Problem 50 - Algebra for JEE ADVANCED

Question

The number of distinct real roots of \(\left|\begin{array}{ccc}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) in the interval \(-\pi / 4 \leq x \leq \pi / 4\) is a. 0 b. 2 c. 1 d. 3

Step-by-Step Solution

Answer

b. 2

Step 1: Understanding the Determinant

The determinant we need to find is \[\left|\begin{array}{ccc}\sin x & \cos x & \cos x \\cos x & \sin x & \cos x \\cos x & \cos x & \sin x\end{array}\right|= 0.\] We are tasked with finding the number of distinct real roots of this equation in the interval \(-\pi / 4 \leq x \leq \pi / 4\).

Step 2: Calculate the Determinant

To proceed, calculate the determinant. We expand by the first row:\[\sin x\left(\sin x(\sin x) - \cos x(\cos x)\right)- \cos x\left(\cos x(\sin x) - \cos x(\cos x)\right)+ \cos x\left(\cos x(\cos x) - \sin x(\cos x)\right).\]Simplifying this, we get \[\sin^3 x - \sin x \cos^2 x - \cos^2 x \sin x + \cos^3 x = (\sin^3 x + \cos^3 x) - 2\sin x \cos^2 x.\]

Step 3: Use Trigonometric Identities

We can use the identity \[\sin^3 x + \cos^3 x = (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x).\]Combine with the identity \[\sin^2 x + \cos^2 x = 1,\]to simplify further:\[(\sin x + \cos x)(1 - \sin x \cos x) = 2\sin x \cos^2 x.\]

Step 4: Setup the Equation

Equating to zero, we have \[(\sin x + \cos x)(1 - \sin x \cos x) = 2\sin x \cos^2 x.\]Simplify to find roots:\[\sin x + \cos x = 0 \quad \text{or} \quad 1 - \sin x \cos x = 0.\]

Step 5: Solve for Roots

For \(\sin x + \cos x = 0\), divide by \(\cos x\) to get \( \tan x = -1\). Thus \(x = -\pi/4, \pi/4\) within the given interval. For \(1 - \sin x \cos x = 0\), use the identity \(\sin(2x) = 2\sin x \cos x\), we get \(\sin(2x) = 1\), or \(2x = \pi/2\). So \(x = \pi/4\), but this is already counted.

Step 6: Conclusion

The distinct real roots from both equations include \(-\pi/4\) and \(\pi/4\). Together they give us 2 distinct roots over the interval specified.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying expressions and solving equations involving trigonometric functions. In this exercise, we specifically use identities to manipulate and simplify the determinant expression. By understanding how these identities work, we can see the relationships between sine and cosine. For example, one fundamental identity is

• \( \sin^2 x + \cos^2 x = 1 \)

This identity allows us to replace one function in terms of the other and greatly simplifies calculations.
Additionally, in the process of expanding the determinant, we utilize an identity involving cubes:

• \( \sin^3 x + \cos^3 x = (\sin x + \cos x)(1 - \sin x \cos x) \)

Thanks to these identities, we can factor expressions and find values of \( x \) that make our determinant equal to zero. This involves evaluating where specific conditions derived from these identities are met within the given interval.

Real Roots

Real roots are the values of \( x \) for which a given equation is satisfied. In our exercise, we aim to find the real roots of the determinant equation within a specific interval. Solving such equations often requires restructuring and simplification, as we saw when using trigonometric identities.
To find real roots, we equate the expression to zero and solve for \( x \). In the case of the exercise, we simplified the original expression to two factors:

• \( \sin x + \cos x = 0 \)
• and \( 1 - \sin x \cos x = 0 \)

By solving these individually, we can identify values where each condition is satisfied, i.e., where the equation equals zero. Consideration of real roots ensures we are focusing on solutions that actually exist on the real number line rather than complex ones.

Interval Analysis

Interval analysis involves examining a function or equation over a specific range of values. In this exercise, the interval given is

• \(-\pi/4 \leq x \leq \pi/4\)

This imposes a constraint on the possible solutions we are interested in, ensuring any roots found lie within these boundaries. This means any solutions to our determinant equation must satisfy this interval condition.
The interval affects how we validate our solutions. For \( \tan x = -1 \), the solutions \( x = -\pi/4 \) and \( x = \pi/4 \) are both valid because they lie within the specified interval. Meanwhile, using the identity \( \sin(2x) = 1 \) led us to \( x = \pi/4 \), which had already been counted. Recognizing distinct solutions requires considering both the equation solutions and the interval constraints.