eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4 and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ? 2yz = 16. How should i determine the order of the coefficient in the form X^2/A+Y^2/B+Z^2/C=1?

formulate an argument to prove that all squares and circles are similar

Assign the coordinates to each point on an Affine plane of order 3.

Consider a rectangular coordinate system in the plane, in the usual sense of analytic geometry. Every point has a pair of coordinates (x,y). For the purposes of this question, let us regard points as indistinguishable from the ordered pairs (x, y) that describe them. Thus every figure, that is, every set of points, becomes a collec- tion of ordered pairs of real numbers. Under what conditions, if any, do the following figures represent functions? (a) a triangle, (b) a single point, (c) a line, (d) a circle, (e) a semicircle, including the endpoints, (f) an ellipse. What, in general, is the geometric condition that a figure in the coordinate plane must satisfy to be a function? (Very rigorous arguments are not required - just give the idea intuitively for each).

Prove Proposition 3.22(SSS Criterion for Congruence). Given triangle ABC and triangle DEF. If AB is congruent to DE, BC is congruent to EF, and AC is congruent to DF, then triangle ABC is congruent to triangle DEF.

(Hint:Use three congruence axioms to reduce to the case where A=D, C=F, and points B and E are on opposite sides of line AC.)

George and Paula are running around a circular track. George starts at the westernmost point of the track, and Paula starts at the easternmost point. The illustration below shows their starting positions and running directions. They start running toward each other at constant speeds. George runs at 8 feet per second. Paula takes 50 seconds to run a lap of the track. George and Paula pass each other after 11 seconds. After running for 3 minutes, how far east of his starting point is George?

Part of Desargue’s Theorem: If two triangles are perspective from a point, then they are perspective from a line.

a. Create a diagram of Desargue’s Theorem in which the point of perspectivity is between the two triangles.

b. Create another diagram in which the point of perspectivity is interior to both triangles.

A container of yogurt has the shape of the frustrum of a cone. With the lower base having diameter length 6 cm and the upper base having diameter length 8 cm, the volume of such a container 7.5 cm tall can be determined by using H ?? 30 cm and h ?? 22.5 cm. Find its volum

A company is creating a new fertilizer additive for lawn seed, which will be a mixture of nitrogen, phosphate, and potash. Based on their research, the total amount fertilizer added must be at least 14 oz. per 5 lb. bag of lawn seed, but should not exceed 20 oz. per 5 lb. bag. At least ¼ oz. of nitrogen must be used for every ounce of phosphate, and at least 1 oz. of potash must be used for every ounce of nitrogen. The costs per ounce of nitrogen, phosphate, and potash are $0.30, $0.18, and $0.54, respectively. Determine the mixture of the three ingredients in a 5 lb. bag that minimizes costs, as well as that cost.

Find the four initial terms of the Maclaurin series for f(x) = tan x

A pilot flew a jet from City A to City B, a distance of 1600 mi. On the return trip, the average speed was 20% faster than the outbound speed. The round-trip took 5 h 20 min. What was the speed from City A to City B?

A wire 370 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

How much will you have accumulated, if you annually invest $1,500 into an IRA at 8% interest compounded monthly for:

a. 5 year

b. 20 years

c. 40 years

d. How long will it take to earn your first million dollars? Your answer should be exact rounded within 2 decimal places. Please use logarithms to solve. (Total of 15 points)

Quest

When proving theorems, mathematicians consider most frequently four statements:

1. the conditional statement

2. the converse to the conditional statement

3. the inverse

4. the contrapositive

Among those four statements, which ones require negation of parts, or all, of the conditional statement? When negating a statement, for which keywords must you search? How are they negated?