3-D, Systems, Miscellaneous Test Review
Name_______________________________
Solve the following systems by hand by converting to row-echelon form. You may check your
solutions using matrices (IF POSSIBLE).


 3x  2y  3z  8

1. Solve the system:  x  3y  6z  3
2x  6y  12z  6





 x  2y  6z  5

2. Solve the system:   x  y  2z  3
 x  4y  2z  1



2x  3y  z  13

x  2y  2w  5

3. Solve the system: x  2z  w  3
2x  3y  z  w  11



3-D, Systems, Miscellaneous Test Review
4. Perform the partial-fraction decomposition of
x 2
.
x2  x
5. Perform the partial-fraction decomposition of
3x2  7x  8
.
x3  4x2  4x
6. A particular math test has three types of questions: multiple choice, true/false, and free response.
Multiple choice questions count 3 points each, true/false questions count 4 points each, and free
response questions count 8 points each. The test contains a total of 40 questions. A student earns a
total score of 190 points. The student earns 18 more points for his free response answers than the
sum of the points he earns for the multiple choice and true/false questions combined. How many
questions of each type does the student answer correctly?
(a) Set up a system of equations.
(b) Set up the system using matrices.
(c) Answer the question – how many questions of each type did the student answer correctly?
3-D, Systems, Miscellaneous Test Review
7. Plot the points in the same three-dimensional coordinate plane.

(a) 0,4, 3


(b) 4,0,4

z
x
y
8. Find the coordinate of the point located in the yz-plane, four units to the right of the xz-plane, and
three units above the xy-plane.


9. Determine the octant in which x,y,z is located if x  0 , y  0 , and z  0 .


10. Find the standard form of the equation of a sphere with diameter endpoints of 2, 2,2 and
 1,4,6 .
3-D, Systems, Miscellaneous Test Review
11. Find the center and radius of the sphere with the following equation:
2x2  2y 2  2z2  2x  6y  4z  5  0
12. Given u  1,3,2 , v  1, 2, 2 , and w  5,0, 5 , find vector z if:
u  v  2w  z  0




13. Find the magnitude of the vector v with initial point 0, 1,0 and terminal point 1,2, 2 .
14.Use vectors to determine whether the points are collinear:

P: 2,7,4


Q: 4,8,1


R: 0,6,7

3 5
v  i  2k
u  i  j k
15. Find a vector orthogonal to
.
2 2 and
3-D, Systems, Miscellaneous Test Review
16. Find a unit vector orthogonal to
u  i  2j
and
v  i  3k
.
17. For each of the following pairs of vectors, plan an O in the blank if they are orthogonal, a P in the
blank if they are parallel, or a N in the blank if they are neither.
_____ (a)
_____ (b)
_____ (c)
u  12i  6j  15k
v  8i  4j  10k
3 1
,  ,2
4 2
v  4,10,1
u  i  3j  k
v  2i  j  5k
u
1
u  1, , 1
_____ (d)
2
v  8, 4,8
18. Find the angle between u and v if:
u  8j  20k
v  10i  5k