EIT Mathematics and Statistics 3

Question 1

The eigenvalues of the matrix [[4,1],[2,3]] are:

Accepted Answer

5 and 2

Answer

The characteristic equation is (4-λ)(3-λ) - 2 = λ² - 7λ + 10 = 0, giving λ = 5 and λ = 2.

Question 2

If f(x) = x³ - 6x² + 9x, at which x-value does a local minimum occur?

Accepted Answer

x = 3

Answer

f'(x) = 3x² - 12x + 9 = 0 gives x = 1 (local max) and x = 3 (local min), verified by the second derivative test.

Question 3

In hypothesis testing, a Type I error is defined as:

Accepted Answer

Rejecting a true null hypothesis

Answer

A Type I error (false positive) occurs when a true null hypothesis is incorrectly rejected; its probability equals α.

Question 4

What is the value of the integral ∫₀^π sin(x) dx?

Accepted Answer

2

Answer

∫₀^π sin(x) dx = [-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.

Question 5

The dot product of vectors A = (3, 4) and B = (1, -2) is:

Accepted Answer

-5

Answer

A · B = (3)(1) + (4)(-2) = 3 - 8 = -5.

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