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
.
