Podcast
Questions and Answers
If Z = -6i, what is Arg(z)?
If Z = -6i, what is Arg(z)?
- θ = 3π/2 (correct)
- θ = π/3
- θ = π/2
- θ = π
The result of (5i + 3)(2 + 7i) is 31 - 29i.
The result of (5i + 3)(2 + 7i) is 31 - 29i.
False (B)
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 __________.
What is the result of i^3 + i^9 + i^6?
What is the result of i^3 + i^9 + i^6?
A = [2 4 6; 1 2 3] is a singular matrix.
A = [2 4 6; 1 2 3] is a singular matrix.
What is the expression for $z^4$ if $z = 8e^{3i}$?
What is the expression for $z^4$ if $z = 8e^{3i}$?
Match the complex operations with their results:
Match the complex operations with their results:
Flashcards
Argument of a complex number
Argument of a complex number
The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. It is measured in radians.
Polar form of a complex number
Polar form of a complex number
The polar form of a complex number represents the number using its distance from the origin (magnitude or modulus) and the angle it makes with the positive real axis (argument).
Singular Matrix
Singular Matrix
A square matrix is singular if its determinant is zero. This means that its inverse does not exist.
Non-singular Matrix
Non-singular Matrix
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Rank of a matrix
Rank of a matrix
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Complex conjugate
Complex conjugate
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Power of a complex number (polar form)
Power of a complex number (polar form)
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Complex numbers
Complex numbers
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Study Notes
Question 1: Choose the correct answer
-
1- Z = -6i then Arg(z) = ...
- Correct option is d) 0 = π/2
-
2- The result of (5i + 3)(2 + 7i)
- Correct option is c) 31 – 29i
-
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))
-
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
-
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]
- 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!
