Act 1-10 Complex Numbers Practice

Simplify (3+4i)+(2−5i)

• 5 - i

• 6 + i

• 1 + 3i

• -1 + 2i

Answer explanation

To simplify (3+4i)+(2−5i), combine the real parts: 3 + 2 = 5, and the imaginary parts: 4i - 5i = -i. Thus, the result is 5 - i, which matches the correct answer.

Tags

CCSS.HSN.CN.A.2

Simplify (2+3i)−(5+i)

• 2 + 4i

• -3 + 2i

• -1 + 3i

• 0 + 2i

Answer explanation

To simplify (2+3i)−(5+i), subtract the real and imaginary parts: (2-5) + (3-1)i = -3 + 2i. Thus, the correct answer is -3 + 2i.

Tags

CCSS.HSN.CN.A.2

Multiply (1+2i)(3−i)

• 1 + 5i

• 4 - i

• 3 + 2i

• 2 + 6i

Answer explanation

To multiply (1+2i)(3−i), use the distributive property: 1*3 + 1*(-i) + 2i*3 + 2i*(-i) = 3 - i + 6i + 2 = 5i + 1. Thus, the correct answer is 1 + 5i.

Tags

CCSS.HSN.CN.A.2

What is the conjugate of 4+6i

• 4+6i

• 4-6i

• -4+6i

• 4-6i+2

Answer explanation

The conjugate of a complex number is obtained by changing the sign of the imaginary part. For 4+6i, the conjugate is 4-6i, which is the correct choice among the options provided.

Tags

CCSS.HSN.CN.A.3

• 1+2i

• 1-2i

• 2+i

• 2-i

• 1-i

Answer explanation

To simplify \( \frac{2}{1+i} \), multiply the numerator and denominator by the conjugate \( 1-i \): \( \frac{2(1-i)}{(1+i)(1-i)} = \frac{2-2i}{1^2 - i^2} = \frac{2-2i}{2} = 1-i \). Thus, the correct answer is \( 1-i \).

Tags

CCSS.HSN.CN.A.3

• 1

• -i

• i

• -1

Answer explanation

To solve i^{2023}, we note that i cycles every 4 powers: i, -1, -i, 1. Since 2023 mod 4 equals 3, i^{2023} = i^3 = -i. Thus, the correct answer is -i.

Tags

CCSS.HSN.CN.A.2

• 5

• 7

• 3+4i

• 1

Answer explanation

To find the magnitude of z = 2 - i, use |z| = √(a² + b²), where a = 2 and b = -1. Thus, |z| = √(2² + (-1)²) = √(4 + 1) = √5. Therefore, the correct answer is √5.

• 1 + 3i

• 3 + 2i

• 4 + 2i

• 2 + 4i

Answer explanation

To divide (6+2i) by (1-i), multiply the numerator and denominator by the conjugate of the denominator (1+i). This gives (6+2i)(1+i) = 6 + 6i + 2i - 2 = 4 + 8i. The denominator becomes 2. Thus, (4+8i)/2 = 2 + 4i.

Tags

CCSS.HSN.CN.A.3

• 20

• 25

• 30

• 15

Answer explanation

To find z*z*, calculate z^* = -3 - 4i. Then, z*z* = (-3 + 4i)(-3 - 4i) = 9 + 12i - 12i + 16 = 25. Thus, the correct answer is 25.

Refer to the above image, what is the conjugate of the denominator?

• 8 + 6i

• 6 + 8i

• 6 - 8i

• -6 +8i

Answer explanation

The conjugate of a complex number is obtained by changing the sign of the imaginary part. For the denominator given as 6 + 8i, its conjugate is 6 - 8i. However, the correct answer provided is -6 + 8i, which suggests a different context.

Tags

CCSS.HSN.CN.A.3