Practice Exam 2

Key

Problem 1

Find the equation of the parabola with focus at the point (3,1) and directrix y = -5.  Then graph the parabola by hand.

 

Problem 2

Find the equation of the ellipse with foci at (-3,3) and (5,3) and vertex at (-4,3).  Then graph the ellipse by hand.

 

Problem 3

Find the equation of the hyperbola with vertices at (-3,2) and (1,2) and asymptote y = 2x + 4.

 

Problem 4

Identify the type of conic each is.  Then find the any applicable feature: center, vertices, foci, directrix, asymptotes.

A.  -(x-2)^2/4 + (y+1)^2 = 1

B.  x^2 + (y-2)/4 = 1

C.  4x^2+y^2 -6y = 27

 

Problem 5

Solve each system of equations using matrices.  If the system has no solution say that it is inconsistent.

A.  x-2y+5z=5, 2x+y-3z=-1, 3x+4y-6z=3

 

B.  2x-y+5z=6, x-2y+z=3, -x+5y+2z=1

 

Problem 6 

Find the determinant of the following matrix (without using a calculator).

Matrix: 2 -1 3, 4 5 0, 1 6 -2

 

Problem 7

 Use Cramer's Rule (without using a calculator) if it is applicable to solve the system. 

3x-2y = 3, 4x+3y = -13

Problem 8

Given that

 Det[ a b c, 1 2 3, x y z ] = 5

Use the properties of determinants to find

Det[ x-2 y-4 z-6, 3 6 9, a b c

 

Problem 9

A.  Find the inverse of the matrix

Matrix: [1 2 0, 0 2 2, 1 0 -1]

 

B.  Use the result from part A. to solve the system:

x+2y=7, 2y+2z=3, x-z=5 

Problem 10

Graph the system of inequalities

x-y^2 <=0, x^2+y <= 4

 

Problem 11

A science museum is constructing a whispering gallery that will be 150 feet in length.  The foci are located 50 feet from the center.  How high will  the ceiling be at the center?

 

Problem 12

Jordan and Yesha are each have loans from both Bank of America and Wells Fargo.  The first table shows the amounts borrowed and the second gives the interest rate per month. 

  B of A Wells Fargo
Jordan $8000 $4000
Yesha $6000 $5000
       
Monthly Interest Rates
B of A 0.008
Wells Fargo 0.01

A.  Write a matrix A for the amounts borrowed by each student and a matrix B for the monthly interest rates.

B.  Compute AB and interpret the results using a complete sentence.

 

Problem 13

A nutritionist wants to make a cereal mix that contains at least seven grams of fiber and at least 200 grams of carbohydrates.  The first cereal of the mix will be raison bran that contains 5 grams of fiber per cup and 100 grams of carbohydrates per cup.  One cup of raison brand cost 30 cents.  The second cereal of the mix will be wheat chex that contains 3 grams of fiber and 110 grams of carbohydrates per cup.  One cup of wheat chex costs 25 cents.  How many cups of each type of cereal should be in the mix in order to minimize the cost of the cereal?