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# Math 121, Practice for Test 4

The test will cover the following topics and sections from the text.

• Sections 9.1 and 9.2: systems of linear equations. Key types of problems included solving
linear systems with 2 or 3 variables. Setting up and solving word problems involving lin-
ear systems of equations. Recognizing inconsistent, dependent and independent systems of
equations.

• Sections 10.1, 10.2, and 10.3: Gaussian elimination , the algebra of matrices, inverses of
matrices. Key types of problems include converting systems of equations to matrix form
and solving the system using elementary row operations . Matrix addition and multiplication ,
writing linear systems in matrix form. Finding inverses of matrices 2 by 2 or 3 by 3. Recognizing
singular matrices that don't have inverses. Solving systems of linear equations using the matrix
inverse.

• Sections 11.1 and 11.5: sequences and summation notation , the binomial theorem . Key
types of problems include, using summation notation, finding terms of sequences , binomial
expansions, and finding specific terms in a binomial expansion.

Some practice problems are as follows:

1. Solve each system of equations if possible.

2. A motorboat traveled a distance of 120 miles in 4 hours while traveling with the current.
Against the current, the same trip took 6 hours. Find the rate of the boat in calm water and
find the rate of the current.

3. A broker invests \$25,000 of a clients' money in two different municipal bonds. The annual
rate of return on one bond is 6%, and the annual rate of return on the second bond is 6.5%.
The investor receives a total annual interest payment of \$1555 from the two bonds. Find the
amount invested in each bond.

4. Solve the following systems of equations.

5. The equation of a (nonvertical) plane can be written in the form z = ax+ by+ c where a, b,
and c are numbers. Find the equation of the plane that contains the points (2, 1, 1), (-1, 2, 12)
and (3, 2, 0).

6. A coin bank contains only nickels, dimes, and quarters. The value of the coins is \$2. There
are twice as many nickels as dimes and one more dime than quarters. Find the number of each
coin
in the bank.

7. Consider the following systems of equations. For each system, find the values of k for which
there (if possible), (i) one solution, (ii) no solution, (iii) infinitely many solutions?

8. Write the systems of equations in 1(a) and 4(a) as augmented matrices, and solve them
using Gaussian elimination.

9. The following are augmented matrices for a systems of equations in the variables
and , find the solutions to the systems of equations.

10. Let and .Find the
following, if possible.

(a) AB (b) BA (c) A + B (d) 5A - 3C (e) AC.

11. Determine whether it is possible to find the product AB for matrices of the given sizes, if
so, determine the size of AB. Same question for BA.

(a) A is a 3 × 3 matrix, B is a 3 ×1 matrix.

(b) A is a 3 × 5 matrix, B is a 3 × 5 matrix.

(c) A is a 5 × 3 matrix, B is a 3 × 5 matrix.

(d) A is a 3 × 2 matrix B is a 2 × 3 matrix.

12. Write the following matrix equation as an equivalent system of equations.

13. Find the inverses of the following matrices (if they exist).

14. Consider the following systems of equations:

Solve these systems using the fact that the inverse of is .

15. Find the first three terms and the 8th term of the sequence whose nth term is .

16. Find the first three terms of the recursively defined sequence = 5, and .
17. (a) Evaluate the sum .

(b) Write 8 + 10 + 12 + 14 + 16 + 18 in summation notation.

18. Expand the binomial .

19. Find the fourth term of (x + 2y)12.

20. Find the term that contains b9 in the expansion of .

Further Practice: See Test 4 from Autumn 2004, Winter 2005, Winter 2006, Autumn 2006.

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