Solve the differential equation by variation of parameters, subject to the initial conditions

y(0) = 1, y'(0) = 0.

y'' + 2y' − 8y = 4e^−3x − e^−x

1. A six-sided die is weighted so that all even numbers have an equal chance of coming up when the die is rolled, all odd numbers have an equal chance of coming up, and each even number is three times as likely to come up as each odd number. This die is rolled once.

What is the probability of rolling a 3?

What is the probability of rolling a 6?

What is the probability of rolling an even number?

2. An eight-sided die is weighted so that all even numbers have an equal chance of coming up when the die is rolled, all odd numbers have an equal chance of coming up, and each odd number is eight times as likely to come up as each even number. This die is rolled once.

What is the probability of rolling the number 5?
What is the probability of rolling the number 4?
What is the probability of rolling a number greater than 5?
Enter each answer as a whole number or a fraction in lowest terms.

3.A gumball machine contains 45 gumballs. Some are purple and the rest are yellow. There are 4 times as many purple gumballs as yellow. Because the purple gumballs are slightly smaller than the yellow, each purple gumball is 3 times as likely to be dispensed as each yellow gumball.
An experiment consists of the machine dispensing one gumball. Let each gumball be considered one outcome.

What weight should be assigned to each purple gumball?
What is the probability of the event that a yellow gumball is dispensed?
Enter your answers as whole numbers or fractions in lowest terms.

2. Find the general solution to the differential equation x^2y''+ y'+y = 0 using the Method of Frobenius and power series techniques.

1. Find the :

(a) Inverse using explicit Gauss-Jordan elimination

(b) The eigenvalues

(c) Respective eigenvectors

(2 0 -2

0 4 0

-2 0 5)

Determine whether the polynomials form a basis for P3:

1 − 2t ^2 , t + 2t^3 , 1 − t + 2t^2

1. Suppose A is (10, 2, 5, 9, 1, 8, 2, 4). Consider the function:

int BBOX(int n, int k)

            if (n <= 0) return 0;

            else if (A[n] < k) return (1+ 2*BBOX(n-1,k+1));

            else return BBOX(n-1,k-2);

            Find BBOX(8, 5)

Use Newton's method to find a solution for the equation in the given interval. Round your answer to the nearest thousandths. ? 3 ? −? = −? + 4; [2, 3] [5 marks] Answer 2.680

Q6. Use the Taylor Polynomial of degree 4 for ln(1 − 4?)to approximate the value of ln(2). Answer: −4? − 8?2 − 64 3 ? 3 − [6 marks]

Q7. Consider the curve defined by the equation 2(x2 + y2 ) 2 = 25(x2 − y2 ). Find the equation of the line tangent to the curve at the point(3, 1). [5 marks]

1. Find the solution to the given linear systems by Jacobi and Gauss Seidel iteration methods.

          2x + 5y = 16                      20x + y – 2z = 17            5x – y +2z = 12

          3x + y = 11                        3x +20y – z = -18           3x +8y -2z = -25

                                                      2x – 3y +20z = 25            x + y +4z = 6

2. Solve the equation Ax = b by using the LU decomposition method given the following linear systems of equations:

a. 3x – 7y -2z = -7

-3x +5y + z = 5

6x – 4y      = 2

b. 2x – y +2z = 1

-6x     -2z = 0

8x – y+5z = 4

Consider the given matrix.

Find the eigenvalues. (Enter your answers as a comma-separated list.)

λ = 1,2,3

Find the eigenvectors. (Enter your answers in order of the corresponding eigenvalues, from smallest eigenvalue to largest.)

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical point of the function f(x, y) = (90)x 2 + (0)xy + (90)y 2 + (−72)x + (96)y + (40), and Q3 = 1 if f has a local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at (Q1, Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4 otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then T = 5 sin2 (100Q)

satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T < 3. — (D) 3 ≤ T < 4. — (E) 4 ≤ T ≤ 5

General Directions: Show all necessary supporting work and box in each answer. If a blank is provided, state the answer in the blank.

Use dimensional analysis to convert. If unable to state an exact answer, round to the nearest tenth.

1. 0.35 mi. =                           yd.                                          2. 182 ft =                              in.

3.143 ¾ in (fraction 3 over 4) in. =             cm                                          4. 5,237,833 in = _____ mi

Consider the “metric staircase” to convert to the specified unit. If unable to state an exact answer, round to the nearest tenth.

5. 24 hm =                              dam                                        6. 18,426 mm =                                  m

7. 650,000 mm =                                km                               8. 50,000 cm =                                     km

Theorem 3.4. Let a and b be integers, not both zero, and suppose that b = aq + r

for some integers q and r. Then gcd(b, a) = gcd(a, r).

a) Suppose that for some integer k > d, k | a and k | r. Show that k | b also. Deduce that k is a common divisor of b and a.

b) Explain how part (a) contradicts the assumption that d = gcd(b, a).

Consider the two savings plans below. Compare the balances in each plan after

1111

years. Which person deposited more money in the​ plan? Which of the two investment strategies is​ better?

YolandaYolanda

deposits

​$450450

per month in an account with an APR of

55​%,

while

ZachZach

deposits

$ 5000$5000

at the end of each year in an account with an APR of

5.55.5​%.

The balance in

YolandaYolanda​'s

saving plan after

1111

years was ​$

.

​(Round the final answer to the nearest cent as needed. Round all intermediate values to seven decimal places as​ needed.)

The following kets name vectors in the Euclidean plane:

|a>, |b>, |c>.

Some inner products: <a|a> = 1, <a|b> = −1, <a|c> = 0, <b|c> = 1, <c|c> = 1

(a) Which of the kets are normalized?

(b) Which of these are an orthonormal basis?

(c) Write the other ket as a superposition of the two basis kets. What is the norm |h·|·i| of this ket (i.e., the length of the vector)? What is the angle between this ket and the two basis kets?

(d) In the same basis, write as a superposition a ket that has the same direction but is normalized.

^{The number of new businesses established in the US since 1990 can be modeled by the function Nx=110.8x^3-5305.5x^2+76,701x+332,892 where x = 0 represents 1990 and the domain is [0, 25].}      

1) What was the average rate of change in the number of new businesses established between 2000 and 2010? Don’t forget to label and interpret the answers.