \alpha = (\cos^2 \alpha)/(\sin^2 \alpha) \). Substitute these into the expression.

Step 7: Simplify and Set z

The expression simplifies to \( 1 - 2 = \frac{1}{2} \left(z^3 + \frac{1}{z^3} \right) \). Hence, \( z^3 + \frac{1}{z^3} = 0 \).

Step 8: Find z

\( z^3 = -\frac{1}{z^3} \) implies that \( (z^3)^2 = -(1) \) indicating \( z^3 = i \) or \( z = i^{1/3} \). The options correspond to trigonometric half-angles, therefore check special half-angle formulas to see if \( z = \cos \frac{\alpha}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They are essential tools in simplifying and solving trigonometric expressions and equations. In this exercise, identities are used to express complex relationships between the elements of the matrix and determinant calculation.
Understanding basic trigonometric identities is crucial. For instance:

• The Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
• Reciprocal identities such as \( \operatorname{cosec} \alpha = 1/\sin \alpha \) and \( \cot \alpha = \cos \alpha / \sin \alpha \).

These identities allow us to convert and simplify the trigonometric terms encountered in matrix computations.
For example, when simplifying \( \operatorname{cosec} \alpha \cot^2 \alpha \), the given identities convert this into a form that can more easily plug into other algebraic expressions.

Matrix Algebra

Matrix algebra is a powerful tool when dealing with linear transformations and systems of equations. In this context, we use matrix algebra to compute the determinant of a matrix.
A matrix is essentially a rectangular array of numbers arranged in rows and columns. The matrix in our exercise is:\[A = \begin{bmatrix} \operatorname{cosec} \alpha & 1 & 0 \1 & 2 \operatorname{cosec} \alpha & 1 \0 & 1 & 2 \operatorname{cosec} \alpha \end{bmatrix}\]The determinant is a special number calculated from its elements that provides useful properties, such as whether the matrix is invertible.
Determinants for 3x3 matrices can be calculated through various methods, but the cofactor expansion, also known as Laplace’s Theorem, is particularly useful when simplifying calculations. Understanding how to use matrix algebra to compute determinants is critical for determining relationships like the one in the exercise.

Cofactor Expansion

Cofactor expansion, or Laplace expansion, is a method used to calculate the determinant of a matrix. It involves expanding the determinant along a row or column, simplifying the process when dealing with complex matrices.
In this exercise, we expand the determinant along the first row of matrix \( A \), as it contains simpler terms:\[\det(A) = \operatorname{cosec} \alpha \cdot \det \left( \begin{bmatrix} 2 \operatorname{cosec} \alpha & 1 \1 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right) - 1 \cdot \det \left( \begin{bmatrix} 1 & 1 \0 & 2 \operatorname{cosec} \alpha \end{bmatrix} \right)\]Each minor matrix determinant is evaluated further using the same principles, reducing the problem to a set of smaller arithmetic calculations and trigonometric simplifications.
Using cofactor expansion effectively requires an understanding of the matrix structure and recognizing how identities simplify elements of the matrix. This method reveals deeper insights into how determinants are related to other expressions.