EIT Mathematics and Statistics 3 — Questions and Answers
Question 1: The eigenvalues of the matrix [[4,1],[2,3]] are:
- 5 and 2 (Correct answer)
- 4 and 3
- 6 and 1
- 7 and 0
Correct answer: 5 and 2
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?
- x = 1
- x = 2
- x = 3 (Correct answer)
- x = 0
Correct answer: x = 3
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:
- Failing to reject a false null hypothesis
- Rejecting a true null hypothesis (Correct answer)
- Accepting a false alternative hypothesis
- Setting α too small
Correct answer: Rejecting a true null hypothesis
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?
- 0
- 1
- 2 (Correct answer)
- π
Correct answer: 2
∫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:
- -5 (Correct answer)
- 5
- 3
- -3
Correct answer: -5
A · B = (3)(1) + (4)(-2) = 3 - 8 = -5.
Question 6: Which of the following describes a Poisson distribution?
- Number of successes in a fixed number of trials
- Number of events in a fixed time interval given a known average rate (Correct answer)
- Continuous probability between two limits
- Probability of exactly one success per trial
Correct answer: Number of events in a fixed time interval given a known average rate
The Poisson distribution models the number of rare, independent events occurring in a fixed interval at a constant mean rate λ.
Question 7: Using Simpson's 1/3 rule with n=2 intervals to approximate ∫₀^2 x² dx (h=1), the result is:
- 2.67 (Correct answer)
- 2.33
- 3.00
- 2.50
Correct answer: 2.67
Simpson's rule gives (h/3)[f(0) + 4f(1) + f(2)] = (1/3)[0 + 4 + 4] = 8/3 ≈ 2.67, matching the exact answer.
EIT Mathematics and Statistics 3
The eigenvalues of the matrix [[4,1],[2,3]] are: