Give examples of sets of three vectors that are:

a) Collinear

b) Coplanar

c) Not coplanar

Find the general power series solution to the following ode:

y"-4xy'+(4x²-2)y=0

Consider the following LOP P.
Max. z = 212x1 −320x2 +273x3 −347x4 +295x5
s.t. −4x1 −2x3 +8x5 ≤ −22
2x1 +3x2 −x4 = 31
−5x2 +3x3 −2x5 ≤ 27
−7x1 −8x3 +6x4 = −38
−9x3 −2x4 +x5 ≤ −40
−x2 −3x4 −5x5 ≤ 42
& x1, x3, x4 ≥ 0
a. Find x∗ and write the Phase 0, I and II pivots that solve P.
b. Use the General Complementary Slackness Theorem to find
the optimal certificate y∗
[do not solve the dual LOP D!].

z² = xy-3x+9 Find the point closest to the origin on the surface.

The equation of parabola is x2 = 4ay . Find the length of the latus rectum of the parabola and length of the parabolic arc intercepted by the latus rectum. (a) Length of latusrectum : 4a; Length of the intercepted arc : 4.29a (b)Length of latusrectum: —4a; Length of the intercepted arc : 4.39a (c) Length of latusrectum: —4a; Length of the intercepted arc : 4.49a (d) Length of latusrectum: 4a; Length of the intercepted arc : 4.59a

A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility function given by Ux,y=x+y-0.05x2-0.05y2 subject to

2x+5y=128

Where $128 is the consumer’s budget and the prices of the two goods are, respectively, 2 and 5. Assuming marginal utilities Ux, Uy > 0,

a. Find the quantity x and y that maximize the utility function.

b. Using bordered Hessian, check utility for a maximum.

c. What is the maximum utility of the consumer?

Benoit believes that he didn’t invent the Mandelbrot set. Are complex numbers invented or discovered? Whichever position you take, please research from reputable sources how complex numbers came about and make sure to include views that support both sides of the argument in your response. (Remember to cite your sources. (Minimum two paragraphs, five complete sentences each.)

A small business owner contributes $3000 at the end of each quarter to a retirement account that earns 8% compounded quarterly.

(a) How long will it be until the account is worth $150,000? (Round your answer UP to the nearest quarter.)

quarters

(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $7000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.)

$

How do I solve this equation? (1/3(5)-1/4(-3))(-3/4(3))

Sketch the graph of the given function. (x^2+x-2) / x^2

Give

a) x intercept

b) y intercept

c) Vertical asymtope

d)Horizontal asymtope

e) first derivative

f)second derivative

g)critical numbers

h)extrema max/min

i) y coordinate of exterma

j) possible point of infletion

h)y coordinate of possible point of inflection

k) table

l)graph

1. In 2003, gross sales at McDonald’s and Burger King totaled $60.9 billion. McDonald’s sales exceeded Burger King’s sales by $22.1 billion.

(a) Write a system of equations whose solution gives the individual sales of each company in billions of dollars.

(b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

2. Solve the system symbolically 3x-2y=5 x+5y=13

3. Answer the following questions

a. What is a matrix?

b. What is a matrix Dimension

c. What is a matrix element?

Consider F and C below.

F(x, y, z) =

2xz + y²

i + 2xy j +

x² + 15z²

k

C: x = t²,    y = t + 2,    z = 4t − 1,    0 ≤ t ≤ 1

(a) Find a function f such that F = ∇f.

f(x, y, z) =

(b) Use part (a) to evaluate

∇f · dr along the given curve C.

4(a) Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t.

x2 y2
(i) By finding a, b ∈ R such that a2 + b2 = 1, show that P is travelling on an elliptical

path. [10 marks] (ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).[8 marks] (iii) How fast is the distance between P and the origin changing when t = π/8?[7 marks]

(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square. The other piece is bent to form an equilateral triangle. Find the dimensions of the two pieces of wire so that the sum of the areas of the square and the triangle is minimized.(25marks)

Two circles intersect at A and B. P is any point on the circumference of one of the circles. PA and PB are joined and produced to meet the circumference of the other circle at C and D respectively. Prove that the tangent at P is parallel to CD.