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# Basic Algebra

### Factor Group and Torsion Group

Heres my problem. Consider the group <R,+> (reals under addition) and its normal subgroups Z (integers) and Q (rationals0. (These are normal because R is abelian, of course.) (i) Find an element of Q/Z of order 350. (ii) Show that Q/Z is the torsion subgroup of R/Z. This problem is quite straightforward if you use the defini

### Actual Values of Logarithmic and Exponential Functions

What is the actual values for each one? 1. f(x) = e^x 2. f(x) = log x.

### Inner / Linear Product : Polynomial Space

In a polynomial space of degree {see attachment} is (p,q)=p(0)q(0)+Ap(1/2)q(1/2)+Bp(1)q(1) for A,B>0. Define the linear product.

### Matrix Algebra - Definite vs. Indefinite

Let A be an nxn symmetric matrix such that det {see attachment}. We know from the matrix algebra that the associated quadratic form {see attachment}, where x = (x1,...xn), is either positive-definite, negative- definite or indefinite. Now assume the diagonal entries of A are all zero. Explain why q(x) is indefinite. (Hint:

### Finding slope and intercept of a line, and solving equations

1. Find the slope and intercept for the line 2y-x-4=0 2. Find the slope and intercept for the line 6/9y+2/3x-5=0 3. For the line y= 3x-3 find a line parallel to it that passes through the point (-1,0) 4. For the line y= 2x+2 find a line parallel to it that passes through the points (2,0) (4,4) 5. For the set of p

### Heat Equation : Non-Homogeneous PDE

Please see the attached file for the fully formatted problem. 2. (30 points) A thin, one-dimensional rod of heat-conducting material with length L = pi is internally and uniformly heated at a rate of alpha degrees per unit time. This means that the temperature of the rod, u(x, t), satisfies the inhomogeneous heat equation ..

### Absolute Value Equations

1. Solve the following absolute value inequality and graph the solution set. 2| 5 - 2x| - 15 5

### Investment allocation

Matrix Algebra Investment Allocation Problem Donald Trump has some pocket change he wants to invest to make \$3380. His stock broker suggested four (4) investment possibilities to accomplish this goal. The investments and their expected yields are listed in the table below. Investment Rate of Return Risk Level Magellan

Find the voltage V where V=V1+V2 using graphical addition and analytical methods, given that V1=3sin(theta+20 degrees) and V2=5sin(theta-40degrees).

### 5 Algebra Problems : Inequalities and Systems of Equations Word Problems

1. Solve the following equation for the variable y. 5x - 3y + 9 = 0 2.Solve the following problem by identifying a variable, writing an equation that describes the situation, and then solving the equation. Find two consecutive even integers that have a sum of 450. 3.Solve the following problem by identifying a variable,

### Stokes's Problem : Calculating drag coefficients around a falling sphere.

Please see the attached problem statement.

### 5 Algebra Problems : Solving Equations

1.Solve the following equation 19 - 3n = - 2n 2.Solve the following equation 2(a - 4) + 4 = 5(9 - a) 3.Solve the following equation and check your answer. (1/2b)- (1/2)= 1/4b 4.Solve the following equation 0.6(x - 50) = 18 - 0.3(40 - 10x) 5.Solve the following equation for the variable x. t - 5x = 4x

### Graph

Refer to the graph given below and identify the graph that represents the corresponding function. Justify your answer. y = 3x y = log3x Plot the graphs of the following functions. 1. f(x)=7 2. f(x)=4 x - 3 3. f(x)=(1/5x 4. f(x)= 3 log x 5. f(x) = log x +2 See Attachment for Graph

### Plotting Graphs of Given Functions

Refer to the graph given below and identify the graph that represents the corresponding function. Justify your answer. y = 3x y = log3x See attachment for Graph Plot the graphs of the following functions. Scan the graphs and plot them. 1. f(x)=7 2. f(x)=4 x - 3 3. f(x)=(1/5x 4. f(x)= 3 log x 5. f(x) = log x

### Exponential and Logarithmic Functions: Real-Life Applications

1. Explain the relevance and application of exponential functions in real-life situations. 2. Explain the relevance and application of logarithmic functions in real-life situations. 3. Think of a real-life situation that can be represented by a logarithmic function, translate the situation to the function, and solve the

### 20 Algebra Problems : Sequences, Series, Combinations and Permutations

Please see the attached file for the fully formatted problems. 1. Write the first four terms of the sequence an = 64(1/4)n , for n=0, 1, 2, 3,... a) 16, 4, 1, ¼ b) 60, 56, 52, 48 c) 64, 16, 4, 1 d) 256, 1024, 16384, 262144 2. Write the given series in expanded form without summation notation. a) -x2-x3

### Algebra : Four Word Problems

1. A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet. 2. A business invests \$10,000 in a saving account for two years. At the beginning of the second year, and additional \$3500 is inve

### Quadratic Equations: Formulation of Real-Life Problems and Graph

Please see the attached file for the fully formatted problems. 1. Can a graph be used to solve any quadratic equation? Why or why not? 2. Look at the graph below and comment on the sign of D or the discriminant. From the quadratic equation based on the information provided and find its solution. 3. Formulate two word

### Algebra: Systems of Equations - 3 Word Problems

1. Joe has a collection of nickels and dimes that is worth \$5.65. If the number of dimes was doubled and the number of nickels was increased by 8, the value of the coins would be \$10.45. How many dimes does he have? 2. An express train and a local train leave GraysLake at 3 P.M. and head for Chicago 50 miles away. The express

### MATLAB : Least Squares - Solving Inexactly Specified Equations in an Approximation

The solution can ONLY be accepted in Matlab. The problem is in the attachment file. Least Square Planetary orbit [2]. The expression z = a + bxy + cy + dx + ey + f is known as a quadratic form. The set of points (x, y) where z = 0 is a conic section. It can be an ellipse, a parabola, or a hyperbola, depending on the si

### Hamilton's Equations

A system with two degrees of freedom has Hamiltonian (see attachment) ? Show that p2 and H will remain constant during the motion. ? If (see attachment), show that at other times (see attachment) ? Show that in the subsequent motion cannot reach the value (see attachment).

### Exponentials, Simple and Compound Interest, Annuities, NPV and Amortization

1. Rational Functions Graph the following function when a=3 and b=2. Develop a generic expression (i.e., as a function of "a" and "b") to find the "x" and "y" intercepts for this function (see attachment) 2. Exponential Functions Once a new automobile enters the market, the manufacturers try to estimate their residual valu

### Nth Term of Expression

Write the 3rd term of the expression : (x+ y) to the seventh power. ( or (x+y)^7 )

### Polynomial expansion terms

(x+y)to the 16th power

### Divisbility

Prove that for all odd integers n, (1^n)+(2^n)+(3^n)...+(n^n) is a proper multiple of 1+2+3+...n

### Irrational roots

If a,b,c are odd integers, show that all real roots of ax^2+bx+c=0 are irrational numbers.

Derive rationalized expressions for the admittances of the two parallel arms. Hence derive an expression for the total circuit admittance.

### Laws of Logarithms

a) log y = 1.2x -1 b) ln y = 1.2x - 1

### Laws of Logarithms

Q a) log y = 1.2x - 1 A y = 0.1(10^1.2)^x Q b) ln y = 1.2x - 1 A y = 1/e(e^1.2)^x

### Algebra Topics: Radical Equation, Complete the Square etc.

See the attached file. 1. Solve the radical equation: {see attachment} 2. Find the equation of the line through the point (-2,-4) and perpendicular to the line {see attachment} 3. Complete the square, find the vertex, the axis of symmetry, all the intercepts, and graph the parabola: {see attachment} 4. ... Find the