Advanced Mathematics

1. Let a, b, c be integers. Suppose that a divides bc and that (a, b) = 1. Prove that a divides c

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

Consider the following One-Dimensional heat Equation for heat flow in a rod of length L:

1. Let T: R¹ → R³ be the linear transformation given by the formula? (a) What is the standard matrix of T? (b) Find all vectors in Rª such that T(7) =

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.

Q 11 Use the graph to find: (a) The numbers, if any, at which f has a local maximum. What are these local maxima? (b) The numbers, if any, at which f has a local minimum. What are these local minima?

Problem 2 A line goes from point A (X=1800.00', Y=1900.00'), to point B (X=2100', Y=-2000.00' *). What are the coordinates of a point C that was established by turning a left angle a=65° from AB, and a right angle ß=46° from BA? Show all calculations. (* Note the minus)

Assume a continuous function f (x) defined on x axis. Consider these points A, B, C with co-ordinates xA, xB and xC. however, these three points are not

2. Where on the path r(t) = (t²-5t)i + (21+1)] +38² k vectors orthogonal? are the velocity and acceleration