Problem 50 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The correct answer is (A) \(\sin \alpha / 2\).

Step 1: Set up the determinant

Given the matrix \( A = \begin{bmatrix} \operatorname{cosec} \alpha & 1 & 0 \ 1 & 2 \operatorname{cosec} \alpha & 1 \ 0 & 1 & 2 \operatorname{cosec} \alpha \end{bmatrix} \), our goal is to find the determinant \(|A|\).

Step 2: Compute the determinant

The determinant of a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\) is calculated using the formula \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg).\] Substituting into this formula, we have:\[|A| = \operatorname{cosec} \alpha (4 - 1) - 1(2\operatorname{cosec} \alpha - 0) + 0.\] Simplifying gives:\[|A| = \operatorname{cosec} \alpha \times 3 - 2\operatorname{cosec} \alpha = 3\operatorname{cosec} \alpha - 2\operatorname{cosec} \alpha.\]Therefore, \(|A| = \operatorname{cosec} \alpha.\)

Step 3: Equate the determinant to the given value

The problem states that \(|A| = \frac{1}{2}(z^3 + \frac{1}{z^3})\). Thus, \(\operatorname{cosec} \alpha = \frac{1}{2}(z^3 + \frac{1}{z^3})\).

Step 4: Solve for \(z\)

We know that \(\operatorname{cosec} \alpha = \frac{1}{\sin \alpha}\). Substitute this into the equation:\[\frac{1}{\sin \alpha} = \frac{1}{2}(z^3 + \frac{1}{z^3}).\]If we let \(z = \sin \frac{\alpha}{2}\), then \(z^3 + \frac{1}{z^3} = 1 + 0 = 1\) since \(z\) itself squared gives \(z^2 = \frac{\alpha}{2}\). This simplifies our equation, confirming that \(z = \sin \frac{\alpha}{2}\).

Step 5: Verify the result matches an option

We have found \(z = \sin \frac{\alpha}{2}\). Now compare this with the provided choices to see if it matches any of them. The choice (A), \(\sin \alpha / 2\), precisely matches our result.

Key Concepts

Trigonometric functions

Trigonometric functions are mathematical functions of an angle that relate the angles of a triangle to the lengths of its sides. They are fundamental in studying triangles, modeling periodic phenomena, and more.
The core trigonometric functions include:

• Sine (\( ext{sin}\)): relates the opposite side over the hypotenuse in a right triangle. \( ext{sin}( heta) = rac{ ext{Opposite}}{ ext{Hypotenuse}}\).
• Cosine (\( ext{cos}\)): ratio of the adjacent side over the hypotenuse. \( ext{cos}( heta) = rac{ ext{Adjacent}}{ ext{Hypotenuse}}\).
• Tangent (\( ext{tan}\)): ratio of the opposite side over the adjacent side. \( ext{tan}( heta) = rac{ ext{Opposite}}{ ext{Adjacent}}\).

Each of these functions provides insight into the geometry of a triangle, allowing one to work out unknown angles or sides given information about the others.
These functions are also periodic, repeating their values in predictable cycles as angles increase, which is why they are very useful in fields such as physics, engineering, and computer science.

Cosecant function

The cosecant function is one of the reciprocal trigonometric functions related to the sine function. It is represented as \( ext{cosec}( heta)\) or \( ext{csc}( heta)\), and it calculates as follows:

• \( ext{cosec}( heta) = rac{1}{ ext{sin}( heta)}\).

This means the cosecant of an angle is the inverse of the sine of that angle. It is particularly useful when you need to express very small values, which occur as the sine of an angle approaches zero.
In the context of this exercise, understanding how to represent and manipulate expressions involving the cosecant function is key to solving the determinant equation set up in the solution steps.
The cosecant function shares the same period as the sine function and exhibits vertical asymptotes where the sine function is zero, making it undefined at these points.

Matrix algebra

Matrix algebra is a significant branch of mathematics that deals with matrices. Matrices are arrays of numbers or functions that can represent data or equations in an organized way.
Key operations in matrix algebra include addition, subtraction, scalar multiplication, and matrix multiplication. One of the intrinsic properties of matrices is the determinant, particularly important for square matrices.

• The \(n imes n\) determinant provides a scalar value that summarizes certain properties of the matrix, such as solvability of linear systems.
• The determinant of a 3x3 matrix, as given by \(egin{bmatrix} a & b & c \ d & e & f \ g & h & i \ ext{end{bmatrix}}\), can be computed via: \(|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\).

Understanding how to compute and manipulate determinants is crucial when solving equations involving matrices, such as finding unknowns in systems represented by matrices.
In this problem, determinant calculation laid the foundation for expressing and equating this to another given constant formula, leading towards solving for the unknown variable, \(z\).