Problem 90 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\theta\) lying between \(\theta=0\) and \(\theta=\frac{\pi}{2}\) and satisfying the equation \(\left|\begin{array}{ccc}1+\sin ^{2} \theta & \cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & 1+\cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0\) is (A) \(\frac{7 \pi}{24}\) (B) \(\frac{5 \pi}{24}\) (C) \(\frac{11 \pi}{24}\) (D) \(\frac{\pi}{24}\)

Step-by-Step Solution

Answer

The value of \( \theta \) is \( \frac{5\pi}{24} \).

Step 1: Determine the Determinant

The problem requires finding \( \theta \) such that the determinant of the given matrix is zero. Let's calculate the determinant of the 3x3 matrix: \[\begin{vmatrix}1+\sin^2\theta & \cos^2\theta & 4\sin4\theta \\sin^2\theta & 1+\cos^2\theta & 4\sin4\theta \\sin^2\theta & \cos^2\theta & 1+4\sin4\theta\end{vmatrix} = 0\]The formula for the determinant of a 3x3 matrix \( A \) is:\[ \det(A) = a(ei-fh)-b(di-fg)+c(dh-eg) \]

Step 2: Assign Values from the Matrix

Let's assign the matrix elements to the variables in the determinant formula:- \( a = 1 + \sin^2\theta \)- \( b = \cos^2\theta \)- \( c = 4\sin4\theta \)- \( d = \sin^2\theta \)- \( e = 1 + \cos^2\theta \)- \( f = 4\sin4\theta \)- \( g = \sin^2\theta \)- \( h = \cos^2\theta \)- \( i = 1 + 4\sin4\theta \)

Step 3: Simplify Determinant Expression

Substitute the assigned values into the determinant equation and simplify:\[ \det(A) = (1+\sin^2\theta)[(1+\cos^2\theta)(1+4\sin4\theta) - 4\sin4\theta\cdot\cos^2\theta] \- \cos^2\theta[\sin^2\theta(1+4\sin4\theta) - 4\sin4\theta\cdot\sin^2\theta] \+ 4\sin4\theta[\sin^2\theta\cdot\cos^2\theta - \cos^2\theta\cdot(1+\cos^2\theta)] \]Simplifying this expression further, we focus on identifying patterns or cancellations, aiming for \( 0 \).

Step 4: Analyze Solutions

By simplifying or applying computational tools, solve for \( \theta \). Recognize this could involve trial and error, using trigonometric identities, or any computational aid to solve \( \det(A) = 0 \). Use the options (A) \( \frac{7\pi}{24} \), (B) \( \frac{5\pi}{24} \), (C) \( \frac{11\pi}{24} \), (D) \( \frac{\pi}{24} \) to test as possible solutions.

Step 5: Verification

To find the correct \( \theta \), substitute each possible solution option back into the simplified determinant expression and verify which one equates the determinant to zero.

Key Concepts

Determinants

A determinant is a special number that can be calculated from a square matrix. In the context of matrix algebra, it represents a scalar value that enables us to answer several important questions about the matrix, such as whether it has an inverse. When the determinant of a matrix equals zero, the matrix is termed non-invertible or singular, meaning it does not have an inverse.
When working with a 3x3 matrix, computing the determinant might seem complex, but it is structured using a specific formula. For a matrix:

• First compute the products of its diagonal elements.
• Then apply a pattern of addition and subtraction to those products.

The formula for the determinant of a 3x3 matrix \( A \) can be written as \( \det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \). Here \( a, b, c, d, e, f, g, h, \) and \( i \) represent the entries of the matrix. It is important to get these products right and maintain the pattern of signs to ensure accuracy in calculations.
Understanding determinants will help you determine crucial properties of matrices and solve systems of linear equations in matrix algebra. In this exercise, the task was to simplify this expression to find when it equaled zero—indicating a specific value of \( \theta \) satisfied the condition.

Matrix Algebra

Matrix algebra is a system of mathematics that deals with matrices, which are rectangular arrays of numbers or expressions arranged in rows and columns. This field of mathematics forms the basis for various applications in science, engineering, and computer graphics.
Matrices can represent complex systems of linear equations and transformations. Basic operations in matrix algebra include addition, subtraction, scalar multiplication, and particularly, multiplication of matrices themselves. Each of these operations follows specific rules:

• For addition and subtraction, matrices must be the same size.
• Scalar multiplication multiplies every element of the matrix by a single number (the scalar).
• Matrix multiplication is more complex and involves taking the dot product of rows and columns.

The exercise involves determining whether the determinant of a particular matrix equals zero, leading to the realization that solving the given trigonometric equation is a key step. Matrix algebra is a critical tool here, as it gives us a way to explore and manipulate sets of linear equations in a neat and organized manner.
Understanding matrices and their properties, including determinants, helps deal with larger systems, reduce computational problems, and solve practical applications efficiently.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. In trigonometry, these identities are particularly useful for simplifying equations and solving complex trigonometric problems.
Some basic trigonometric identities include:

• Pythagorean identities, like \( \sin^2\theta + \cos^2\theta = 1 \).
• Angle addition identities, such as \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \).
• Double angle identities, like \( \sin2\theta = 2\sin\theta\cos\theta \).

These identities are key to tackling trigonometric equations by reducing them down to more manageable forms. In the given exercise, recognizing and applying these identities are crucial for checking solution options to determine which one satisfies the given determinant condition.
By testing the values provided (i.e., \( \frac{7\pi}{24}, \frac{5\pi}{24}, \frac{11\pi}{24}, \frac{\pi}{24} \)), trigonometric identities help reveal the correct angle \( \theta \) that makes the trigonometric equation true. Mastery of these identities can greatly enhance your problem-solving skills in mathematics.