Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

The complex numbers \(\sin x+i \cos 2 x\) and \(\cos x-\) \(i \sin 2 x\) are conjugate to each other, for (A) \(x=n \pi\) (B) \(x=0\) (C) \(x=\left(n+\frac{1}{2}\right) \pi\) (D) no value of \(x\)

Step-by-Step Solution

Answer

Correct answer: (D) no value of \(x\).

Step 1: Understand Conjugate Complex Numbers

For two complex numbers to be conjugates, the real part of one should equal the real part of the other, and the imaginary part should be the negative of the imaginary part of the other. Here, this means \[\sin x = \cos x \quad \text{and} \quad \cos 2x = -\sin 2x.\]

Step 2: Analyze the Real Parts

From the equality \(\sin x = \cos x\), this holds when:\[\tan x = 1.\] That happens when \[x = \frac{\pi}{4} + n\pi,\ n \in \mathbb{Z}.\]

Step 3: Analyze the Imaginary Parts

For \(-\cos 2x = \sin 2x\), we find:\[\tan(2x) = -1.\] This occurs when:\[2x = \frac{3\pi}{4} + m\pi \quad\Rightarrow\quad x = \frac{3\pi}{8} + \frac{m\pi}{2},\ m \in \mathbb{Z}.\]

Step 4: Find Common Solutions

To find values of \(x\) that satisfy both conditions, equate:\[\frac{\pi}{4} + n\pi = \frac{3\pi}{8} + \frac{m\pi}{2}\]Solving the equation will determine if there is a solution that fits both equations.

Step 5: Evaluate Possibilities

By solving the equation:\[2n\pi + \frac{1}{2} = m\pi + \frac{3}{8}\]No simplification resolves to natural numbers simultaneously, so there is no common solution under these transformations.

Step 6: Conclusion

There is no single value for \(x\) that satisfies both conditions in natural arithmetic solutions, recognizing the relationship constraints without a simple overlap in values.

Key Concepts

Conjugate Complex Numbers

In the world of complex numbers, conjugates are fascinating pairings. To understand conjugate complex numbers, imagine two numbers: if one is represented as \(a + bi\), its conjugate will be \(a - bi\). The rule for conjugates is straightforward: the real parts must be identical, while the imaginary parts are opposites. This pattern helps simplify many mathematical equations and operations.

• Real parts are equal.
• Imaginary parts are negatives of each other.

While dealing with complex numbers given in the exercise, we identify that for them to be conjugates, certain trigonometric constraints must hold. If we consider \(\sin x + i \cos 2x\) and \(\cos x - i \sin 2x\), then their real parts \(\sin x\) and \(\cos x\) must be equal, and imaginary parts \(\cos 2x\) and \(-\sin 2x\) must be opposites.
Understanding these principles lays a foundation in complex numbers, making it easier to grasp more advanced concepts in mathematics.

Trigonometric Equations

Trigonometric equations can seem intimidating at first, but they are just equations that include trigonometric functions like sine, cosine, and tangent. In our context, solving trigonometric equations involves determining conditions where two trigonometric expressions are equal or opposite.

• \(\sin x = \cos x\) highlights where the trigonometric identity \(\tan x = 1\) holds true.
• \(\cos 2x = -\sin 2x\) shows where \(\tan(2x) = -1\) is satisfied.

These equations help us find specific values for \(x\). For example, \(\tan x = 1\) will imply \(x = \frac{\pi}{4} + n\pi\) for integer values of \(n\), meaning \(x\) is a multiple of \(\pi\) away from \(\frac{\pi}{4}\). Similarly, solving \(\tan(2x) = -1\) lets us find another set of solutions.
These trigonometric equations are crucial for finding common solutions where both conditions are met.

Imaginary and Real Parts Analysis

Analyzing imaginary and real parts separately is a valuable technique when dealing with complex equations. For a complex number, both components hold essential information.

• Real part analysis involves equating function outputs, like \(\sin x = \cos x\).
• Imaginary part analysis focuses on negating outputs, such as \(\cos 2x = -\sin 2x\).

Breaking down a complex equation into real and imaginary components simplifies the process of finding solutions. It allows us to apply specific mathematical tools to separate trigonometric conditions, making it more manageable to solve complex equations.
Understanding this division proves useful not only in exercises involving trigonometric identities but also in broader applications such as signal processing and quantum mechanics.