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Complex Numbers Practice Tests

These Grade 11 Math practice tests develop fluency with the imaginary unit, complex-number operations, conjugates, modulus, the complex plane, quadratic equations with nonreal roots, and polynomial root relationships.

Skills covered

  • Using powers of the imaginary unit and identifying real and imaginary parts
  • Adding, subtracting, multiplying, dividing, and simplifying complex numbers
  • Using conjugates, modulus, distance, midpoint, and complex-plane coordinates
  • Solving quadratic equations with negative discriminants
  • Applying conjugate roots, Vieta’s formulas, factorization, and the Fundamental Theorem of Algebra

How to use these practice tests

Replace i² with -1, group real and imaginary terms, use a conjugate when rationalizing a denominator, and verify roots by substitution or factoring.

For more Grade 11 Math practice, return to the Grade 11 Math category.

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2. Test

Grade 11 Complex Numbers Practice Tests

Practice Grade 11 Math questions about the imaginary unit, complex-number operations, conjugates, modulus, the complex plane, quadratic equations, and polynomial roots.

Choose the best answer. Use i^2 = -1, combine real and imaginary parts separately, and check restrictions and roots carefully.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

Simplify (3+2i)+(4−5i)(3+2i)+(4-5i).

Please select an answer first.

Correct answer: A

The correct answer is 7−3i7-3i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Add real parts and imaginary parts: (3+4) + (2-5)i = 7 - 3i.
  3. Check the result: The correct answer is 7−3i7-3i.

Add real parts and imaginary parts: (3+4) + (2-5)i = 7 - 3i.

2. QuestionMultiple choice

Simplify (6−i)−(2+3i)(6-i)-(2+3i).

Please select an answer first.

Correct answer: B

The correct answer is 4−4i4-4i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Subtract the real and imaginary parts: (6-2) + (-1-3)i = 4 - 4i.
  3. Check the result: The correct answer is 4−4i4-4i.

Subtract the real and imaginary parts: (6-2) + (-1-3)i = 4 - 4i.

3. QuestionMultiple choice

Simplify (2+i)(3−i)(2+i)(3-i).

Please select an answer first.

Correct answer: C

The correct answer is 7+i7+i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Use distribution: 6 - 2i + 3i - i^2 = 6 + i + 1 = 7 + i.
  3. Check the result: The correct answer is 7+i7+i.

Use distribution: 6 - 2i + 3i - i^2 = 6 + i + 1 = 7 + i.

4. QuestionMultiple choice

Simplify (1+2i)(1−2i)(1+2i)(1-2i).

Please select an answer first.

Correct answer: D

The correct answer is 5.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: The factors are conjugates, so the product is 1^2 + 2^2 = 5.
  3. Check the result: The correct answer is 5.

The factors are conjugates, so the product is 1^2 + 2^2 = 5.

5. QuestionMultiple choice

Simplify (4+3i)(2−i)(4+3i)(2-i).

Please select an answer first.

Correct answer: A

The correct answer is 11+2i11+2i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Distribute: 8 - 4i + 6i - 3i^2 = 8 + 2i + 3 = 11 + 2i.
  3. Check the result: The correct answer is 11+2i11+2i.

Distribute: 8 - 4i + 6i - 3i^2 = 8 + 2i + 3 = 11 + 2i.

6. QuestionMultiple choice

Simplify (2+i)2(2+i)^2.

Please select an answer first.

Correct answer: B

The correct answer is 3+4i3+4i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Expand: 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i.
  3. Check the result: The correct answer is 3+4i3+4i.

Expand: 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i.

7. QuestionMultiple choice

Simplify 3+i1−i\frac{3+i}{1-i}.

Please select an answer first.

Correct answer: C

The correct answer is 1+2i1+2i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Multiply by the conjugate 1+i: (3+i)(1+i)/2 = (2+4i)/2 = 1+2i.
  3. Check the result: The correct answer is 1+2i1+2i.

Multiply by the conjugate 1+i: (3+i)(1+i)/2 = (2+4i)/2 = 1+2i.

8. QuestionMultiple choice

What is the conjugate of 7−4i7-4i?

Please select an answer first.

Correct answer: D

The correct answer is 7+4i7+4i.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: The conjugate changes the sign of the imaginary part and keeps the real part unchanged.
  3. Check the result: The correct answer is 7+4i7+4i.

The conjugate changes the sign of the imaginary part and keeps the real part unchanged.

9. QuestionMultiple choice

What is (3+2i)(3−2i)(3+2i)(3-2i)?

Please select an answer first.

Correct answer: A

The correct answer is 13.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: A complex number times its conjugate is a^2+b^2: 3^2+2^2=13.
  3. Check the result: The correct answer is 13.

A complex number times its conjugate is a^2+b^2: 3^2+2^2=13.

10. QuestionMultiple choice

Simplify 2−i2−i\frac{2-i}{2-i}.

Please select an answer first.

Correct answer: B

The correct answer is 1.

  1. Identify the complex-number structure: Determine the real part, imaginary part, powers of i, conjugate, modulus, operation, or equation involved.
  2. Apply the complex-number relationship: Any nonzero quantity divided by itself equals 1; 2 - i is nonzero.
  3. Check the result: The correct answer is 1.

Any nonzero quantity divided by itself equals 1; 2 - i is nonzero.

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