EIT Mathematics and Statistics 2 — Questions and Answers
Question 1: What is ∫(2x + 5)dx?
- x² + 5x + C (Correct answer)
- 2x² + 5x + C
- x² + 5 + C
- 2x + C
Correct answer: x² + 5x + C
Integrating term by term: ∫2x dx = x² and ∫5 dx = 5x, giving x² + 5x + C.
Question 2: The eigenvalues of the matrix [[3,1],[0,2]] are:
- 1 and 2
- 3 and 2 (Correct answer)
- 3 and 1
- 2 and 4
Correct answer: 3 and 2
For an upper triangular matrix, eigenvalues are the diagonal entries: 3 and 2.
Question 3: What does the Central Limit Theorem state?
- All distributions are normal
- The sample mean approaches a normal distribution as sample size increases (Correct answer)
- Variance decreases with sample size
- The population must be normal for statistics to work
Correct answer: The sample mean approaches a normal distribution as sample size increases
The Central Limit Theorem states that sample means approach a normal distribution as sample size grows, regardless of the population distribution.
Question 4: What is the Taylor series expansion of e^x about x=0?
- 1 + x + x²/2 + x³/6 + ... (Correct answer)
- 1 + x + x² + x³ + ...
- x + x²/2 + x³/6 + ...
- 1 - x + x²/2 - x³/6 + ...
Correct answer: 1 + x + x²/2 + x³/6 + ...
The Taylor series of e^x = Σ(xⁿ/n!) = 1 + x + x²/2! + x³/3! + ...
Question 5: If two random variables X and Y have a correlation coefficient of 0, they are:
- Perfectly correlated
- Perfectly anticorrelated
- Uncorrelated (but not necessarily independent) (Correct answer)
- Identical
Correct answer: Uncorrelated (but not necessarily independent)
A correlation of 0 means no linear relationship exists, but the variables could still have nonlinear dependence.
Question 6: The gradient of a scalar function f(x,y) = x² + y² is:
- 2x + 2y
- (2x, 2y) (Correct answer)
- 2xy
- x² + y²
Correct answer: (2x, 2y)
The gradient ∇f = (∂f/∂x, ∂f/∂y) = (2x, 2y) is a vector pointing in the direction of steepest ascent.
EIT Mathematics and Statistics 2
What is ∫(2x + 5)dx?