Quick recall · 161 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
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The \( \lim_{x \to 0} \frac{\sin x}{x} \) is equal to:
B · 1
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If \( y = \sin x + \cos x \), find \( \frac{dy}{dx} \).
A · \( \cos x - \sin x \)
Derivative of \( \sin x \) is \( \cos x \), derivative of \( \cos x \) is \( -\sin x \). So \( \frac{dy}{dx} = \cos x - \sin x \). This matches option A.
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Evaluate the definite integral \( \int_0^1 \frac{1}{\sqrt{1 + x^2}} \, dx \).
A · \( \frac{\pi}{4} \)
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The area bounded by the curve y = x^2, the line y = 4, and the y-axis is to be found using definite integration.
A · \( \frac{16}{3} \)
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The general solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) is:
A · A) \( y = kx \)
PYQ · 2017
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The Laplace transform \( L(e^{at}) \) is:
B · \( \frac{1}{s-a} \)
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Which of the following is NOT a type of matrix based on its shape?
C · Symmetric matrix
Symmetric matrix is classified based on element properties, not shape. Shape-based types include square and rectangular matrices.
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A matrix with all elements zero except the main diagonal is called a:
C · Diagonal matrix
A diagonal matrix has non-zero elements only on the main diagonal and zeros elsewhere.
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Which matrix type must be square and satisfies \( A = A^T \)?
B · Symmetric matrix
A symmetric matrix is square and equal to its transpose.
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If \( A \) is a \( 2 \times 3 \) matrix and \( B \) is a \( 3 \times 2 \) matrix, what is the order of the product \( AB \)?
A · \( 2 \times 2 \)
The product \( AB \) is defined and has order \( 2 \times 2 \) since inner dimensions match.
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Refer to the diagram below showing matrices \( A \) and \( B \). What is the transpose of \( A + B \)?
A · \( A^T + B^T \)
The transpose of a sum of matrices equals the sum of their transposes.
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Which of the following properties is TRUE for matrix multiplication?
B · Matrix multiplication is associative
Matrix multiplication is associative but not commutative in general.
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Given \( A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \), find \( A^T \).
A · \( \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} \)
Transpose of a matrix is obtained by interchanging rows and columns.
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Refer to the diagram below showing the stepwise multiplication of two matrices. What is the element at position (1,2) in the product matrix?
B · 22
Element (1,2) is calculated as sum of products of row 1 of first matrix and column 2 of second matrix.
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Which of the following is the determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix}a & b \\ c & d\end{bmatrix} \)?
B · \( ad - bc \)
The determinant of a 2x2 matrix is \( ad - bc \).
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Which property of determinants states that swapping two rows changes the sign of the determinant?
B · Row interchange property
Swapping two rows of a determinant changes its sign (row interchange property).
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Refer to the diagram below showing the expansion of a \( 3 \times 3 \) determinant along the first row. Which term corresponds to the cofactor of element \( a_{12} \)?
A · \( -a_{12} \times \) minor of \( a_{12} \)
Cofactor of \( a_{12} \) is \( (-1)^{1+2} a_{12} \) times its minor, which is negative.
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What is the determinant of the matrix \( \begin{bmatrix}2 & 0 & 1 \\ 3 & 0 & 0 \\ 5 & 1 & 1\end{bmatrix} \)?
A · 1
Expanding along the second row or using cofactor expansion yields determinant = 1.
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If the determinant of matrix \( A \) is 5, what is the determinant of \( 3A \) where \( A \) is a \( 2 \times 2 \) matrix?
B · 45
Determinant of \( kA \) for \( n \times n \) matrix is \( k^n \det(A) \). Here, \( 3^2 \times 5 = 45 \).
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Which method is commonly used to find the inverse of a matrix using determinants?
B · Adjugate and determinant method
The inverse is found by dividing the adjugate matrix by the determinant.
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Refer to the diagram below showing the calculation steps for the inverse of matrix \( A \). What is the determinant of \( A \)?
D · -4
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If \( A \) is invertible and \( \det(A) = 7 \), what is \( \det(A^{-1}) \)?
B · \( \frac{1}{7} \)
Determinant of inverse matrix is reciprocal of determinant of original matrix.
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Which of the following conditions is necessary for a matrix \( A \) to have an inverse?
B · \( \det(A) eq 0 \)
A matrix has an inverse only if its determinant is non-zero.
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Refer to the diagram below showing the adjugate matrix calculation. Which element corresponds to the cofactor \( C_{21} \)?
B · Element at (2,1)
Cofactor \( C_{21} \) corresponds to element at row 2, column 1.
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In engineering, which application commonly uses matrices to solve systems of linear equations?
A · Structural analysis
Structural analysis uses matrices to solve equilibrium equations for forces and displacements.
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Which of the following engineering problems can be solved using determinants to check system solvability?
A · Determining stability of a structure
Determinants help check if a system of linear equations (e.g., equilibrium equations) has unique solutions, important in stability analysis.
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Refer to the matrix equation below representing nodal analysis in electrical circuits. If \( \det(A) = 0 \), what does it imply about the circuit?
B · Infinite or no solutions (dependent system)
Zero determinant indicates dependent equations, leading to infinite or no solutions.
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Which matrix operation is used in computer graphics to perform rotation transformations?
C · Matrix multiplication
Rotation transformations are applied using multiplication by rotation matrices.
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Refer to the diagram below showing a system of linear equations represented by matrix \( A \) and vector \( b \). If \( \det(A) eq 0 \), which method can be used to find \( x \)?
A · Cramer's rule
Cramer's rule applies when determinant is non-zero to find unique solutions.
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Which of the following statements is TRUE about the inverse of a singular matrix?
B · It does not exist
Singular matrices have zero determinant and no inverse.
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Given \( A = \begin{bmatrix}4 & 7 \\ 2 & 6\end{bmatrix} \), find \( A^{-1} \) using the adjugate and determinant method.
A · \( \frac{1}{10} \begin{bmatrix}6 & -7 \\ -2 & 4\end{bmatrix} \)
Determinant is \( 4 \times 6 - 7 \times 2 = 10 \). Adjugate is \( \begin{bmatrix}6 & -7 \\ -2 & 4\end{bmatrix} \).
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Refer to the diagram below illustrating the stepwise calculation of \( A^{-1} \). Which step corresponds to finding the adjugate matrix?
C · Step 3: Transpose cofactor matrix
Adjugate matrix is the transpose of the cofactor matrix.
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In a mechanical system, the stiffness matrix \( K \) is invertible if:
B · \( \det(K) eq 0 \)
Invertibility requires non-zero determinant, ensuring unique displacement solutions.
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Which of the following statements about determinants is FALSE?
D · Determinant of the sum of two matrices equals the sum of their determinants
Determinant of sum is not equal to sum of determinants in general.
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Refer to the diagram below showing matrices \( A \) and \( B \). If \( A \) is invertible, which expression represents \( B \) in terms of \( A \) and \( C \) given \( AB = C \)?
A · \( B = A^{-1} C \)
Multiplying both sides by \( A^{-1} \) on the left gives \( B = A^{-1} C \).
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Which of the following statements correctly describes the relationship between the rank of a matrix and its invertibility?
B · Matrix is invertible if rank equals order
A square matrix is invertible if its rank equals its order (full rank).
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Refer to the diagram below showing a matrix and its determinant expansion along the second row. What is the sign of the cofactor corresponding to element \( a_{23} \)?
B · Negative
Cofactor sign is \( (-1)^{2+3} = -1 \), so negative.
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Which of the following is NOT a type of matrix based on its elements?
C · Orthogonal matrix
Orthogonal matrix is defined based on matrix multiplication properties (A\(A^T\) = I), not just element arrangement, unlike diagonal, symmetric, and skew-symmetric matrices.
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A matrix with all elements zero except the main diagonal elements is called a:
B · Diagonal matrix
A diagonal matrix has non-zero elements only on the main diagonal and zero elsewhere.
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If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \), what is \( A + B \)?
A · \( \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \)
Matrix addition is element-wise: add corresponding elements of A and B.
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Which of the following operations is NOT defined between two matrices \( A \) and \( B \) of order \( 2 \times 3 \)?
C · Multiplication
Matrix multiplication requires the number of columns of \( A \) to equal the number of rows of \( B \). For two \( 2 \times 3 \) matrices, multiplication is not defined.
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If \( A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \), what is \( 3A \)?
A · \( \begin{bmatrix} 6 & 0 \\ 3 & 9 \end{bmatrix} \)
Scalar multiplication multiplies each element by the scalar value.
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Which property of determinants states that swapping two rows changes the sign of the determinant?
B · Row interchange property
The row interchange property states that swapping two rows multiplies the determinant by \(-1\).
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The determinant of a triangular matrix (upper or lower) is equal to:
B · Product of diagonal elements
The determinant of a triangular matrix is the product of its diagonal elements.
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Calculate the determinant of \( A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \).
A · 10
Determinant of 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc \). Here, \(3 \times 4 - 2 \times 1 = 12 - 2 = 10\).
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \). What is the determinant of \( A \)?
A · 22
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If the determinant of a matrix \( A \) is zero, which of the following statements is TRUE?
B · Matrix \( A \) is singular
A matrix with zero determinant is singular and does not have an inverse.
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Which of the following is a necessary condition for a square matrix \( A \) to have an inverse?
C · Determinant of \( A \) is non-zero
A matrix is invertible if and only if its determinant is non-zero.
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Refer to the diagram below showing the steps to find the inverse of \( A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \) using the adjoint method. What is the value of \( \det(A) \)?
A · 10
Determinant \( = 4 \times 6 - 7 \times 2 = 24 - 14 = 10 \).
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Which special matrix satisfies \( A^T = -A \)?
B · Skew-symmetric matrix
A skew-symmetric matrix satisfies \( A^T = -A \).
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Which of the following matrices is both symmetric and diagonal?
B · \( \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)
A diagonal matrix has zero off-diagonal elements and is symmetric by default.
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If a matrix \( A \) has rank less than its order, which of the following is TRUE?
C · Determinant of \( A \) is zero
If rank is less than order, determinant is zero and matrix is singular.
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Refer to the diagram below showing a matrix \( A \) undergoing an elementary row operation: swapping row 1 and row 2. If \( \det(A) = 5 \), what is \( \det(A') \) after the operation?
B · -5
Swapping two rows changes the sign of the determinant.
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Which of the following statements about the determinant is FALSE?
C · Adding a multiple of one row to another changes the determinant
Adding a multiple of one row to another does NOT change the determinant.
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Which of the following matrices has an inverse?
C · \( \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)
Matrix C is diagonal with non-zero diagonal elements, so invertible. Others have zero determinant.
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 0 & 0 \\ 5 & 1 & 1 \end{bmatrix} \). Calculate the determinant of \( A \).
B · -1
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Which of the following is TRUE about the rank of a matrix?
B · Rank is the maximum number of linearly independent rows or columns
Rank is defined as the maximum number of linearly independent rows or columns.
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If \( A \) is a \( 3 \times 3 \) matrix with \( \det(A) = 4 \), what is \( \det(3A) \)?
B · 108
For an \( n \times n \) matrix, \( \det(kA) = k^n \det(A) \). Here, \( 3^3 \times 4 = 27 \times 4 = 108 \). So correct answer is 108, option B.
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 4 & 0 & 5 \end{bmatrix} \). What is the determinant of \( A \)?
A · 15
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Which of the following is TRUE about the inverse matrix \( A^{-1} \)?
B · \( A \times A^{-1} = I \)
By definition, \( A \times A^{-1} = I \), the identity matrix.
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Refer to the diagram below illustrating the adjoint method for finding \( A^{-1} \) of \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \). What is the adjoint matrix \( \text{adj}(A) \)?
A · \( \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \)
Adjoint matrix is the transpose of the cofactor matrix. Cofactors are \( C_{11} = 4, C_{12} = -3, C_{21} = -2, C_{22} = 1 \), so adjoint is \( \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \).
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Which of the following row operations multiplies the determinant by a scalar factor?
C · Multiplying a row by a non-zero scalar
Multiplying a row by scalar \( k \) multiplies the determinant by \( k \).
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If \( A \) is a \( 3 \times 3 \) matrix with \( \det(A) = 7 \), what is \( \det(A^{-1}) \)?
C · \( \frac{1}{7} \)
Determinant of inverse matrix is reciprocal of determinant: \( \det(A^{-1}) = \frac{1}{\det(A)} \).
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \). Find the inverse \( A^{-1} \) using the adjoint method.
A · \( \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \)
Inverse \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \).
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Which of the following matrices is skew-symmetric?
A · \( \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \)
A skew-symmetric matrix satisfies \( A^T = -A \), which is true for option A.
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If \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix} \), what is the rank of \( A \)?
C · 3
Matrix \( A \) is full rank (3) since its determinant is non-zero.
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Which of the following statements about elementary row operations and determinants is TRUE?
B · Swapping two rows multiplies determinant by -1
Swapping two rows changes the sign of the determinant.
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Which of the following matrices is an identity matrix?
A · \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
Identity matrix has 1s on the main diagonal and 0 elsewhere.
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If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), what is the inverse \( A^{-1} \)?
A · \( \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \)
Determinant \( = 2 \times 4 - 3 \times 1 = 8 - 3 = 5 \). Adjoint matrix is \( \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \). So inverse is \( \frac{1}{5} \) times adjoint.
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Which of the following is TRUE regarding the product of two matrices \( A \) and \( B \)?
B · Order of multiplication matters
Matrix multiplication is not commutative; order matters.
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \). If row 2 is replaced by row 2 minus 3 times row 1, what is the determinant of the new matrix?
D · 1
Adding a multiple of one row to another does not change the determinant.
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If \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \), then \( A \) is called:
B · Identity matrix
Matrix with 1s on the diagonal and zeros elsewhere is the identity matrix.
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Which of the following is NOT true about the determinant of a matrix?
C · Determinant is additive over matrix addition
Determinant is NOT additive over matrix addition.
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Refer to the diagram below showing matrix \( A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \). Calculate \( \det(A) \).
A · 49
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Which of the following matrices is singular?
B · \( \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \)
Matrix B has determinant zero (2*2 - 4*1 = 4 - 4 = 0), so it is singular.
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Which of the following is an application of determinants?
B · Solving system of linear equations
Determinants are used in Cramer's rule to solve linear equations.
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Refer to the diagram below showing a system of equations represented by matrix \( A \) and vector \( b \). If \( \det(A) = 0 \), what can be concluded about the system?
B · No solution or infinite solutions exist
Zero determinant means matrix is singular, so system has no unique solution.
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Which of the following is TRUE about the product of determinants of two square matrices \( A \) and \( B \)?
B · \( \det(AB) = \det(A) \times \det(B) \)
Determinant of product equals product of determinants.
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Refer to the diagram below showing the stepwise transformation of matrix \( A \) by multiplying row 1 by 3. If original \( \det(A) = 5 \), what is the determinant after this operation?
B · 15
Multiplying a row by scalar \( k \) multiplies determinant by \( k \). So determinant becomes \( 3 \times 5 = 15 \).
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What is the Laplace Transform of the function \( f(t) = 1 \)?
A · \( \frac{1}{s} \)
The Laplace Transform of a constant function \( f(t) = 1 \) is \( \mathcal{L}\{1\} = \frac{1}{s} \) for \( s > 0 \).
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Which of the following is NOT a property of the Laplace Transform?
D · Integration in the s-domain corresponds to differentiation in the t-domain
Integration in the s-domain corresponds to integration in the t-domain, not differentiation. Differentiation in the t-domain corresponds to multiplication by \( s \) in the s-domain.
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The Laplace Transform of \( f(t) = e^{at} \) is:
A · \( \frac{1}{s - a} \)
The Laplace Transform of \( e^{at} \) is \( \frac{1}{s - a} \) for \( s > a \).
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If \( \mathcal{L}\{f(t)\} = F(s) \), then the Laplace Transform of \( t f(t) \) is given by:
A · \( -\frac{d}{ds}F(s) \)
Multiplying the time function by \( t \) corresponds to differentiating its Laplace Transform with respect to \( s \) and taking the negative: \( \mathcal{L}\{t f(t)\} = -\frac{d}{ds}F(s) \).
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Which of the following is the correct Laplace Transform of the unit step function \( u(t - a) \)?
A · \( \frac{e^{-as}}{s} \)
The Laplace Transform of the unit step function \( u(t - a) \) is \( \frac{e^{-as}}{s} \).
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Find the inverse Laplace Transform of \( \frac{1}{s^2 + 4} \).
A · \( \sin 2t \)
The inverse Laplace Transform of \( \frac{1}{s^2 + a^2} \) is \( \frac{\sin at}{a} \). For \( a=2 \), it is \( \sin 2t \).
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Which method is commonly used to find the inverse Laplace Transform when \( F(s) \) is a rational function?
A · Partial Fraction Decomposition
Partial fraction decomposition is used to break down rational functions into simpler terms whose inverse Laplace Transforms are known.
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The Laplace Transform of \( f(t) = t^n \) where \( n \) is a positive integer is:
A · \( \frac{n!}{s^{n+1}} \)
The Laplace Transform of \( t^n \) is \( \frac{n!}{s^{n+1}} \) for \( s > 0 \).
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Using the linearity property, find \( \mathcal{L}\{3t + 5\} \).
A · \( \frac{3}{s^2} + \frac{5}{s} \)
By linearity, \( \mathcal{L}\{3t\} = \frac{3}{s^2} \) and \( \mathcal{L}\{5\} = \frac{5}{s} \). Sum is \( \frac{3}{s^2} + \frac{5}{s} \).
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Refer to the diagram below showing a block diagram of a control system with transfer function \( G(s) = \frac{5}{s+2} \). What is the Laplace Transform of the output \( Y(s) \) if the input \( R(s) = \frac{1}{s} \)?
A · \( \frac{5}{s(s+2)} \)
Output \( Y(s) = G(s) \times R(s) = \frac{5}{s+2} \times \frac{1}{s} = \frac{5}{s(s+2)} \).
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Find the inverse Laplace Transform of \( \frac{s+3}{(s+1)(s+2)} \) using partial fraction decomposition.
A · \( 2e^{-t} - e^{-2t} \)
Partial fractions: \( \frac{s+3}{(s+1)(s+2)} = \frac{2}{s+1} - \frac{1}{s+2} \). Inverse Laplace is \( 2e^{-t} - e^{-2t} \).
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Which theorem relates the initial value of \( f(t) \) to the limit of \( sF(s) \) as \( s \to \infty \)?
A · Initial Value Theorem
The Initial Value Theorem states \( \lim_{t \to 0^+} f(t) = \lim_{s \to \infty} sF(s) \).
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The Final Value Theorem can be applied only if:
A · All poles of \( sF(s) \) lie in the left half-plane except possibly at \( s=0 \)
Final Value Theorem requires all poles of \( sF(s) \) to be in the left half-plane except possibly at \( s=0 \) to ensure convergence.
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Solve the ODE \( \frac{dy}{dt} + 3y = 6 \) with initial condition \( y(0) = 0 \) using Laplace Transform. What is \( Y(s) \)?
A · \( \frac{6}{s(s+3)} \)
Taking Laplace: \( sY(s) - y(0) + 3Y(s) = \frac{6}{s} \) \( \Rightarrow (s+3)Y(s) = \frac{6}{s} \) \( \Rightarrow Y(s) = \frac{6}{s(s+3)} \).
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Refer to the graph below showing \( f(t) = e^{-2t}u(t) \). What is the Laplace Transform \( F(s) \)?
A · \( \frac{1}{s+2} \)
The Laplace Transform of \( e^{-at}u(t) \) is \( \frac{1}{s+a} \). Here, \( a=2 \).
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Which of the following is the Laplace Transform of the derivative \( \frac{df}{dt} \) given \( f(0) = f_0 \)?
A · \( sF(s) - f_0 \)
The Laplace Transform of \( \frac{df}{dt} \) is \( sF(s) - f(0) \).
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Find the inverse Laplace Transform of \( \frac{2s + 5}{s^2 + 4s + 5} \).
A · \( 2e^{-2t} \cos t + e^{-2t} \sin t \)
Denominator roots: \( s^2 + 4s + 5 = (s+2)^2 + 1 \). Numerator matches form \( A(s+a) + B \). Inverse is \( 2e^{-2t} \cos t + e^{-2t} \sin t \).
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Which of the following is the Laplace Transform of \( f(t) = t e^{3t} \)?
A · \( \frac{1}{(s - 3)^2} \)
Using the formula \( \mathcal{L}\{t e^{at}\} = \frac{1}{(s - a)^2} \), here \( a=3 \).
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Refer to the circuit diagram below consisting of a resistor \( R = 2\Omega \) and inductor \( L = 1H \) in series with input voltage \( V(t) \). What is the Laplace Transform of the circuit's impedance \( Z(s) \)?
A · \( 2 + s \)
Impedance of resistor is \( R = 2 \), inductor is \( sL = s \times 1 = s \). Total impedance \( Z(s) = 2 + s \).
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Using the shifting theorem, the Laplace Transform of \( f(t - a) u(t - a) \) is:
A · \( e^{-as} F(s) \)
The time shifting theorem states \( \mathcal{L}\{f(t - a) u(t - a)\} = e^{-as} F(s) \).
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Find the Laplace Transform of \( f(t) = \cos 3t \).
A · \( \frac{s}{s^2 + 9} \)
The Laplace Transform of \( \cos at \) is \( \frac{s}{s^2 + a^2} \). Here, \( a=3 \).
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Solve the ODE \( \frac{d^2y}{dt^2} - 4 \frac{dy}{dt} + 3y = 0 \) with \( y(0) = 2 \) and \( y'(0) = 0 \) using Laplace Transform. What is \( Y(s) \)?
A · \( \frac{2(s - 4)}{(s - 3)(s - 1)} \)
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Which of the following statements is TRUE about the Laplace Transform?
A · It converts differential equations into algebraic equations
Laplace Transform converts differential equations into algebraic equations in the s-domain, simplifying their solution.
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Refer to the diagram below showing a step input \( u(t) \) applied to an RC circuit with \( R = 1\Omega \) and \( C = 1F \). What is the Laplace Transform of the output voltage \( V_o(s) \)?
A · \( \frac{1}{s(s+1)} \)
Transfer function of RC circuit is \( \frac{1}{RC s + 1} = \frac{1}{s + 1} \). Input \( U(s) = \frac{1}{s} \). Output \( V_o(s) = \frac{1}{s} \times \frac{1}{s+1} = \frac{1}{s(s+1)} \).
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The inverse Laplace Transform of \( \frac{1}{s(s+2)} \) is:
A · \( \frac{1}{2} (1 - e^{-2t}) \)
Partial fractions: \( \frac{1}{s(s+2)} = \frac{1/2}{s} - \frac{1/2}{s+2} \). Inverse is \( \frac{1}{2} (1 - e^{-2t}) \).
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Which of the following is the Laplace Transform of the function \( f(t) = \delta(t - a) \), where \( \delta \) is the Dirac delta function?
A · \( e^{-as} \)
The Laplace Transform of \( \delta(t - a) \) is \( e^{-as} \).
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Using the initial value theorem, find \( \lim_{t \to 0^+} f(t) \) if \( F(s) = \frac{5}{s+3} \).
B · 5
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Which of the following is the Laplace Transform of \( f(t) = t^2 \)?
A · \( \frac{2}{s^3} \)
The Laplace Transform of \( t^n \) is \( \frac{n!}{s^{n+1}} \). For \( n=2 \), \( 2! = 2 \), so \( \frac{2}{s^3} \).
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Solve the ODE \( \frac{dy}{dt} + 2y = 4e^{-t} \) with \( y(0) = 1 \) using Laplace Transform. What is \( Y(s) \)?
A · \( \frac{4}{(s+1)(s+2)} + \frac{1}{s+2} \)
Taking Laplace: \( sY(s) - 1 + 2Y(s) = \frac{4}{s+1} \) \( \Rightarrow (s+2)Y(s) = \frac{4}{s+1} + 1 \) \( \Rightarrow Y(s) = \frac{4}{(s+1)(s+2)} + \frac{1}{s+2} \).
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Which of the following is TRUE about the linearity property of Laplace Transform?
A · \( \mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s) \)
Linearity means the Laplace Transform of a linear combination is the same linear combination of the transforms.
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Refer to the block diagram below of a feedback control system with forward path \( G(s) = \frac{10}{s+5} \) and feedback \( H(s) = 1 \). What is the closed-loop transfer function \( T(s) \)?
A · \( \frac{10}{s+15} \)
Closed-loop transfer function \( T(s) = \frac{G(s)}{1 + G(s)H(s)} = \frac{10/(s+5)}{1 + 10/(s+5)} = \frac{10}{s+5 + 10} = \frac{10}{s+15} \).
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Which of the following is the Laplace Transform of the integral \( \int_0^t f(\tau) d\tau \)?
A · \( \frac{F(s)}{s} \)
The Laplace Transform of the integral of \( f(t) \) is \( \frac{F(s)}{s} \).
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Find the inverse Laplace Transform of \( \frac{3}{(s+1)^2} \).
A · \( 3t e^{-t} \)
Inverse Laplace of \( \frac{1}{(s+a)^2} \) is \( t e^{-a t} \). Multiplying by 3 gives \( 3t e^{-t} \).
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Which of the following is TRUE about the final value theorem?
A · \( \lim_{t \to \infty} f(t) = \lim_{s \to 0} s F(s) \)
The final value theorem states \( \lim_{t \to \infty} f(t) = \lim_{s \to 0} s F(s) \), given poles conditions are met.
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Using partial fraction decomposition, express \( \frac{5s + 7}{(s+1)(s+3)} \) as:
A · \( \frac{1}{s+1} + \frac{4}{s+3} \)
Solving for A and B: \( 5s + 7 = A(s+3) + B(s+1) \). Substituting \( s = -1 \), \( A = 1 \); \( s = -3 \), \( B = 4 \).
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Find the Laplace Transform of \( f(t) = t^3 e^{-2t} \).
A · \( \frac{6}{(s+2)^4} \)
Laplace of \( t^n e^{-at} = \frac{n!}{(s+a)^{n+1}} \). For \( n=3 \), \( 3! = 6 \), so \( \frac{6}{(s+2)^4} \).
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Refer to the diagram below showing the graph of \( f(t) = t e^{-t} u(t) \). What is the Laplace Transform \( F(s) \)?
A · \( \frac{1}{(s+1)^2} \)
Laplace Transform of \( t e^{-a t} u(t) = \frac{1}{(s+a)^2} \). Here, \( a=1 \).
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Which of the following is the Laplace Transform of the convolution \( (f * g)(t) \)?
A · Product of Laplace Transforms \( F(s) G(s) \)
The Laplace Transform of the convolution of two functions is the product of their Laplace Transforms.
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Find the inverse Laplace Transform of \( \frac{1}{s^2 (s+1)} \) using partial fractions.
A · \( t - 1 + e^{-t} \)
Partial fraction decomposition: \( \frac{1}{s^2 (s+1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s+1} \). Solving gives inverse \( t - 1 + e^{-t} \).
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Which of the following is the correct definition of the Laplace transform of a function \( f(t) \)?
B · \( \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) dt \)
The Laplace transform is defined as \( \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) dt \), where \( s \) is a complex variable.
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Which property of Laplace transform states that \( \mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n}F(s) \), where \( F(s) = \mathcal{L}\{f(t)\} \)?
C · Differentiation in s-domain
The differentiation property in the s-domain states that multiplying \( f(t) \) by \( t^n \) corresponds to differentiating \( F(s) \) \( n \) times with respect to \( s \) with a factor \( (-1)^n \).
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If \( \mathcal{L}\{f(t)\} = F(s) \), which of the following is true for the linearity property of Laplace transform?
A · \( \mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s) \)
The Laplace transform is linear, so \( \mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s) \) where \( a, b \) are constants.
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Which of the following functions has the Laplace transform \( \frac{1}{s-a} \), where \( s > a \)?
A · \( e^{at} \)
The Laplace transform of \( e^{at} \) is \( \frac{1}{s-a} \) for \( s > a \).
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Refer to the diagram below showing the graph of \( f(t) = e^{-2t} \). What is the Laplace transform \( F(s) \) of this function?
A · \( \frac{1}{s+2} \)
The Laplace transform of \( e^{-at} \) is \( \frac{1}{s+a} \) for \( s > -a \). Here \( a=2 \), so \( F(s) = \frac{1}{s+2} \).
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Which method is commonly used to find the inverse Laplace transform of a rational function when the denominator can be factored into linear terms?
A · Partial fraction decomposition
Partial fraction decomposition is used to break down rational functions into simpler fractions whose inverse Laplace transforms are known.
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The inverse Laplace transform of \( \frac{s+3}{(s+1)(s+2)} \) can be found by partial fraction decomposition. Which of the following represents the correct decomposition?
A · \( \frac{A}{s+1} + \frac{B}{s+2} \) where \( A=2, B=1 \)
By equating \( \frac{s+3}{(s+1)(s+2)} = \frac{A}{s+1} + \frac{B}{s+2} \), solving gives \( A=2, B=1 \).
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Which of the following is the correct inverse Laplace transform of \( \frac{1}{s^2 + 4s + 5} \)?
A · \( e^{-2t} \sin t \)
Completing the square: \( s^2 + 4s + 5 = (s+2)^2 + 1 \). The inverse Laplace of \( \frac{1}{(s+a)^2 + b^2} \) is \( e^{-at} \sin bt \).
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Which theorem allows the Laplace transform of \( f(t-a)u(t-a) \) (where \( u(t) \) is the unit step function) to be expressed as \( e^{-as}F(s) \)?
B · Second shifting theorem
The second shifting theorem states that the Laplace transform of a delayed function \( f(t-a)u(t-a) \) is \( e^{-as}F(s) \).
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Which of the following is the initial value theorem for Laplace transforms?
A · \( \lim_{t \to 0^+} f(t) = \lim_{s \to \infty} sF(s) \)
The initial value theorem states that \( f(0^+) = \lim_{s \to \infty} sF(s) \), provided the limits exist.
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The convolution theorem states that the Laplace transform of the convolution of two functions \( f(t) * g(t) \) is equal to:
B · \( F(s) G(s) \)
The convolution theorem states \( \mathcal{L}\{f(t) * g(t)\} = F(s) G(s) \).
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Which of the following is the Laplace transform of \( \sin(at) \)?
A · \( \frac{a}{s^2 + a^2} \)
The Laplace transform of \( \sin(at) \) is \( \frac{a}{s^2 + a^2} \).
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Which of the following statements is true regarding the first shifting theorem in Laplace transforms?
A · If \( \mathcal{L}\{f(t)\} = F(s) \), then \( \mathcal{L}\{e^{at} f(t)\} = F(s-a) \)
The first shifting theorem states that multiplying \( f(t) \) by \( e^{at} \) shifts \( F(s) \) to \( F(s-a) \).
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Consider the ODE \( \frac{dy}{dt} + 3y = 6e^{-2t} \) with initial condition \( y(0) = 0 \). Using Laplace transform, what is the Laplace transform \( Y(s) \) of \( y(t) \)?
A · \( Y(s) = \frac{6}{(s+3)(s+2)} \)
Taking Laplace transform: \( sY(s) - y(0) + 3Y(s) = \frac{6}{s+2} \) \( \Rightarrow (s+3)Y(s) = \frac{6}{s+2} \) \( \Rightarrow Y(s) = \frac{6}{(s+3)(s+2)} \).
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Which of the following is the correct inverse Laplace transform of \( \frac{1}{s(s+1)} \)?
A · \( 1 - e^{-t} \)
Using partial fractions: \( \frac{1}{s(s+1)} = \frac{1}{s} - \frac{1}{s+1} \). Inverse Laplace gives \( 1 - e^{-t} \).
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Refer to the flowchart below illustrating the steps to solve an ODE using Laplace transform. Which step comes immediately after applying Laplace transform to the differential equation?
C · Apply initial conditions
After applying Laplace transform, initial conditions are applied to simplify the algebraic equation before solving for \( Y(s) \).
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Which of the following is true about the final value theorem for Laplace transforms?
A · \( \lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s) \), if poles of \( sF(s) \) lie in the left half-plane
The final value theorem holds if all poles of \( sF(s) \) are in the left half-plane (stable system).
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Which of the following is the Laplace transform of the unit step function \( u(t-a) \)?
A · \( \frac{e^{-as}}{s} \)
The Laplace transform of \( u(t-a) \) is \( \frac{e^{-as}}{s} \).
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Which of the following functions corresponds to the Laplace transform \( \frac{s}{(s+1)^2 + 4} \)?
A · \( e^{-t} \cos 2t \)
The Laplace transform of \( e^{-at} \cos bt \) is \( \frac{s+a}{(s+a)^2 + b^2} \). Here \( a=1, b=2 \).
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Which of the following is the Laplace transform of the derivative \( \frac{df}{dt} \) assuming \( f(0) = 0 \)?
A · \( sF(s) \)
The Laplace transform of \( \frac{df}{dt} \) is \( sF(s) - f(0) \). Given \( f(0) = 0 \), it is \( sF(s) \).
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Refer to the diagram below showing the convolution of two functions \( f(t) \) and \( g(t) \). Which of the following expressions represents the convolution \( (f * g)(t) \)?
C · Both A and B
Convolution is commutative, so both integrals represent the convolution \( (f * g)(t) \).
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Which of the following is the Laplace transform of the function \( t^n \) where \( n \) is a positive integer?
A · \( \frac{n!}{s^{n+1}} \)
The Laplace transform of \( t^n \) is \( \frac{n!}{s^{n+1}} \).
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Which of the following is the correct Laplace transform of the integral \( \int_0^t f(\tau) d\tau \)?
A · \( \frac{F(s)}{s} \)
The Laplace transform of the integral of \( f(t) \) is \( \frac{F(s)}{s} \).
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Which of the following is the inverse Laplace transform of \( \frac{2}{s^2 + 9} \)?
A · \( \frac{2}{3} \sin 3t \)
The inverse Laplace transform of \( \frac{a}{s^2 + a^2} \) is \( \sin at \). Here \( a=3 \), so the function is \( \frac{2}{3} \sin 3t \).
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Refer to the diagram below showing the graph of \( F(s) = \frac{1}{s^2 + 4} \). What is the corresponding time domain function \( f(t) \)?
A · \( \sin 2t \)
The Laplace transform \( \frac{1}{s^2 + a^2} \) corresponds to \( \sin at \). Here \( a=2 \).
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Which of the following is the Laplace transform of the function \( f(t) = t e^{3t} \)?
A · \( \frac{1}{(s-3)^2} \)
The Laplace transform of \( t e^{at} \) is \( \frac{1}{(s - a)^2} \). Here \( a=3 \).
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Which of the following is the correct Laplace transform of the function \( f(t) = \cosh(at) \)?
A · \( \frac{s}{s^2 - a^2} \)
The Laplace transform of \( \cosh(at) \) is \( \frac{s}{s^2 - a^2} \) for \( s > |a| \).
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Which of the following is the Laplace transform of the function \( f(t) = t^2 \)?
A · \( \frac{2}{s^3} \)
The Laplace transform of \( t^n \) is \( \frac{n!}{s^{n+1}} \). For \( n=2 \), \( 2! = 2 \), so \( \frac{2}{s^3} \).
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Which of the following is the correct inverse Laplace transform of \( \frac{1}{s(s+2)} \)?
A · \( \frac{1}{2} - \frac{1}{2} e^{-2t} \)
Using partial fractions: \( \frac{1}{s(s+2)} = \frac{A}{s} + \frac{B}{s+2} \), solving gives \( A=\frac{1}{2}, B=-\frac{1}{2} \). Inverse Laplace gives \( \frac{1}{2} - \frac{1}{2} e^{-2t} \).
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Which of the following is the Laplace transform of the function \( f(t) = e^{-3t} \sin 4t \)?
A · \( \frac{4}{(s+3)^2 + 16} \)
The Laplace transform of \( e^{-at} \sin bt \) is \( \frac{b}{(s+a)^2 + b^2} \). Here \( a=3, b=4 \).
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Refer to the block diagram below of a control system with transfer function \( G(s) = \frac{5}{s(s+2)} \). If the input is a unit step \( U(s) = \frac{1}{s} \), what is the Laplace transform of the output \( Y(s) \)?
A · \( \frac{5}{s^2 (s+2)} \)
Output \( Y(s) = G(s) U(s) = \frac{5}{s(s+2)} \times \frac{1}{s} = \frac{5}{s^2 (s+2)} \).
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Which of the following is the Laplace transform of the function \( f(t) = t e^{-2t} \)?
A · \( \frac{1}{(s+2)^2} \)
The Laplace transform of \( t e^{at} \) is \( \frac{1}{(s - a)^2} \). Here \( a = -2 \), so \( \frac{1}{(s+2)^2} \).
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Which of the following is the correct application of the initial value theorem to find \( f(0^+) \) if \( F(s) = \frac{5}{s+4} \)?
A · \( \lim_{s \to \infty} s \times \frac{5}{s+4} = 5 \)
Initial value theorem: \( f(0^+) = \lim_{s \to \infty} sF(s) = \lim_{s \to \infty} \frac{5s}{s+4} = 5 \).
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Which of the following is the Laplace transform of the function \( f(t) = u(t-3) \), where \( u(t) \) is the unit step function?
A · \( \frac{e^{-3s}}{s} \)
The Laplace transform of \( u(t-a) \) is \( \frac{e^{-as}}{s} \). Here \( a=3 \).
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Which of the following is the correct inverse Laplace transform of \( \frac{3s + 5}{(s+1)(s+2)} \)?
A · \( 2 e^{-t} + e^{-2t} \)
Partial fraction decomposition gives \( \frac{3s+5}{(s+1)(s+2)} = \frac{2}{s+1} + \frac{1}{s+2} \). Inverse Laplace gives \( 2 e^{-t} + e^{-2t} \).
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Refer to the diagram below showing the graph of \( f(t) = u(t-2) \). What is the Laplace transform of this function?
A · \( \frac{e^{-2s}}{s} \)
The Laplace transform of the delayed unit step function \( u(t-a) \) is \( \frac{e^{-as}}{s} \). Here \( a=2 \).
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Which of the following is the Laplace transform of \( f(t) = t^3 \)?
A · \( \frac{6}{s^4} \)
The Laplace transform of \( t^n \) is \( \frac{n!}{s^{n+1}} \). For \( n=3 \), \( 3! = 6 \), so \( \frac{6}{s^4} \).
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Which of the following is the Laplace transform of the function \( f(t) = e^{2t} \cos 3t \)?
A · \( \frac{s-2}{(s-2)^2 + 9} \)
The Laplace transform of \( e^{at} \cos bt \) is \( \frac{s - a}{(s - a)^2 + b^2} \). Here \( a=2, b=3 \).
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Which of the following is the correct Laplace transform of the function \( f(t) = \int_0^t e^{-\tau} d\tau \)?
A · \( \frac{1}{s(s+1)} \)
The integral \( \int_0^t e^{-\tau} d\tau = 1 - e^{-t} \). Laplace transform is \( \frac{1}{s} - \frac{1}{s+1} = \frac{1}{s(s+1)} \).
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For the function \(f(t) = e^{-2t} \sin(5t) u(t)\), find the Laplace transform \(F(s)\). Then, compute \(H(s) = \frac{d}{ds} \left( s F(s) \right)\). What is the inverse Laplace transform \(h(t)\)?
A · \(h(t) = -t e^{-2t} \sin(5t) u(t) + e^{-2t} \cos(5t) u(t)\)
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Given \(F(s) = \frac{s^2 + 4s + 5}{(s+1)^3}\), find the inverse Laplace transform \(f(t)\). Which of the following is correct?
B · \(f(t) = (t^2 + 2t + 1) e^{-t} u(t)\)
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A system is described by \(\frac{dy}{dt} + 2 y = \int_0^t e^{-3(t-\tau)} u(\tau) d\tau\) with zero initial conditions. Find \(Y(s)\) and determine \(y(t)\). Which is correct?
A · \(y(t) = \frac{1}{5} (e^{-2t} - e^{-3t}) u(t)\)
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Find the Laplace transform of \(f(t) = t e^{-t} \int_0^t e^{\tau} \sin(2 \tau) d\tau\). Which of the following is correct?
B · \(F(s) = \frac{2}{(s+1)^2 + 4} \cdot \frac{1}{(s+1)^2}\)
