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Algebra

Solving Equations

I) f(x) = 2 - (square root x) for x>= 0 Given that |f(x)| = k has two distinct roots, determine the possible values of the constant k. ii) Solve for 0 < x < 90, the equation 3sin6xcosec2x = 4

Let a commutative ring R be generated by {a_1, a_2, ..., a_n}

Let a commutative ring R be generated by {a_1, a_2, ..., a_n} such that [a_1, a_2, ... , a_n] = {(a_1xr_1) + (a_2xr_2) + ... + (a_nxr_n) for r_1, ..., r_n in set of Reals}. I need to show this set is an ideal. Do I just need to show that it satisfies the commutative properties of the ideal?

Word Problem

The village green is the shape of a square. Along the diagonal of the square is a path through the park. The path is 80 feet long. Find the area of the park. Round to the nearest tenth, if necessary.

Recurrence relation

1. If a person invests in a tax-sheltered annuity, the money invested, as well as the interest earned, is not subject to taxation until withdrawn from the account. Assume that a person invests $2000 each year in a tax-sheltered annuity at 10 percent interest compounded annually. Let An represent the amount at the end of n year

Fibonacci Sequences

Let (f_k) be the Fibonacci sequence, show that: a) For every integer n>= 0, we have f_4(n+1) = 3f_4n+1 + 2f_4n b) Use a) in order to prove by induction that &#8704;n &#1028; N, 3 | f_4n

Writing Equations from Word Problems : Speed and Distance

A pilot flies from Toledo to Chicago, which lies 280 mi directly west. Her plane can fly at 120 mi/hr. She ignores the wind and heads directly west. However, there is a 25 mi/hr wind blowing from the south. a) Write the equation that describes the effect of the wind b) Write the equation that describes the plane's contr

Factor Groups of Non-Abelian Groups

Let G be a nonabelian group and Z(G) be its center. Show that the factor group G/Z(G) is not a cyclic group. We know if G is abelian, Z(G)=G. But now if it is not abelian, can we simply say because G is not cyclic, then any factor group will not be cyclic either? or is there more to it?

Higher-Order Taylor Methods

Use Taylor's method with h = 0.05 to approximate the solution, and compare it with, actual values of y. See attached file for full problem description.

R-modules, Ideals and Submodules

Let R=Z[X] and let M=(2,X) be the ideal generated by 2 and X considered as a submodule of R. Show that {2,X} is not a basis of M.

Finding the Sum of a Convergent Series

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. Please see the attached file for the fully formatted problems.

Lines and Planes

Find all points of intersection of the following planes, and describe what shape the intersection makes in 3 space 2z -10 =0 x +y +z -5 =0 -3x -3y =0

Lines and planes

Show that the following two planes are parallel and find the minimum distance between them: x-2y+5z+4=0 -2x +4y -10z +9 =0

Systems of Congruences : Simultaneous Solutions

1. Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)]. 2. Find the simultaneous solutions of the following congruences: 2x &#8801; 1(mod 5) 3x &#8801; 9 (mod 6) 4x &#8801; 1 (mod 7) 5x &#8801; 9 (mod 11)

Fibonacci Sequences

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined

Solving Equations

Solve for R. (distance, rate, & time). See attached file for full problem description.

Solving Equations

1/a + 1/b = 1/f for a P1V1 P2V2 ______ = ______ for P2 (note: the 2 is a tiny #2 below the P) T1 T2

Solving Equations

Three problems 5/x + 6/x =12 a-1/a^2-4 = 1/a-2 = a+4/a+2 z/z+1 - 1/z+2 = 2z+5/z^2 + 3z +2

Solve: Lowest Common Denominator

Please address the following question: Convert each rational expression into an equivalent rational expression with the LCD as the denominator. y x ___ , _____ 4x 6y