EIT Mathematics and Statistics 5

Question 1

The Fourier series of a periodic function requires which condition on the function over its period?

Accepted Answer

The function must satisfy Dirichlet conditions (bounded, finite extrema, finite discontinuities)

Answer

Dirichlet conditions — bounded variation, finite number of maxima/minima and discontinuities — are sufficient for Fourier series convergence.

Question 2

For a binomial distribution with n = 10 trials and p = 0.3, the mean number of successes is:

Accepted Answer

3.0

Answer

The mean of a binomial distribution is μ = np = 10 × 0.3 = 3.0.

Question 3

What is the rank of the matrix [[1,2,3],[2,4,6],[0,0,0]]?

Accepted Answer

1

Answer

Row 2 is twice row 1, and row 3 is zero, so only one row is linearly independent, giving rank = 1.

Question 4

The trapezoidal rule for numerical integration approximates the area under a curve as a series of:

Accepted Answer

Trapezoids connecting consecutive data points

Answer

The trapezoidal rule connects consecutive function values with straight lines, forming trapezoids whose areas are summed.

Question 5

The gradient of f(x,y) = x²y + 3y² evaluated at (1, 2) is:

Accepted Answer

(4, 13)

Answer

∂f/∂x = 2xy = 4, ∂f/∂y = x² + 6y = 1 + 12 = 13, so ∇f(1,2) = (4, 13).

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