Complex Numbers

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Q1. What is the conjugate of the complex number 5 - 2i? (2020)

Solution:

The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 5 - 2i is 5 + 2i.

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Q2. If z = 2 - 3i, what is the conjugate of z? (2020)

Solution:

The conjugate of z = 2 - 3i is 2 + 3i.

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Q3. What is the product of the complex numbers (1 + i) and (1 - i)? (2023) 2023

Solution:

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

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Q4. Find the real part of the complex number 4 + 5i. (2023)

Solution:

The real part of the complex number 4 + 5i is 4.

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Q5. If z = 1 + √3i, what is the argument of z? (2015)

Solution:

The argument is given by tan^(-1)(√3/1) = π/3.

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Q6. If z = 1 + i, what is the value of z²? (2022)

Solution:

z² = (1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.

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Q7. What is the square root of the complex number -4? (2020)

Solution:

The square root of -4 is √(-1) * √4 = 2i.

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Q8. What is the argument of the complex number -1 + 0i? (2021)

Solution:

The argument of -1 + 0i is π (180 degrees) since it lies on the negative x-axis.

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Q9. What is the square root of the complex number -1? (2021)

Solution:

The square root of -1 is defined as i.

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Q10. If z = 2 + 2i, what is |z|^2? (2021)

Solution:

|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.