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Q1. What is the value of (2 + 3i) - (1 + 2i)? (2014)
Solution:
(2 + 3i) - (1 + 2i) = (2 - 1) + (3i - 2i) = 1 + i.
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Q2. What is the modulus of the complex number 3 + 4i? (2023)
Solution:
The modulus is calculated as √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
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Q3. If z = 3 + 4i, what is the value of z + z̅? (2022)
Solution:
z + z̅ = (3 + 4i) + (3 - 4i) = 6.
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Q4. 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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Q5. Which of the following is the conjugate of the complex number 5 - 3i? (2020)
Solution:
The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 5 - 3i is 5 + 3i.
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Q6. If z = 4 + 3i, what is the real part of z? (2023)
Solution:
The real part of the complex number z = 4 + 3i is 4.
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Q7. 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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Q8. What is the value of (3 + 4i) / (1 + i)? (2021)
Solution:
Multiplying numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 - i) / (1 + i)(1 - i) = (3 - 3i + 4i + 4) / 2 = (7 + i) / 2 = 2 + i.
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Q9. If z = 2 + 2i, what is |z|^2? (2021)
Solution:
|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.
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Q10. What is the square of the modulus of the complex number 1 + i? (2020)
Solution:
The modulus is √(1^2 + 1^2) = √2, and the square of the modulus is 2.
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