Problem 35 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The number of distinct real roots of the equaiton, \(\left|\begin{array}{lll}\cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\\ \sin x & \sin x & \cos x\end{array}\right|=0\) in the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]\) is : [OnlineApril 9, 2016] (a) 1 (b) 4 (c) 2 (d) 3

Step-by-Step Solution

Answer

The number of distinct real roots is 2.

Step 1: Setup the Determinant

We are given the determinant \( \left| \begin{array}{ccc} \cos x & \sin x & \sin x \ \sin x & \cos x & \sin x \ \sin x & \sin x & \cos x \end{array} \right| = 0 \). This is a 3x3 matrix, and we need to calculate its determinant and set it equal to zero.

Step 2: Calculate the Determinant

The determinant of a 3x3 matrix \(A\) can be calculated using the formula:\[ \det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]For our matrix:\[ \det = \cos x ((\cos x)(\cos x) - (\sin x)(\sin x)) - \sin x ((\sin x)(\cos x) - (\sin x)(\sin x)) + \sin x ((\sin x)(\sin x) - (\cos x)(\sin x)) \].

Step 3: Simplify the Determinant

After applying the formula, we simplify:\[ = \cos x(\cos^2 x - \sin^2 x) - \sin x(\cos x \sin x - \sin^2 x) + \sin x(\sin^2 x - \sin x \cos x) \]This simplifies to:\[ = \cos x (\cos^2 x - \sin^2 x) \]Using the identity \( \cos^2 x - \sin^2 x = \cos(2x) \), the determinant becomes:\[ \cos x \cos(2x) \].

Step 4: Set the Determinant to Zero

To find the distinct real roots, set the determinant equal to zero:\[ \cos x \cos(2x) = 0 \].Thus, we have two equations:1. \( \cos x = 0 \)2. \( \cos(2x) = 0 \).

Step 5: Solve \( \cos x = 0 \)

In the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\), \( \cos x = 0 \) occurs at \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \). However, neither of them is within the interval. So, no solutions from \( \cos x = 0 \).

Step 6: Solve \( \cos(2x) = 0 \)

\( \cos(2x) = 0 \) implies \( 2x = \frac{\pi}{2} + n\pi \), so;\( x = \frac{\pi}{4} + \frac{n\pi}{2} \).Within \([-\frac{\pi}{4}, \frac{\pi}{4}] \):- \( x = \frac{\pi}{4} \)- \( x = -\frac{\pi}{4} \)Thus, there are two solutions, \( x = \frac{\pi}{4} \) and \( x = -\frac{\pi}{4} \).

Step 7: Conclude the Number of Distinct Real Roots

In the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\), there are two distinct real roots. Hence, choice (c) 2 is correct.

Key Concepts

3x3 matrix determinant

When it comes to calculating the determinant of a 3x3 matrix, there is a structured formula to follow. A determinant is a scalar attribute of a matrix that is significant in linear transformations and solving systems of equations. The determinant of a matrix helps in understanding the properties of the matrix and can determine if the matrix is invertible or not.

For a matrix:

• \[ \left| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33} \end{array} \right| \]
• The determinant is calculated using:\[ det(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22}) \]

In simpler terms, for each element in the first row, multiply it by its corresponding cofactor (which involves calculating a 2x2 determinant after ignoring the row and column containing the element) and sum up these values. This formula is structured to work methodically, ensuring that each required equation step is covered.

In our case, once we apply this method on the given trigonometric matrix, we simplify to find \(cos x \cdot \cos(2x) = 0\). This simplification is achieved by using another important mathematical tool: trigonometric identities.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold for all values of the variables within their domains. They are crucial for simplifying trigonometric equations and finding unknown angles and sides in trigonometry. One of the most used identities is the Pythagorean identity, which states:

• \[\cos^2 x + \sin^2 x = 1\]

These identities are beneficial in rewriting expressions, proving other identities, or solving equations.

In our solution, the identity \(\cos^2 x - \sin^2 x = \cos(2x)\) was used to simplify the determinant expression. This identity relates the squares of sine and cosine to the double angle of cosine, providing a simpler form to work with when solving the equation.

Understanding such identities not only makes solving equations quicker but also enhances comprehension of trigonometric transformations. Knowing these identities allows for a range of manipulations that provide insights into otherwise complex-looking expressions.

Roots of trigonometric equations

Finding the roots of trigonometric equations means determining the angle values that satisfy the equation. These roots may occur over various intervals and are crucial in fields like engineering and physics to model periodic phenomena.

In our problem, the roots are found by setting the simplified determinant expression to zero: \(\cos x \cdot \cos(2x) = 0\). This gives us two separate equations to solve:

• \(\cos x = 0\)
• \(\cos(2x) = 0\)

The solution of \(\cos x = 0\) involves angles where the cosine value is zero. However, within the specified interval of \([-\frac{\pi}{4}, \frac{\pi}{4}]\), this does not yield valid roots. For \(\cos(2x) = 0\), the solutions are obtained by recognizing that cosine hits zero at angles like \(\pi/2\), and this requires considering the doubled angle.

Therefore, manipulating the equation \(2x = \frac{\pi}{2} + n\pi \,\), where \( n\) is any integer, helps find: \( x = \frac{\pi}{4}\ + \frac{n\pi}{2}\). When focusing on the interval, distinct roots \( x = \frac{\pi}{4} \) and \( x = -\frac{\pi}{4} \) emerge as solutions, showcasing the importance of restricting the analysis to the given domain.