PreCalculus
Final Exam
December 2007
Name: ______________________________
Directions: Show all work for all problems.
1.
  1  n 1 
Write the first 4 terms of the sequence 6  
  2  
2.
Find the sum of the sequence
5
 (k  10)
k 1
 x  2 4
3.
Expand and simplify:
4.
In a student survey, 109 students indicated that they speak Spanish, 33 students indicated they speak
French, 9 students indicated they speak both
Spanish and French, and 129 students indicated that they
speak neither. How many students participated in the survey?
5.
1

tan  cos 1 
3

6.
Recall: cos(   )  cos   cos   sin   sin 
Find the exact value of cos(   ) under the given conditions
sin  
12

21

, 0    Solve the equation in the given domain. (NOTE: x
, 0    ; cos  
13
2
29
2
is measured in radians)
7.
5 sin( x)  2
8.
Use v  2i  3 j  3k , and w  3i  2 j  2k to find v  ( v  w ).
9.
Write the complex number 1  3i in polar form:
x  [2,4]
10.
Recall: If u and v are two nonzero vectors, the angle  , 0     between u and v is
determined by the formula:
uv
cos 
u  v
A pilot is doing stunts with her plane at an air show. At times t  0 sec and t  10 sec, her position
vectors from the control tower are
a  7i  4 j  3k
b  10i  20 j  5k
How large is the angle between a and b ?
11.
10
( x  2) 2 ( y  1) 2

1
Graph
4
9
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
12.
Find the center, foci, and vertices of
2
center____________________
2
x
y

1
49 25
foci______________________
vertices___________________
13.
Determine the angle of rotation and the rotation formulas that would be used to eliminate the xy-term.
You do not need to go any further.
9 x 2  24 xy  16 y 2  90 x  130 y  0
14.
x  0
y  0

Maximize z  2 x  5 y subject to 
.
x  2 y  6
9 x  3 y  27
10
9
8
7
6
5
4
3
2
1
1
 3  5
 and use it to find the solution to:
 2 7 
15.
Find the inverse of 
16.
Solve the system of equations
x  y  z  2

 x  y  5 z  12
4 x  y  z  4

2
3
4
5
3x  5 y  21

 2 x  7 y  25
6
7
8
9 10