Advanced Mathematics

1. Determine whether the line x = 3+8t, y = 4+ 5t, z=-3-1 is parallel to the plane x-3y + 5z = 12.

Problem 5 Classify (local and global) the stationary points of f: R² → R, ƒ(x) = x¹Qx+b¹x where a/2 Q = (a/2 °1²) = and b = (1, 2) for a € R. Consider all possibilities depending on a.

Consider the One-dimensional Wave Equation for vibrations on a string of length L with a free end and linear damping.

2. Explain why f(x) = x + 1 is not a linear transformation from R → R.

3. Find the standard matrix of the linear transformation from R² → R²: (a) Reflection across the x-axis. (b) Clockwise rotation by π/2 radians.

5. For the given periodic signal, a) find trigonometric Fourier Series coefficients. (ao, an's and be 's) b) Find Exponential Fourier Series Coefficients. (Dn's) c) Find the amplitudes and phases of (Co. C₁. C₁, C₁, C₁, and Cs).

2. In how many ways can two red and four blue rooks be placed on an 8-by-8 chessboard so that no two rooks can attack one another?

Let A1 be X and Y sets. Furthermore, let p(x, y) and q (x, y) be properties that are valid for all x € X and for all y € Y are defined and may or may not be fulfilled. Negate the statements and give the result in such a way that negations (if available) are only directly in front of the properties p(x, y) and q(x, y). (a) Vx € X 3y € Y: -p(x, y) (b) Vx € X 3y € Y : p(x, y) Aq(x, y) (c) Vx € X:-(3y € Y : p(x, y))

A 4 In the student flat of Alan, Kurt, Sophie, Ada and Évariste is one morning the otherwise so abundantly filled refrigerator is empty. The WG meets for a crisis- and he wants to expose the wrongdoer(s). Every resident comments on incident. However, the statements should be treated with caution: all residents lie. The negation of the statements is true. Alan: 'That wasn't one of us." Kurt: ,, It wasn't me and neither was Sophie." Sophie: ,,If Kurt was there, then so was Evariste." Ada: ,,It was either Sophie or it was Kurt." Évariste: ,,That was Alan or Ada." Who robbed the refrigerator? Note: First, formalize the individual statements with appropriate information- gevariables and junctors.

11. A footrace takes place among four runners. If ties are allowed (even all four runners finishing at the same time), how many ways are there for the race to finish?