Complex Numbers Practice Exam – Complete Solutions | PinoyBIX

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You finished the 25 item Complex Numbers Practice Exam. Now check your work. Every item below follows the same Given, Find, Solution format used throughout this series, with no step skipped. If you got an item wrong, do not just read the answer. Find the exact step where your process broke down, then solve a similar problem from scratch before you move on.

Part A — Forms and the j-Operator (Items 1 to 5)

Full lesson on this topic is in Part 1 — Forms and the j-Operator.

Item 1. Evaluate $j^{63}$ .

Choices: (A) $1$ (B) $-1$ (C) $j$ (D) $-j$

Given: $j^{63}$

Find: The value of the expression.

Solution:

Step 1: Recall the cycle of four for powers of $j$ .

$j^1 = j \qquad j^2 = -1 \qquad j^3 = -j \qquad j^4 = 1$

Step 2: Divide the exponent by 4 and check the remainder.

$63 \div 4 = 15 \text{ remainder } 3$

Step 3: Match the remainder to the cycle. Remainder 3 corresponds to $j^3$ .

$j^{63} = j^3 = -j$

✓ ANSWER: D — $j^{63} = -j$

Examiner note: This is a one step problem once you know the cycle of four. It should take you under 10 seconds. Students who lose points here stopped at the quotient instead of checking the remainder.

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What to Do With Your Score

Go back to the Complex Numbers Series Index and check your raw score against the score interpretation table there. If you missed three or more items in any single part, that part is your weak spot. Go back to that part’s full post, not just this solutions page, and work through one additional similar problem from scratch, not from memory and not by rereading the answer.

If you are reviewing for ECE or EE, give extra weight to Part 4. Impedance and phasor problems carry the most points on those two boards, and every one of them builds directly on what you just worked through in Parts 1 through 3.

Series navigation: Part 1 — Forms and j-Operator | Part 2 — Operations | Part 3 — De Moivre’s Theorem | Part 4 — AC Circuits and Applications

The next series on PinoyBIX covers Matrices and Determinants, another high frequency topic in Engineering Mathematics tested across all nine engineering boards. Follow PinoyBIX on Facebook to get notified when it goes live.

Published by PinoyBIX.org — Engineering Education for Every Filipino Student. Electronics · Mathematics · Board Exam Review · Free for Everyone.

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