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Practice Questions for College Math Test 4, Exams of Algebra

A set of practice questions for college math test 4, including systems of equations, matrix operations, inverses, sequences, binomial theorem, and word problems. It covers various topics that are typically studied in a first-year college math course.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Math 121, Some Practice Questions for Test 4
1. The augmented matrix
1 2 0 3
0 1 3 0
0 0 k4
represents a system of equations in the vari-
ables x,y,z.
(a) For what values of kis there no solution?
(b) For what values of kis there exactly one solution?
(c) Find the solution to this system of equations if k=2.
2. Let A=21 4
3 1 0 ,B=1 3
21, and C=31 1
2 2 0 . Find the following,
if possible, if an operation is not possible, state why it is not possible.
(a) BC (b) C B (c) 2A2C
3. Consider the following system of equations:
x+y+ 2z= 1
2x+ 3y+ 3z=2
3x+ 3y+ 7z= 3
.
Solve this system using the fact that the inverse of
1 1 2
2 3 3
3 3 7
is
12 13
5 1 1
3 0 1
.
4. Consider the matrices A=
12 13
5 1 1
3 0 1
and B=
203 2
1 3 3 0
2 4 1 2
(a) Let C=AB. Determine the dimensions of C, and find c23 (you don’t need to compute the
entire product).
(b) Does D=BA exist? If so, determine the dimensions of Dand find d31 .
5. (a) Let Abe a 2 by 3 matrix, how many rows and how many columns must Bhave so that
C=AB is a 2 by 8 matrix?
(b) If Ais an mby nmatrix and Bis an nby pmatrix, what are the dimensions of the product
AB?
(c) If Ais an mby nmatrix and Bis a rby smatrix. Under what condition(s) will AB exist?
Under what condition(s) will BA exist?
pf2

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Math 121, Some Practice Questions for Test 4

  1. The augmented matrix

0 0 k − 4

 (^) represents a system of equations in the vari-

ables x, y, z. (a) For what values of k is there no solution? (b) For what values of k is there exactly one solution? (c) Find the solution to this system of equations if k = −2.

  1. Let A =

[ 2 − 1 4

]

, B =

[ 1

]

, and C =

[ 3 − 1 1

]

. Find the following, if possible, if an operation is not possible, state why it is not possible. (a) BC (b) CB (c) 2A − 2 C

  1. Consider the following system of equations:

x + y + 2 z = 1 2 x + 3 y + 3 z = − 2 3 x + 3 y + 7 z = 3

Solve this system using the fact that the inverse of

 (^) is

  1. Consider the matrices A =

 (^) and B =

(a) Let C = AB. Determine the dimensions of C, and find c 23 (you don’t need to compute the entire product). (b) Does D = BA exist? If so, determine the dimensions of D and find d 31.

  1. (a) Let A be a 2 by 3 matrix, how many rows and how many columns must B have so that C = AB is a 2 by 8 matrix? (b) If A is an m by n matrix and B is an n by p matrix, what are the dimensions of the product AB? (c) If A is an m by n matrix and B is a r by s matrix. Under what condition(s) will AB exist? Under what condition(s) will BA exist?
  1. The augmented matrix

 (^) represents a system of equations in the vari-

ables x, y, z. Find the solution to this system of equations.

  1. Find the inverse of A =

 (^) if it exists.

  1. Consider the sequence whose nth term is defined by an = (−1)n−^1 3(2n). Find a 1 , a 2 , a 3 and a 9.
  2. Evaluate

∑^5

k=

(−1)k 2 k+^.

  1. Find the term that contains y^9 in (2x^2 − y^3 )^11
  2. Use the binomial theorem to expand (x^2 − 2 y)^6.
  3. Represent 278 − 1681 + 24332 − 729 64 in summation notation.
  4. Find the equation of the circle whose graph passes through the points (5, 3), (− 1 , −5), and (− 2 , 2). (Hint: use the equation x^2 + y^2 + ax + by + c = 0.)
  5. A goldsmith has two gold alloys. The first alloy is 40% gold; the second alloy is 60% gold. How many grams of each should be mixed to produce 20 grams of an alloy that is 52% gold.
  6. Solve the system of equations

2 x − y − z = − 1 −x + 3 y − z = − 3 − 5 x + 5 y + z = − 1

Note. Answers to these questions may be found on your instructor’s website at http://faculty.lasierra.edu/∼jvanderw/classes/m121a06/m121prac4a06ans.pdf Other old tests and review questions are also available at your instructor’s website.