Podcast
Questions and Answers
What value of $ heta$ will lead to a unique solution for the system of equations: $ heta x + 2y - 2z = 1$, $4x + heta y - z = 2$, $6x + 6y + z = 3$?
What value of $ heta$ will lead to a unique solution for the system of equations: $ heta x + 2y - 2z = 1$, $4x + heta y - z = 2$, $6x + 6y + z = 3$?
Which of the following represents the necessary condition for $ rac{ rac{ ext{partial}^{2}f}{ ext{partial }x^{2}} + rac{ ext{partial}^{2}f}{ ext{partial }y^{2}} = 0$ for $f(x, y) = ext{log} ext{sqrt}(x^{2} + y^{2})$ to hold?
Which of the following represents the necessary condition for $ rac{ rac{ ext{partial}^{2}f}{ ext{partial }x^{2}} + rac{ ext{partial}^{2}f}{ ext{partial }y^{2}} = 0$ for $f(x, y) = ext{log} ext{sqrt}(x^{2} + y^{2})$ to hold?
For the function $f(x,y) = rac{xy(x^{2} - y^{2})}{x^{2} + y^{2}}$ when checking the limit as $(x,y) o (0,0)$, which limit approach is crucial?
For the function $f(x,y) = rac{xy(x^{2} - y^{2})}{x^{2} + y^{2}}$ when checking the limit as $(x,y) o (0,0)$, which limit approach is crucial?
What is the geometric multiplicity of an eigenvalue if the associated modal matrix does not have enough linearly independent eigenvectors?
What is the geometric multiplicity of an eigenvalue if the associated modal matrix does not have enough linearly independent eigenvectors?
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Which of the following indicates that the system of equations is consistent: $x_1 + 2x_2 + x_3 = 2$, $2x_1 + 4x_2 + 2x_3 = 4$, $3x_1 + x_2 - 2x_3 = 1$, $4x_1 - 3x_2 - x_3 = 3$?
Which of the following indicates that the system of equations is consistent: $x_1 + 2x_2 + x_3 = 2$, $2x_1 + 4x_2 + 2x_3 = 4$, $3x_1 + x_2 - 2x_3 = 1$, $4x_1 - 3x_2 - x_3 = 3$?
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Study Notes
Q3 - Attempt Any Two Questions
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Question 1: Find the value of λ such that the system of equations Ax + 2y - 2z = 1, 4x + λy - z = 2, 6x + 6y + λz = 3 has a unique solution.
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Question 2: Prove that the function f(x, y) = log√(x² + y²) satisfies the given equation.
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Question 3: Examine if the limit lim (x,y)→(0,0) (x³y-y⁴)/(x⁴+y⁴) exists for x ≠ 0, y ≠ 0.
Q4 - Compulsory
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Part A:
- A square matrix A is defined by A = [4 1] [2 3]
- Find the modal matrix P and the resulting diagonal matrix D of A.
- Determine the algebraic and geometric multiplicities of the eigenvalues.
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Part B:
- OR: Check if the following system of equations is consistent using Gaussian elimination:
- x₁ + 2x₂ + x₃ = 2
- 2x₁ + 4x₂ + 2x₃ = 4
- 3x₁ + x₂ - 2x₃ = 1
- 4x₁ - 3x₂ - x₃ = 3
- OR: Test the continuity of f(x, y) = {(xy(x²-y²))/(x²+y²), when x ≠ 0, y ≠ 0} {0, when x = 0, y = 0}
- OR: Check if the following system of equations is consistent using Gaussian elimination:
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Description
This quiz consists of questions focused on linear algebra concepts, including finding values for unique solutions, proving functions, and investigating limits. It also covers matrix operations, eigenvalues, and Gaussian elimination techniques.
