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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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5. 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

What are the roots of x2+1x^2+1?

Please select an answer first.

Correct answer: A

The correct answer is ii and −i-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: The equation x^2 = -1 has the complex roots i and -i.
  3. Check the result: The correct answer is ii and −i-i.

The equation x^2 = -1 has the complex roots i and -i.

2. QuestionMultiple choice

If 2+i2+i is a root of a polynomial with real coefficients, what else must be a root?

Please select an answer first.

Correct answer: B

The correct answer is 2−i2-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: Nonreal roots of polynomials with real coefficients occur in conjugate pairs.
  3. Check the result: The correct answer is 2−i2-i.

Nonreal roots of polynomials with real coefficients occur in conjugate pairs.

3. QuestionMultiple choice

What polynomial has roots 1+i1+i and 1−i1-i?

Please select an answer first.

Correct answer: C

The correct answer is x2−2x+2x^2-2x+2.

  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 (x-(1+i))(x-(1-i)) = (x-1)^2+1 = x^2-2x+2.
  3. Check the result: The correct answer is x2−2x+2x^2-2x+2.

Multiply (x-(1+i))(x-(1-i)) = (x-1)^2+1 = x^2-2x+2.

4. QuestionMultiple choice

Which factorization represents x2+4x^2+4?

Please select an answer first.

Correct answer: D

The correct answer is (x−2i)(x+2i)(x-2i)(x+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: The roots are ±2i, so the factors are (x-2i)(x+2i).
  3. Check the result: The correct answer is (x−2i)(x+2i)(x-2i)(x+2i).

The roots are ±2i, so the factors are (x-2i)(x+2i).

5. QuestionMultiple choice

What is the sum of the roots of x2−5x+6=0x^2-5x+6=0?

Please select an answer first.

Correct answer: A

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: Vieta's formula gives the sum -b/a = -(-5)/1 = 5.
  3. Check the result: The correct answer is 5.

Vieta's formula gives the sum -b/a = -(-5)/1 = 5.

6. QuestionMultiple choice

If a polynomial with real coefficients has root −i-i, which root must also occur?

Please select an answer first.

Correct answer: B

The correct answer is 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: The conjugate of -i is i, and conjugate roots occur for real-coefficient polynomials.
  3. Check the result: The correct answer is i.

The conjugate of -i is i, and conjugate roots occur for real-coefficient polynomials.

7. QuestionMultiple choice

What is the value of p(i)p(i) for p(x)=x2+1p(x)=x^2+1?

Please select an answer first.

Correct answer: C

The correct answer is 0.

  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: Substitute i: p(i) = i^2+1 = -1+1 = 0.
  3. Check the result: The correct answer is 0.

Substitute i: p(i) = i^2+1 = -1+1 = 0.

8. QuestionMultiple choice

If z=3+4iz=3+4i, what is ∣z∣|z|?

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 modulus is sqrt(3^2+4^2) = 5.
  3. Check the result: The correct answer is 5.

The modulus is sqrt(3^2+4^2) = 5.

9. QuestionMultiple choice

How many complex roots does a degree-n polynomial have when multiplicity is counted?

Please select an answer first.

Correct answer: A

The correct answer is n.

  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 Fundamental Theorem of Algebra guarantees n complex roots counting multiplicity.
  3. Check the result: The correct answer is n.

The Fundamental Theorem of Algebra guarantees n complex roots counting multiplicity.

10. QuestionMultiple choice

What is the factored form of x2−2x+2x^2-2x+2?

Please select an answer first.

Correct answer: B

The correct answer is (x−(1+i))(x−(1−i))(x-(1+i))(x-(1-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: The roots are 1 ± i, so the factored form is built from the corresponding linear factors.
  3. Check the result: The correct answer is (x−(1+i))(x−(1−i))(x-(1+i))(x-(1-i)).

The roots are 1 ± i, so the factored form is built from the corresponding linear factors.

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