# Upper-Level Maths Questions & Answers

Recent questions in Upper Level Math
Meera Sunker 2022-05-20

### 1a.  Show from first principles, i.e., by using the definition of linear independence,that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrixA with associated eigenvector v = u + iw, then the two real solutionsY(t) = eat(u cos bt − wsin bt)andZ(t) = eat(u sin bt + wcos bt)are linearly independent solutions of ˙X = AX. 1b. Use (a) to solve the system (see image)

Meera Sunker 2022-05-20

### Identify each of the following statements as true or false in relation to confidence intervals (CIs). Note: 0.5 marks will be taken away for each incorrect answer. The minimum score is 0.    TrueFalseA 95% CI is a numerical interval within which we are 95% confident that the true mean μ$\mu$ lies.  A 95% CI is a numerical interval within which we are 95% confident that the sample mean x¯¯¯$\overline{x}$ lies.  The true mean μ$\mu$ is always inside the corresponding confidence interval.  For a sample size n=29$n=29$, the number of degrees of freedom is n=30$n=30$.  If we repeat an experiment 100 times (with 100 different samples) and construct a 95% CI each time, then approximately 5 of those 100 CIs would not$not$ contain the true mean 𝜇.

Niraj Prajapati 2022-04-15

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Jane 2022-03-26

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