Question

If \( z_1=4i^{40}-5i^{35}+6i^{17}+2,\ z_2=-1+i \), then \( |z_1+z_2|= \)

• A. \(13\)
• B. \(5\)
• C. \(15\)
• D. \(12\)

Hint:

For powers of \(i\), divide the exponent by 4 and use the remainder to simplify.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question involves simplifying powers of the imaginary unit \( i \) to find the complex number \( z_1 \), then adding it to \( z_2 \), and finally calculating the modulus of the result.
Step 2: Key Formula or Approach:
1. Remember that \( i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \).
2. Any \( i^n \) where \( n \) is a multiple of 4 is 1. Divide high powers by 4 to find the remainder.
3. Modulus \( |x + iy| = \sqrt{x^2 + y^2} \).
Step 3: Detailed Explanation:

Simplify \( z_1 \):
\( i^{40} = (i^4)^{10} = 1^{10} = 1 \).
\( i^{35} = (i^4)^8 \cdot i^3 = 1 \cdot (-i) = -i \).
\( i^{17} = (i^4)^4 \cdot i = 1 \cdot i = i \).

Substitute back into \( z_1 \):
\( z_1 = 4(1) - 5(-i) + 6(i) + 2 \)
\( z_1 = 4 + 5i + 6i + 2 = 6 + 11i \).

Now add \( z_1 \) and \( z_2 \):
\( z_1 + z_2 = (6 + 11i) + (-1 + i) \)
\( z_1 + z_2 = (6 - 1) + (11 + 1)i = 5 + 12i \).

Calculate the modulus:
\( |z_1 + z_2| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).

Step 4: Final Answer:
The value of \( |z_1 + z_2| \) is 13.