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Sample Final Exam Questions - College Algebra | MATH 116, Exams of Algebra

Material Type: Exam; Class: College Algebra; Subject: Mathematical Sciences; University: University of Wisconsin - Milwaukee; Term: Spring 2008;

Typology: Exams

2009/2010

Uploaded on 02/25/2010

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Math 116, Sample Final Exam Questions, Spring 2008
1. Consider the augmented matrix
6|200
5|100
8|331
7|122
.
(a) Using the variables x, y and z , what system does this augmented matrix
represent?
(b) Either solve the system of equations or explain why it has no solution.
2. Consider the system of equations
72
=
+
yx
632
=
+
yx
(a)
Solve this system of equations by Gauss-Jordan elimination, and check your
answer.
(b)
Solve this system of equations by writing it in the form AX = B. Find the inverse
of A and use it to solve the equation AX = B. Show your calculations.
(c)
Solve this system of equations by using Cramer’s rule. Show your calculations.
(d)
What method do you find easiest and why?
(e)
What method would you choose if there were 8 equations and 8unknowns? Why?
(f)
Suppose that AX = B denotes a system of N equations in N unknowns. What does
the determinant of A tell you about the number of solutions of this system of
equations?
3.
The point P has coordinates (-1, -4) and the point Q has coordinates (-9, 2). Let
PQ the line segment with endpoints P and Q.
(a) Give the coordinate-free definition of a circle.
(b) Find the standard equation of the circle for which PQ is a diameter.
(c) Give the coordinates of the points where this circle intersects the line
0)1(3)5(4
=
+
+
+
yx
(d)
Find the equation of the line tangent to this circle and passing
through the point P.
4.
Some More Conic Sections.
(a)
Give the coordinate-free definition of an ellipse.
(b) Find an equation of the ellipse whose minor axis has length 24 and its
focal points are at (3, 6) and (
-3
,
-
2). Leave your answer in terms of radicals.
(c) Give the coordinate-free definition of a parabola.
(d) Use the definition of a parabola and the distance formula to derive the standard
equation of the parabola whose focus is at (3, 5) and whose directrix is the line
71
=
y
.
pf2

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Math 116, Sample Final Exam Questions, Spring 2008

  1. Consider the augmented matrix

(a) Using the variables x, y and z , what system does this augmented matrix represent? (b) Either solve the system of equations or explain why it has no solution.

  1. Consider the system of equations x+ 2 y= 7 2 x+ 3 y= 6 (a) Solve this system of equations by Gauss-Jordan elimination, and check your answer. (b) Solve this system of equations by writing it in the form AX = B. Find the inverse of A and use it to solve the equation AX = B. Show your calculations. (c) Solve this system of equations by using Cramer’s rule. Show your calculations. (d) What method do you find easiest and why? (e) What method would you choose if there were 8 equations and 8unknowns? Why? (f) Suppose that AX = B denotes a system of N equations in N unknowns. What does the determinant of A tell you about the number of solutions of this system of equations?
  2. The point P has coordinates (-1, -4) and the point Q has coordinates (-9, 2). Let

PQ the line segment with endpoints P and Q.

(a) Give the coordinate-free definition of a circle. (b) Find the standard equation of the circle for which PQ is a diameter. (c) Give the coordinates of the points where this circle intersects the line 4 ( x+ 5 )+ 3 (y+ 1 )= 0 (d) Find the equation of the line tangent to this circle and passing through the point P.

  1. Some More Conic Sections. (a) Give the coordinate-free definition of an ellipse. (b) Find an equation of the ellipse whose minor axis has length 24 and its focal points are at (3, 6) and (-3, - 2). Leave your answer in terms of radicals. (c) Give the coordinate-free definition of a parabola. (d) Use the definition of a parabola and the distance formula to derive the standard equation of the parabola whose focus is at (3, 5) and whose directrix is the line y =− 71.

(e) Give the coordinate-free definition of a hyperbola. (f) Find the center, vertices, foci, and asymptotes of the hyperbola x 2 − y^2 − 10 x+ 10 y− 1 = 0 and sketch its graph.

  1. Sequences and Series. (a) Let a 1 , a 2 ,a 3 ,...,an,...be an arithmetic sequence with a 1 = 6 and a 6 = 26. Give the value of a 20 and express the value a 1 + L +a 20 as a rational number.

(b) Simplify

998

4

    • +⋅⋅⋅+. You need not simplify the exponential term in

your answer. (c) Use the binomial theorem to simplify (^) ( 1 + 2 i )^5. Simplify all powers of i and collect like terms according to i. (d) A pizza parlor offers a choice of 15 different toppings. How many five-topping pizzas are possible? Express your answer as a product of integers.

  1. Functions. (a) Define function in terms of ordered pairs. (b) If the function f is given by {(-7, 3), (8, 0), (9, -1)}, what is f −^1.

(c) If x must be real number, what is the domain of the function 4 3

x x

x f x.

Find the partial fraction decomposition of f (x).

(d) Graph the function

2

3

−  

x f x. Include the horizontal and vertical

asymptotes, if any, x- and y-intercepts, if any, and clearly label 3 points on the graph with their coordinates. (e) Give the domain, range, and the rule of the inverse of the preceding function. (f) A bottle of beer is initially at 35 degrees C. It is placed into a water bath kept at a constant temperature of 10 degrees C. According to Newton’s law of cooling, the temperature of the bottle, B( t), will follow the rule B (t )= 10 + Aektwhere t is time measured in minutes. After 30 minutes in the water bath the temperature of the bottle is 19 degrees C. What was the temperature of the bottle after only 15 minutes in the water bath? Express your answer without exponents or logarithms.

  1. Consider the polynomial P ( x)= 4 x^3 − 64 x^2 + 231 x− 225.

(a) Use the rational zero theorem to list all the possible rational zeros of P( x). (b) Explain why P ( x)< 0 if x < 0. (c) Find all three solutions of P ( x)= 0. Which of these solutions, if any, lie in the interval (0, 11/2)? (d) An open box is to be made from a rectangular piece of cardboard that measures 21 by 11 inches, by cutting out squares of the same size from each corner and bending up the sides. Note that the height of the box will be x. If the volume of the box to be 225 cubic inches, is x = 3 is the only possible choice? Explain.