Engineering Mathematics & Analytical Skills Flashcards

The general solution of the second-order ODE y'' - 4y = 0 is:

Answer: y = C₁e^(2x) + C₂e^(-2x)

The characteristic equation r² - 4 = 0 gives r = ±2, leading to y = C₁e^(2x) + C₂e^(-2x).

Stokes' theorem relates a surface integral of the curl to:

Answer: A line integral over the boundary curve of the surface

Stokes' theorem states ∬(∇×F)·dS = ∮F·dr, converting a surface integral of curl to a line integral around the boundary.

If A is an n×n invertible matrix, which statement is FALSE?

Answer: A has n linearly independent eigenvectors

An invertible matrix need not have n linearly independent eigenvectors; defective matrices are invertible but not diagonalizable.

The Cauchy-Schwarz inequality for vectors u and v states:

Answer: |u·v| ≤ |u||v|

The Cauchy-Schwarz inequality states |u·v| ≤ ‖u‖‖v‖, with equality when u and v are parallel.

In the method of least squares for curve fitting, the objective is to minimize:

Answer: The sum of squared residuals

Least squares minimizes Σ(yᵢ - ŷᵢ)², the sum of squared differences between observed and fitted values.

For a discrete random variable X with E[X] = μ, the variance Var(X) is defined as:

Answer: E[(X-μ)²]

Variance = E[(X-μ)²] = E[X²] - μ², measuring the expected squared deviation from the mean.

The Taylor series expansion of cos(x) about x=0 up to x⁴ is:

Answer: 1 - x²/2! + x⁴/4!

cos(x) = 1 - x²/2! + x⁴/4! - ..., where 2!=2 and 4!=24, so the terms match option B.