Complex Operations Assessment
\(\mathbb{C}\)
Complex Numbers & Operations
Algebra 2 & Honors Assessment • Form A
Total Score
/ 100
Name:
Date:
Period:
Score:
Directions: Express all solutions in standard complex form \(a + bi\) where \(a, b \in \mathbb{R}\). Show all algebraic steps clearly in the provided blank workspaces. No credit will be awarded for unsupported answers.
1 Part I: Powers of the Imaginary Unit & Radicals
16 Points
1. Evaluate each power of \(i\): [4 pts]
a) \(i^{35} =\)
b) \(i^{102} =\)
2. Which is equivalent to \(\sqrt{-72}\)? [4 pts]
A \(-6\sqrt{2}\)
B \(6i\sqrt{2}\)
C \(12i\sqrt{2}\)
D \(-36i\sqrt{2}\)
3. Simplify to \(a + bi\) form: [4 pts]
\(-3\sqrt{-25} + 2\sqrt{-64}\)
Answer:
4. Conceptual Analysis: [4 pts]
Explain why \(\sqrt{-4} \cdot \sqrt{-9} \neq \sqrt{(-4)(-9)}\). Show the correct evaluation.
2 Part II: Addition, Subtraction & Scalar Multiplication
16 Points
5. Find the sum: [4 pts]
\((7 - 4i) + (-11 + 9i)\)
Answer:
6. Subtract and simplify: [4 pts]
\((5 - 2i) - (8 - 6i) + 4\)
Answer:
7. Distribute and combine: [4 pts]
\(3(4 - 5i) - 2(1 + 3i)\)
Answer:
8. Geometric Application: [4 pts]
A triangle has side lengths \(a = 3 + 2i\), \(b = 5 - i\), and \(c = 4 + 6i\). Find its perimeter in \(a + bi\) form.
Perimeter:
Algebra 2 & Honors Assessment • Complex Numbers Page 1 of 3
Student Name:
Part III – Part V • Operations & Quadratics
Form A
3 Part III: Multiplication & Powers of Binomials
18 Points
9. Multiply the complex binomials: [4 pts]
\((4 - 3i)(2 + 5i)\)
Answer:
10. Expand and simplify: [4 pts]
\((3 - 2i)^2\)
Answer:
11. Product of conjugates: [5 pts]
\((6 + 7i)(6 - 7i)\)
Answer:
12. Multi-step product: [5 pts]
\(2i(3 - i)(1 + 4i)\)
Answer:
4 Part IV: Complex Conjugates & Division
18 Points
13. Monomial divisor: [5 pts]
\(\dfrac{10}{2i}\)
Answer:
14. Binomial divisor: [6 pts]
\(\dfrac{4 + 2i}{3 - i}\)
Answer:
15. Rationalize: [7 pts]
\(\dfrac{5 - 3i}{2 + 4i}\)
Answer:
5 Part V: Solving Quadratic Equations with Complex Roots
14 Points
16. Solve by square root property: [6 pts]
\(3x^2 + 48 = 0\)
Solutions:
17. Solve via Quadratic Formula: [8 pts]
\(2x^2 - 4x + 5 = 0\)
Solutions:
Algebra 2 & Honors Assessment • Complex Numbers Page 2 of 3
Student Name:
Part VI – Part VII • Complex Plane & Honors Challenge
Form A
6 Part VI: The Complex Plane & Modulus (Absolute Value)
18 Points
18. Plot & label on the complex plane: [6 pts]
\(A: z_1 = 3 + 4i\)
\(B: z_2 = -4 - 2i\)
\(C: z_3 = -5i\)
\(D: z_4 = 2 - 3i\)
Re Im 5 -5 5i -5i
19. Calculate the modulus \(|z|\): [6 pts]
Recall: \(|a + bi| = \sqrt{a^2 + b^2}\)
a) \(|3 + 4i| =\)
b) \(|-5 + 12i| =\)
20. Distance in the complex plane: [6 pts]
Find the distance between \(z_1 = 1 + 2i\) and \(z_2 = 4 + 6i\) by evaluating \(|z_2 - z_1|\).
Distance:
7 Part VII: Honors & Advanced STEM Applications Honors
18 Points
21. Factoring over \(\mathbb{C}\): [5 pts]
Factor completely over the complex numbers:
\(9x^2 + 64\)
Factors:
22. Higher-Degree Zeros: [6 pts]
Find all 3 solutions (real & complex) by factoring the difference of cubes:
\(x^3 - 8 = 0\)
Roots:
23. AC Circuit Impedance: [7 pts]
In series, total impedance is \(Z_T = Z_1 + Z_2\). Given \(Z_1 = 8 + 6i\,\Omega\) and \(Z_2 = 4 - 11i\,\Omega\):
a) Find \(Z_T\) in standard form:
b) Find magnitude \(|Z_T|\):
Algebra 2 & Honors Assessment • Complex Numbers Page 3 of 3 • End of Assessment
