1- Number and describe at least five (5) factors that determine your credit score ?

2-Explain why having a strong credit score is critically important?

1. Let G be the symmetry group of a square and let H be the subgroup generated by a rotation by 180 degrees. Find all left H-cosets.

9. For the differential equation: ?′′ + 2?′ + 5? = ?(?)

(a) Write down the auxiliary polynomial associated to this differential equation and find the roots of the auxiliary polynomial.

(b) Find a basis for the real solution set of the associated homogeneous differential equation ?′′ + 2?′ + 5? = 0.

(c) For each ?(?) that follows, write down the trial particular solution to ?′′ + 2?′ + 5? = ?(?), with undetermined coefficients. Do NOT solve the constants that arise in your trial solution.

(i) ?(?) = 120?^−?

(ii) ??(??) = 68 cos(2?)

(iii) ??(??) = 3?

(iv) ??(??) = 75?^−? sin(2?)

(d) Find a particular solution to ?′′ + 2?′ + 5? = 100 cos(2?) (show your work)

Write the following regarding solutions of a system
→ x′= A → x: 1. The definition of a Fundamental Matrix Φ(t), and apply it in one example of your choosing (for a constant matrix A that you pick). 2. On your example in part 1. calculate the function y(t) = detΦ(t) (the determinant of Φ(t) and check that y(t) solves the first order ODE y′(t) = (trA)y(t) (the trace (trA) is the sum of the entries on the main diagonal of A). 3. Choose a constant 2×2 matrix A different from the one used before. By solving the ODE y′(t) = (trA)y(t) conclude that y(t) is either identically zero, or is never zero.

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0.

a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???( (h(?)).

b) Prove that congruence classes modulo h(?) are either disjoint or identical.

Let (s_n) be a sequence that converges.

(a) Show that if s_n ≥ a for all but finitely many n, then lim s_n ≥ a.

(b) Show that if s_n ≤ b for all but finitely many n, then lim s_n ≤ b.

(c) Conclude that if all but finitely many sn belong to [a,b], then lim s_n belongs to [a, b].

Prove the Weierstrass M-test for uniform convergence of Series of Functions.

What real world applications exist for Euler Circuits?

Matlab project:

Solve using Matlab three problems:

One using the combination formula

One using the permutation of n objects formula

One using the permutation of r objects out of n objects

You can pick these problems from the textbook or you can make up your own questions.

A detailed answer will be appreciate.

6. To prove that for all x1, x2, ..., x9 ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there exists a

value of x10 for the check digit in the code ISBN-10.

7. To prove that for every x1, x2, ..., x12 ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there exists a

value of x13 for the check digit in the code ISBN-13.

Solve this Initial Value Problem using the Laplace transform.

x''(t) + 6 x'(t) + 25x(t) = cos(t),

x(0) = 0,

x'(0) = 1

Consider the equation y''− (sin x)y = 0.

Find the general solution as a power series centered at x = 0. Write the first six nonzero terms of the solution. Write the two linearly independent solutions that form the general solution.

Differential Equations

Q.1 Marks 2 (CLO 3)

From the following data, you are required to: -

(a) Fit the regression line Y on X and predict Y if x = 20

(b) Fit the regression line X on Y and predict X if y = 10

Note : Answers should be in Word or Excel Format

Minimize ?(?,?)=3?−2?+6f(x,y)=3y−2x+6 subject to ?≥−3,    ?≤4,    4?+7?≤23,    8?+7?≥−31.

Let D be a division ring, and let M be a right D-module. Recall that a subset S ⊂ M is linearly independent (with respect to D) if for any finite subset T ⊂ S, and elements a_t ∈ D for t ∈ T, if sum of ta_t = 0, then all the a_t = 0.

(a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and that U is a basis for M (that is,M is a free D-module).

(b) Suppose that S is a generating set (that is, for every element m ∈ M, there exists a finite subset T ⊂ S and a_t ∈ D such that m = sum of ta_t). Show that there exists a subset U ⊂ S that is a basis for M.

(c)* Bonus for proving that all bases of M have the same cardinality, if it has a finite basis. (It is also true for infinite bases.)