Let ∆ABC be a triangle in R2 . Show that the centroid G is located at the

average position of the three vertices:

G = 1/3(A + B + C).

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod 13 (b) 10x ≡ 4 mod 6 (c) 6x ≡ 8 mod 12

1. Seven thieves try to share a hoard of gold bars equally between themselves. Unfortunately, six bars are left over, and in the fight over them, one thief is killed. The remaining six thieves, still unable to share (all) the bars equally since two are left over, again fight, and another is killed. When the remaining five share (all) the bars, one bar is left over, and it is only after yet another thief is killed that an equal sharing is possible. What is the minimum number of bars which allows this to happen? [Hint: Be carefully to check that the conditions for the Chinese Remainder Theorem apply before using it.]

Differential Geometry

Open & Closed Sets, Continuity

(1) Prove (2,4) is open

(2) Prove [2,4) is not open

(3) Prove [2,4] is closed

Solve the following differential equations:

1.) y"(x)+ y(x)=4e^x ; y(0)=0, y'(0)=0

2.) x"(t)+3x'(t)+2x(t)=4t^2 ; x(0)=0, x'(0)=0

A Cartesian vector can be thought of as representing magnitudes along the x-, y-, and z-axes multiplied by a unit vector (i, j, k). For such cases, the dot product of two of these fectors {a} and {b} corresponds to the product of their magnitudes and the cosine of the angle between their tails as in {a}⋅ {b} = abcos(theta)

The cross product yields another vector, {c} = {a} × {b} , which is perpendicular to the plane defined by {a} and {b} such that its direction is specified by the right-hand rule. Develop and M-file function that is passed two such vectors and returns Theta, {c} and the magnitude of {c}, and generates a three-dimensional plot of the three vectors {a}, {b}, and {c} with their origins at zero. Use dashed lines for {a} and {b} and a solid line for {c}. Test your function using the following cases:

A. a = [ 6 4 2 ]; b = [ 2 6 4 ];

B. a = [ 3 2 -6 ]; b = [ 4 -3 1];

C. a = [ 2 -2 1 ]; b = [ 4 2 -4 ];

D. a = [ -1 0 0 ]; b = [ 0 -1 0 ];

I know how to find theta, {c}, and the magnitude of {c}, I just don't know how to plot a 3-dimensional graph so if someone could help me with that part of the code for MATLAB

Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R, xRy if and only if sin2 x + cos2 y = 1. Prove that R is an equivalence relation and for any fixed x ∈ R, find the equivalence class x

prove or disppprove.

Suppose A & B are sets.

(1) A function f has an inverse iff f is a bijection.

(2) An injective function f:A->A is surjective.

(3) The composition of bijections is a bijection.

A tank originally contains 100 gal of fresh water. Then water containing

lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of

6 gal/min,

with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (Round your answer to two decimal places.)

\int \frac{x^6+5x^4+4x-3}{\left(x^2+3x+5\right)\left(x^2+7x+12\right)}dx

Use laplace transforms to solve the following problem.

y' + 20y = 6sin 2x; y(0) = 6

Find all possible homomorphisms between Z and Z5.

Can you supply examples of logical operations you apply to your everyday experience?

Use matlab code for bisection method and regula falsi. Thank you!