Credit if these directions aren't followed. for full credit show all work " i the answer in a box is not easily seen. you will not get problems from book (kreyszig, advanced engineering mathematics) (20 pts total) linear algebra: a) (4 pts) write the 2 x 2 identity matrix 12 22 b) (4 pts) find the determinant of the 2 x 2 matrix d = c) (4 pts) write the system of linear equations given below in matrix form ax=b 5x2 + 2.5x2 - 1x3 = 2.5 10x1 - 20x2 + 5x3 = 50 3x1 + 7x2 + 8xz = 10 d) (4 pts) if a system of equations can be written in the form ax=1x where a is a square matrix of size nxn, x is a vector of length n, and is a scalar, either real or complex then: x is called an: (circle one) eigenvector eigenvalue eigenspace idk 1 is called an: (circle one) eigenvector eigenvalue eigenspace idk . annel
2) (20 pts total) laplace transformations a) (5 pts) true or false (circle one): the laplace transform f(s) of a function f(t) is defined as: f(c) = | e-stf(t)dt jteo b) (6 pts) what is the laplace transform of f(t) = eat
c) (5 pts) which of the following is not a property of the laplace transform? linearity integration in time f(t)dt = f(s)/s multiplication f(t)g(t) + f(t)g(t) frequency shifting: eat f(t) + f(s -a)
3) (20 pts total) vector calculus: a) (8 pts) the gradient of a function f(x, y,z) written as vf or grad f is found as: sf sf. sf vf = x + 5y + ezk where is the unit vector in the x-direction, is the unit vector in the y direction, and unit vector in the 2-direction. what is vf when f = 4xy3 - 2y*z?
cuuu very uuu cenu 4) (20 pts total) complex numbers a) (2 pts) which form is better for adding complex numbers, rectangular or polar? b) (6 pts) add z = 2ers and z2 = 5e" together. write your result in both rectangular and polar form indicate if your angle is in degrees or radians. c) (2 pts) which form is better for multiplying complex numbers, rectangular or polar? d) (6 pts) multiply z = -5 + 2 and z = 3 + 9 together. write your result in both rectangular and polar form indicate if your angle is in degrees or radians.
(20 pts total) probability and combinatorics: rs: the last 3 are a) (8 pts) assume a license plate has 6 characters; the first 3 are capital letters, the las decimal numbers. how many possible different license plates are there? b) (8 pts) given the license plate structure above, what is the probability of having the character combination aaa 123?