SCE Engineering Mathematics 1 â Questions and Answers
Question 1: What is the Laplace transform of the function f(t) = e^(-at)?
- 1/(s+a) (Correct answer)
- 1/(s-a)
- a/(s+a)
- s/(s+a)
Correct answer: 1/(s+a)
The Laplace transform of e^(-at) is 1/(s+a) for s > -a, which is a fundamental transform used extensively in control systems and circuit analysis.
Question 2: A matrix A has eigenvalues 2 and 5. What is the determinant of A?
- 10 (Correct answer)
- 7
- 3
- 25
Correct answer: 10
The determinant of a matrix equals the product of its eigenvalues. Therefore det(A) = 2 Ă 5 = 10.
Question 3: Which numerical method uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n)?
- Newton-Raphson method (Correct answer)
- Bisection method
- Secant method
- Euler's method
Correct answer: Newton-Raphson method
The Newton-Raphson method iteratively refines a root estimate using the function value and its derivative, converging quadratically near simple roots.
Question 4: What is the divergence of the vector field F = (x², xy, xz)?
- 2x + x + x = 4x (Correct answer)
- 2x + y + z
- x² + xy + xz
- 2x + x + x
Correct answer: 2x + x + x = 4x
Divergence is âFâ/âx + âFâ/ây + âFâ/âz = 2x + x + x = 4x. This measures the rate of flux expansion at a point.
Question 5: For the differential equation dy/dx + 2y = e^(-x), what is the integrating factor?
- e^(2x) (Correct answer)
- e^(-2x)
- 2x
- e^(x)
Correct answer: e^(2x)
For a first-order linear ODE dy/dx + P(x)y = Q(x), the integrating factor is e^(âŤP(x)dx) = e^(âŤ2dx) = e^(2x).
Question 6: What does the Fourier series represent?
- A periodic function as a sum of sines and cosines (Correct answer)
- A polynomial approximation of any function
- The Taylor expansion around zero
- A matrix decomposition technique
Correct answer: A periodic function as a sum of sines and cosines
A Fourier series decomposes a periodic function into a sum of sine and cosine terms (harmonics), essential in signal processing and vibration analysis.
SCE Engineering Mathematics 1
What is the Laplace transform of the function f(t) = e^(-at)?