Complex Numbers Quiz

Grade 11 · Math · Quiz

10 questions on complex arithmetic, powers of i, modulus, and conjugates.

Questions

1. Simplify: sqrt(-36).

   1. 6
   2. -6
   3. 6i
   4. 36i
   Show answer

   Answer: 6i

   sqrt(-36) = sqrt(36) * sqrt(-1) = 6i.

2. What is (5 + 2i) - (3 - 4i)?

   1. 2 - 2i
   2. 2 + 6i
   3. 8 - 2i
   4. 8 + 6i
   Show answer

   Answer: 2 + 6i

   (5-3) + (2-(-4))i = 2 + 6i.

3. Multiply: (1 + i)(1 - i).

   1. 0
   2. 2
   3. 1 - i^2
   4. 2i
   Show answer

   Answer: 2

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

4. What is the complex conjugate of -4 + 7i?

   1. 4 - 7i
   2. -4 - 7i
   3. 4 + 7i
   4. 7 - 4i
   Show answer

   Answer: -4 - 7i

   Conjugate of a + bi is a - bi. Conjugate of -4 + 7i is -4 - 7i.

5. What is i^12?

   1. i
   2. -1
   3. 1
   4. -i
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   Answer: 1

   12 mod 4 = 0, so i^12 = (i^4)^3 = 1^3 = 1.

6. Divide: (6 + 2i) / (1 + i). Multiply by the conjugate (1 - i).

   1. 4 - 2i
   2. 4 + 2i
   3. 2 + 4i
   4. 2 - 4i
   Show answer

   Answer: 4 - 2i

   Numerator: (6+2i)(1-i) = 6-6i+2i-2i^2 = 6-4i+2 = 8-4i. Denominator: (1+i)(1-i) = 2. Result: (8-4i)/2 = 4-2i.

7. What is the modulus of 5 - 12i?

   1. 7
   2. 13
   3. 17
   4. sqrt(17)
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   Answer: 13

   |5 - 12i| = sqrt(25 + 144) = sqrt(169) = 13.

8. Simplify: i^5 * i^3.

   1. i^2
   2. 1
   3. -1
   4. i
   Show answer

   Answer: 1

   i^5 = i (5 mod 4=1), i^3 = -i (3 mod 4=3). Product: i*(-i) = -i^2 = -(-1) = 1.

9. Which quadrant of the complex plane contains -3 + 4i?

   1. Quadrant I
   2. Quadrant II
   3. Quadrant III
   4. Quadrant IV
   Show answer

   Answer: Quadrant II

   Real part -3 < 0 (left), imaginary part 4 > 0 (up) -> Quadrant II.

10. Multiply: (2 + i)(3 - 2i).

    1. 6 - i
    2. 8 - i
    3. 6 - 4i + 3i - 2i^2 = 8 - i
    4. 4 + 7i
    Show answer

    Answer: 8 - i

    (2+i)(3-2i) = 6-4i+3i-2i^2 = 6-i-2(-1) = 6-i+2 = 8-i.

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