Podcast
Questions and Answers
If Z = -6i, what is Arg(z)?
If Z = -6i, what is Arg(z)?
The result of (5i + 3)(2 + 7i) is 31 - 29i.
The result of (5i + 3)(2 + 7i) is 31 - 29i.
False
What is the rank of the matrix A = [0 4 5; 0 0 0]?
What is the rank of the matrix A = [0 4 5; 0 0 0]?
1
If $z = \frac{1 + 7i}{(2 - i)^2}$, the polar form is __________.
If $z = \frac{1 + 7i}{(2 - i)^2}$, the polar form is __________.
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What is the result of i^3 + i^9 + i^6?
What is the result of i^3 + i^9 + i^6?
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A = [2 4 6; 1 2 3] is a singular matrix.
A = [2 4 6; 1 2 3] is a singular matrix.
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What is the expression for $z^4$ if $z = 8e^{3i}$?
What is the expression for $z^4$ if $z = 8e^{3i}$?
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Match the complex operations with their results:
Match the complex operations with their results:
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Study Notes
Question 1: Choose the correct answer
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1- Z = -6i then Arg(z) = ...
- Correct option is d) 0 = π/2
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2- The result of (5i + 3)(2 + 7i)
- Correct option is c) 31 – 29i
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3- Find the rank A = [0 4 5 / 1 0 2 / 0 0 0]
- Rank is 2
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4. Write z = -4 + 4i in polar form.
- Correct option is b) 4√2(cos(5π/4) + i sin(5π/4))
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5- If z = (1 + 7i) / (2 - i)² the polar form is
- Correct option is b) √2e^(3π/4)i
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6- i³ + i⁰ + i⁶ = ...
- Result is -1
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7- Determine A = [3 6 9 / 1 2 3] is....
- A is singular
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8- if z = 8e^(3πi/2) where z¹?
- z¹ = e^(πi)
Question 2: Solve part A (3 Marks) and part B is bonus
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(a) Prove by mathematical induction that for all positive integers n: 1⁴ + 2⁴ + 3⁴ + ... + n⁴ = n(n + 1)(2n + 1)(3n² + 3n – 1) / 30
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(b) Expand (1 - 3x)³ using the binomial theorem.
- The expansion is 1 - 9x + 27x² - 27x³
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Find the coefficient of x³ in the expansion.
- The coefficient of x³ is -27
Question 3: Show your steps (5 Marks)
- Find the inverse of the matrix A = [2 1 3 / 1 4 2 / 3 1 1] using row operations.
Question 4: Solve for each case (4 Marks)
- Consider the following augmented matrix: [1 2 3 6 / 2 k 4 12 / 3 1 k 9]
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Determine the value(s) of k for which:
- The system has a unique solution.
- The system has infinite solutions.
- The system has no solution.
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Description
Test your knowledge on complex numbers, including polar forms, arithmetic, and applications of mathematical induction. This quiz covers various aspects such as ranks of matrices and binomial expansion. Challenge yourself with these practical problems!
