Let L₁ be the line passing through the point P₁=(−3, −1, 1) with direction vector →d₁=[1, −2, −1]^T, and let L₂ be the line passing through the point P₂=(8, −3, 1) with direction vector →d₂=[−1, 0, 2]^T.
Find the shortest distance d between these two lines, and find a point Q₁ on L₁ and a point Q₂ on L₂ so that d(Q₁,Q₂) = d. Use the square root symbol '√' where needed to give an exact value for your answer.

Q₁ =
Q₂ =

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Convert to standard maximum form and apply two iterations of simplex process using slack form.

Maximize

2x1 -6x3

Subject to

x1 + x2 – x3 <= 7

3x1 – x2 >= 8

-x1 + 2x2 + 2x3 >= -2

x2, x3 >=0

Please write the answer very clearly.

Find general solutions of the following systems using undetermined coefficients.

X′ =(2x2 matrix) ( 2 2; 3 1 ) X + (column matrix) ( e^−4t; 0 )

1. Find general solutions of the following systems using undetermined coefficients.

a.) x′ = −5x + 6y + 1

y′ = −7y + t .

b.) x′ = 6x − 5y + e^5t

y′ = x + 4y .

c.) x′ = −6x − 3y + te^2t

y′ = 4x + y .

Consider the function, ? (?) = 20?² + 28? - 17

Determine both real roots of the quadratic equation as follows:

i. Graphically and determine the intervals for both roots to be used in iii & iv.

ii. Using the quadratic formula

iii. Using the bisection method until ?? is less than ?? = 0.5%

iv. Using the false-position method until ?? is less than ?? = 0.5%

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show that A or is a ring with division or
A is a ring with a prime number of elements in which a · b = 0 for any, b ∈ A.

Approximately how many flops are needed to find the LU factorization of an n x n matrix using Doolittle’s method? If a computer requires 1 second to find an LU factorization of a 500 x 500 matrix, what would you estimate is the largest matrix that could be factored in less than 1 hour?

For each of the following statements, determine whether it is true or false and justify your answer.
a. If the function f + g: IR --> IR is continuous, then the functions f :IR --> IR and g :IR --> IR also are continuous.
b. If the function f^2 : IR --> R is continuous, then so is the function f :R --> IR.
c. If the functions f + g: IR and g: IR --> IR are continuous, then so is the function f :IR --> IR.
d. Every function f: N --> IR is continuous, where N denotes the set of natural numbers.

For each of the following statements, determine whether it is true or false and justify your answer. a. Every function f : [0, 1] ~ lR has a maximum. b. Every continuous function f :[a, b] ~ lR has a minimum. c. Every continuous function f : (0, 1) ~ lR has a maximum. d. Every continuous function f : (0, 1) ~ lR has a bounded image. e. If the image of the continuous function f: (0, 1) ~ lR is bounded below, then the function has a minimum.

Humid air at 155 kPa, 40°C, and 70 percent relative humidity is cooled at constant pressure in a pipe to its dew-point temperature. Calculate the heat transfer, in kJ/kg dry air, required for this process. Use data from the tables.

The heat transfer is  kJ/kg dry air.

Prove the following:

(a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal.

(b) If A/J is a field, then J is a maximal ideal.

2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .