Question

The complex number \( \frac{1 + 7i}{(2 - i)^2 \) lies in}

• A. Quadrant 4
• B. Quadrant 3
• C. Quadrant 1
• D. Quadrant 2

Hint:

To determine the quadrant of a complex number, observe the signs of its real and imaginary parts. If the real part is negative and the imaginary part is positive, it lies in Quadrant 2.

Answer 1

The Correct Option is D

Step 1: Simplify the given expression.
The given complex number is:
\[ \frac{1 + 7i}{(2 - i)^2} \]
First, simplify the denominator \( (2 - i)^2 \). Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), we get:
\[ (2 - i)^2 = 2^2 - 2(2)(i) + i^2 = 4 - 4i - 1 = 3 - 4i \]
So, the given expression becomes: \[ \frac{1 + 7i}{3 - 4i} \]

Step 2: Multiply by the conjugate of the denominator.
To simplify the fraction, multiply both the numerator and the denominator by the conjugate of the denominator, \( 3 + 4i \):
\[ \frac{1 + 7i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i} = \frac{(1 + 7i)(3 + 4i)}{(3 - 4i)(3 + 4i)} \]

Step 3: Simplify the denominator.
The denominator is a difference of squares: \[ (3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25 \]

Step 4: Simplify the numerator.
Now expand the numerator:
\[ (1 + 7i)(3 + 4i) = 1(3) + 1(4i) + 7i(3) + 7i(4i) = 3 + 4i + 21i + 28i^2 \]
Since \( i^2 = -1 \), we get: \[ 3 + 4i + 21i - 28 = -25 + 25i \]

Step 5: Final expression.
Now, the complex number is: \[ \frac{-25 + 25i}{25} = -1 + i \]

Step 6: Plot on the complex plane.
The complex number \( -1 + i \) is in the second quadrant, where the real part is negative and the imaginary part is positive.

Step 7: Conclusion.
Therefore, the complex number lies in Quadrant 2, and the correct answer is option (D).