Complex Numbers

If , then the equation represents

Let . The given equation is .

Substituting , we get .

Taking the magnitude of both sides gives:

Squaring both sides:

This represents the y-axis.

Answer: (D) y-axis

Let
Then,




which represents axis.

so, The correct option is(D): Y-axis.

We are asked to solve the equation .

Step 1: Simplify the base

To simplify the fraction, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is :

Calculate the numerator:

Calculate the denominator:

So, the fraction simplifies to:

Step 2: Rewrite the equation

Substituting the simplified base back into the original equation, we get:

Step 3: Solve for x

We need to find the values of x for which .

Recall the powers of :

The powers of repeat in a cycle of 4. We see that whenever x is a multiple of 4.

Therefore, the general solution for integer x is , where is any integer ( ).

Step 4: Compare with the given options

The options provide expressions for x in terms of n, where (natural numbers, typically ).

• x = 4n + 1; n ∈ N (e.g., 5, 9, 13, ...) →
• x = 2n + 1, n ∈ N (e.g., 3, 5, 7, ...) → or
• x = 2n; n ∈ N (e.g., 2, 4, 6, 8, ...) → or
• x = 4n; n ∈ N (e.g., 4, 8, 12, ...) →

The option correctly describes the positive integer values of x that are multiples of 4, satisfying the condition .

The correct option is x = 4n; n ∈ N.

The correct answer is: (C):

We are tasked with finding the modulus of the complex number:

Step 1: Simplify the numerator

First, expand :

Now, multiply this result by :

Step 2: Simplify the denominator

Now, expand :

Step 3: Find the modulus of the fraction

The complex number is now:

We want to find the modulus of this complex number. The modulus of a complex number is given by:

First, calculate the modulus of the numerator :

Next, calculate the modulus of the denominator :

Step 4: Final simplification

The modulus of the complex number is:

Conclusion:
The modulus of the complex number is , so the correct answer is (C): .

1. Rationalize the denominator:

To determine when the expression is purely real, we need to eliminate the imaginary part from the denominator. We do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator:

2. Expand and Simplify:

Expanding the numerator and the denominator:

Numerator:

Denominator:

3. Combine and Separate Real and Imaginary Parts:

Therefore, the expression becomes:

4. Condition for Purely Real:

For the expression to be purely real, the imaginary part must be zero. This means:

5. Solve for α:

This implies that . The solutions to this equation are:

, where n is an integer.

6. Consider the domain (n ∈ N):

Since n ∈ N (natural numbers), the solution remains .

Conclusion: The real value of 'α' for which the given expression is purely real is , where n is a natural number.

Correct Answer: (C) , n ∈ N