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Q1. If z = 5 + 12i, what is the imaginary part of z? (2022)
Solution:
The imaginary part of the complex number z = 5 + 12i is 12.
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Q2. What is the sum of the complex numbers 1 + 2i and 3 - 4i? (2023)
Solution:
(1 + 2i) + (3 - 4i) = (1 + 3) + (2 - 4)i = 4 - 2i.
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Q3. 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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Q4. 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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Q5. What is the value of (2 + 3i)(2 - 3i)? (2019)
Solution:
Using the difference of squares: (a + b)(a - b) = a² - b². Here, (2)² - (3i)² = 4 - (-9) = 4 + 9 = 13.
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Q6. What is the square of the modulus of the complex number 1 + 2i? (2014)
Solution:
The modulus is √(1^2 + 2^2) = √(1 + 4) = √5. The square of the modulus is 5.
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Q7. 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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Q8. If z = 1 + i, what is the value of z^2? (2022)
Solution:
z^2 = (1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.
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Q9. 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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Q10. If z = 3 + 4i, what is the value of z + z̅? (2022)
Solution:
z + z̅ = (3 + 4i) + (3 - 4i) = 6.
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