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Common Core Standard HSN-CN.B.5 Questions

(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

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Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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What is the product of z1 and z2?
  1. 8+8i
  2. 8-8i
  3. -16+8i
  4. 3+7i
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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What is the modulus, r, and argument, θ, of z1? List them as (r,θ).
  1. (4, 90°)
  2. (2√2, 90°)
  3. (4, 45°)
  4. (4√2, 45°)
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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What is the modulus, r, and argument, θ, of z2? List them as (r,θ).
  1. (√2, -19.5°)
  2. (√10, 108.4°)
  3. (2, 161.4°)
  4. (√10, -71.5°)
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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What is the modulus, r, and argument, θ, of the product of z1 and z2? List them as (r, θ).
  1. (8√2, 45°)
  2. (8√2, -45°)
  3. √58, 66.8°)
  4. (8√5, 153.4°)
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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What is the relationship between the answers in parts b and c and the answer in part d?
  1. The moduli of z1 and z2 multiplied together equals the modulus of z1z2, and the arguments of z1 and z2 added together equals the argument of z1z2.
  2. The moduli of z1 and z2 multiplied together equals the modulus of z1z2, but the arguments have no special relationship.
  3. The moduli have no special relationship, but the arguments of z1 and z2 added together equals the argument of z1z2.
  4. There is no relationship.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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For n=2, show that the equation is true. Hint: use trigonometric identities.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5

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Which of the following gives the best reasoning of how, for n>3, this equation can be proved true?
  1. There are more complicated trigonometric identities that deal with higher powers.
  2. As was done in part b, by using the answer from the previous integer, each successive integer can be shown to be true using the same trigonometric identities (as used in part b).
  3. Having shown it true for two different cases, it can be assumed true for all other cases.
  4. For even powers of n, the process will be similar to n=2, while for odd powers of n, the process of showing this equation is true will be similar to n=3, but increasingly more complicated.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5
One method for adding or subtracting complex numbers in the complex plane is to look at the numbers as vectors, and then add or subtract these vectors as is done with vectors in the real plane. A vector is defined as something having both a direction and a magnitude. Why can complex numbers in the complex plane be represented as vectors? Choose all correct answers.
  1. They can't really, but it's a useful tool to use when adding or subtracting.
  2. Because, as seen in polar form, they have a magnitude, r, and direction, θ.
  3. Because they are represented by two "coordinates", a real and imaginary value, which is similar to the component form of a vector.
  4. Because complex numbers and vectors are identical.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5
André is working on a math problem where he has to subtract two complex numbers graphically. They are z1=3-8i and z2=-5+i. If he is using the translation method, and is subtracting z1 from z2, which complex number is considered the starting point, and then how is it translated?
  1. z1 is the starting point, and is translated 5 units left and 1 unit up.
  2. z1 is the starting point, and is translated 5 units right and 1 unit down.
  3. z2 is the starting point, and is translated 3 units right and 8 units down.
  4. z2 is the starting point, and is translated 3 units left and 8 units up.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5
If z=-4+2i is plotted in the complex plane, how can multiplying by i be described geometrically?
  1. It rotates z 90° counterclockwise about the origin.
  2. It rotates z 90° clockwise about the origin.
  3. It reflects z across the imaginary axis.
  4. There is no geometric interpretation.
Grade 11 Complex Numbers CCSS: HSN-CN.B.5
If z=-2+7i is plotted in the complex plane, how can multiplying by -1 be described geometrically? Choose all correct answers.
  1. It rotates z 90° clockwise about the origin.
  2. It rotates z clockwise 180° about the origin.
  3. It reflects z across the imaginary and real axis.
  4. It reflects z across the real axis.
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