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Solve the inequalities. Give your answer in interval notation, and indicate the answer geometrically on the real-number line. a. t + 6 ≤ 2 + 3t
b. 3(2 – 3x) > 4(1 – 4x)
In the following problems, perform the operations and simplify as much as possible.
a. (x2 + 2x)/(3x2 – 18x + 24) ÷ (x2 – x – 6)/(x 2 – 4x + 4)
b. (x2 + 6x + 9)/x/(x + 3)
c. 1/(3x – 1) + x/(x2 – 9)
Let the function f : R → R and g : R → R be defined by f(x) 2x + 3 and g(x) = -3x + 5.
a. Show that f is one-to-one and onto.
b. Show that g is one-to-one and onto.
c. Determine the composition function g o f
d. Determine the inverse functions f -1 and g -1 .
e. Determine the inverse function (g o f) -1 of g o f and the composite f -1 o g -1 .
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 matrix
A with associated eigenvector v = u + iw, then the two real solutions
Y(t) = eat(u cos bt − wsin bt)
and
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX
1b.Use (a) to solve the system
˙X =
(
3 1
−8 7
)
X.
NB: Real solutions are required.
Reduce the system
(D2 + 1)[x] − 2D[y] = 2t
(2D − 1)[x] + (D − 2)[y] = 7.
to an equivalent triangular system of the form
P1(D)[y] = f1(t)
P2(D)[x] + P3(D)[y] = f2(t)
and solve.
Given the following quadratic form involving three variables,
Q (x1, x2, x3) = 5x21 + 8x1x3 + 3x22 - 6x2x3 + 10x23
a. Derive the symmetric matrix associated with Q
b. Determine the definiteness of the matrix you derived in a
- Telephone manufacturers now offer 1000 different choices for a telephone (as combinations of color, type, options, portability, etc.). A company is opening a large regional office, and each of its 200 managers is allowed to order his or her own choice of telephone. Assuming independence of choices and that each of the 1000 choices is equally likely, what is the probability that a particular choice will be made by none, one, two, or three of the managers?
- The mean number of power outages in the city is four (4) per year. Find the probability that in a given year, there are exactly three (3) outages.
- If a student randomly guesses at five multiple-choice questions, find the probability that the student gets exactly three correct. Each question has five possible choices.
- Say 40% of the class is female. What is the probability that 6 of the first 10 students walking in will be female?
Which of the following is the power set of the set S = {a, b}?
Find Larange’s interpolating polynomial passing through set of points
(0,2) (2,-2),(3,-1),Use it to find
at x = 2
