Math 121. Practice Problems from Chapters 9, 10, 11 Fall 2016

Transcription

1 Math 121. Practice Problems from Chapters 9, 10, 11 Fall 2016 Topic 1. Systems of Linear Equations in Two Variables 1. Solve systems of equations using elimination. For practice see Exercises 1, Solve systems of equations using substitution. For practice see Exercises 3, Classify systems of equations as independent, inconsistent or dependent. For practice see Exercises 5, Applications of systems of equations. For practice see Exercises 7 (mixing prices), 8 (mixing alloys), 9 (rates of motion), 10 (mixing interest rates). Topic 2. Systems of Linear Equations in More than Two Variables 1. Solve systems of equations. For practice see Exercises 1, Recognize when systems have one, many or no solutions. For practice see Exercises 3, Applications of systems. For practice see Exercises 5 (curve fitting), 6 (coin bank). Topic 3. Gaussian Elimination Method 1. Use Gaussian elimination to solve systems of equations. For practice see Exercises 1, 2, 3, Interpret an augmented matrix as a system of equations, and recognize when systems have one, no or many solutions. For practice see Exercises 5, 6, 7. Topic 4. The Algebra of Matrices 1. Addition, subtraction and scalar multiplication of matrices. For practice see Exercises 1, Matrix multiplication. For practice see Exercises 3, 4, 5, Systems of equation and matrix multiplication. For practice see Exercises 7

2 Topic 5. The Inverse of a Matrix 1. Find the inverse of a 2 by 2 matrix or determine that it does not exist. For practice see Exercises 1, Solve linear systems in two variable using the inverse matrix. For practice see Exercises 3, Find the inverse of a 3 by 3 matrix or determine that it does not exist. For practice see Exercises 5, Use a matrix inverse to solve a system of equations in three variables. For practice see Exercise 7. Topic 6. Infinite Sequences and Summation Notation 1. Find specific terms of a sequence whose nth term expression is given. For practice see Exercises 1, Expand a sum given in summation notation. For practice see Exercises Find the nth term expression for a sequence. For practice see Exercise Express a sum in summation notation. For practice see Exercise 5. Topic 7. The Binomial Theorem 1. Expand a binomial raised to a power. For practice see Exercise Simplify a complex number raised to a power using a binomial expansion. For practice see Exercise Use the binomial theorem to find a specific term in a binomial expansion. For practice see Exercise 3, 4, 5 Page 2

3 1 Systems of Linear Equations in Two Variables 1. Use the method of elimination to solve the following system of equations. 2x 11y = 16 and 5x + 2y = Use the method of elimination to solve the following system of equations. 2x + 5y = 8 and 5x + 4y = 3 3. Use the method of substitution to solve the following system of equations. 3x + y = 8 and 5x 4y = Use the method of substitution to solve the following system of equations. 4x + 3y = 2 and x 5y = 8 5. Solve each of the following systems of equations, or state that there is no solution. Then classify the system as independent, dependent, or inconsistent. (a) 8x 9y = 3 and 16x + 18y = 6 (b) 2x + 18y = 2 and x + 9y = 3 6. Determine which of the following systems of linear equations are independent. (You should be able to do this quickly without solving the system.) (a) 7x + 2y = 3 and 28x 8y = 12 (b) x + 9y = 3 and x + 9y = 3 (c) 7x 2y = 3 and x 2y = 12 (d) 9x + y = 3 and 9x y = 3 7. A metallurgist made two purchases. The first purchase, which cost $474, included 14 kilograms of an iron alloy and 10 kilograms of a lead alloy. The second purchase, at the same prices, cost $823 and included 28 kilograms of the iron alloy and 15 kilograms of the lead alloy. Solve a system of linear equations to determine the price per kilogram of each alloy. 8. A goldsmith has two gold alloys. The first alloy is 30% gold; the second alloy is 62% gold. Set-up and solve a system of two linear equations to determine how many grams of each alloy should be mixed to produce 80 grams of an alloy that is 52% gold? 9. A motorboat traveled a distance of 280 km in 5 hours while traveling with the current. Against the current, the return trip covering the same distance took 7 hours. Find the rate of the boat in calm water and find the rate of the current. 10. An investment adviser invested $11500 in two accounts. One investment earned 5% annual simple interest, and the other investment earned 7% annual simple interest. The amount of interest earned for one year was $661. Set-up and solve a system of equations to determine how much money was invested in each account? Page 3

4 2 Systems of Linear Equations in More than Two Variables 1. Use algebra to solve the following system of equations. 2x 5y + 5z = 4 x + 3y 3z = 2 2x + 4y + 5z = Solve the following systems of equations. { x 6y + 6z + 5w = 7 (a) y 6w = 5 x 5y + 7z + 6w = 6 (b) 2x + 5y 7z 7w = 0 x + w = 6 3. Consider the following system of equations. x + 3y 1z = 3 y + 4z = 2 kz = 0 (a) Find the values of k for which (if possible) the system: (i) one solution, (ii) no solution, (iii) infinitely many solutions. (b) Solve the system, if possible, when k = 8. (c) Solve the system, if possible, when k = Consider the following system of equations. x + 3y 5z = 3 y + 4z = 2 (k 2 10k + 9)z = k 2 1 (a) Determine values of k for which this system has no solution, if possible. (b) Determine values of k for this this system has infinitely many solutions, if possible. (c) Determine values of k for this this system has a unique solution, if possible. 5. Find the equation of a parabola y = ax 2 + bx + c that passes through the points ( 3, 39), ( 1, 21) and (2, 24). Then check that your equation works. 6. A coin bank contains only nickels, dimes and quarters. There are 7 fewer dimes than 3-times the number of nickels. The are 4 more quarters than 2-times the number of dimes. If the coin bank has a total of $28.25 in it, how many of each type of coin does it contain? Page 4

5 3 Gaussian Elimination Method 1. Use Gaussian elimination to solve the following system of equations 4x 9y 14z = 3 3x + 9y + 5z = 1 x 2y 4z = 0 2. Use Gaussian elimination to solve the following system of equations 3x 10y 10z = 2 3x + 11y + 8z = 5 x 3y 4z = 1 3. Use Gaussian elimination to solve the following system of equations 4x 9y 15z = 3 5x + 13y + 17z = 2 x 2y 4z = 1 4. Use Gaussian elimination to solve the following system of equations 4x 5y 10z = 2 4x + 7y + 5z = 1 x 1y 3z = 1 4y + 8z = The augmented matrix k k variables x, y, z where k is a real number. (a) Write down the system of equations. represents a system of equations in the (b) Solve the system when k = The augmented matrix represents a system of equations in 0 0 k k(k + 5) the variables x, y, z where k is a real number. (a) Write down the system of equations. (b) Solve the system (if possible) when k = 5. (c) Solve the system (if possible) when k = The augmented matrix represents a system of equations 0 0 k(k 5) k 2 25 in the variables x, y, z where k is a real number. (a) Write down the system of equations. (b) For which value(s) of k are there (i) infinitely many solutions, (ii) no solution, and (iii) exactly one solution? Page 5

6 4 The Algebra of Matrices 1. Let A = [ [ and B = (a) Find 4A [ [ Let A = and B = (a) Find 3A (b) Find B 3A. (c) Find 3A 2B (b) Find 4A B. 3. Let the matrices A and B be defined by [ 2 2 A = 3 3 and B = [ (a) Find the product AB if it exists, or explain why the product doesn t exist. (b) Find the product BA if it exists, or explain why the product doesn t exist. 4. Let the matrices A and B be defined by 1 1 A = 3 1 and B = 0 3 [ (a) Find the product AB if it exists, or explain why the product doesn t exist. (b) Find the product BA if it exists, or explain why the product doesn t exist. 5. Let the matrices A and B be defined by A = and B = (a) Find the product AB. (b) Find A 2. [ 6. Let E = (a) What are the dimensions of E? (b) Find the entry e 25 in the product. (Don t find the whole product, and write DNE if the entry does not exist). Page 6

7 7. (a) Find the following matrix product (b) Write the following matrix equation as an equivalent system of equations x y = z 6 (c) Write the system x y z 7x + 3y 6z = 7 2x + 2z = 3 7x + 3y 3z = 6 in matrix AX = B form. Page 7

8 5 The Inverse of a Matrix 1. Use matrix row operations on an augmented matrix to find the inverse of [ 5 7 A = 1 3 or show that the inverse does not exist. 2. Use matrix row operations on an augmented matrix to find the inverse of [ 5 1 A = 15 3 or show that inverse does not exist. [ a b 3. The inverse of A = is A 1 = c d [ 1 ad bc d c (a) Use this formula to find the inverse of the matrix [ 1 9 A = 1 7 b a. (b) Use your answer in (a) and the method of matrix inverses to solve the system of equations x 9y = 6 and 1x + 7y = 4 [ a b 4. The inverse of A = is A 1 = c d [ 1 ad bc d c (a) Use this formula to find the inverse of the matrix [ 5 7 A = 3 4 b a (b) Use the inverse you found in (a) to solve the system of equations. 5x + 7y = 2 and 3x + 4y = 3 (c) Solve the following system of equations using the method of your choice. 5x + 7y = 1 and 3x + 4y = 1 5. Find the inverse of the matrix , if it exists. Show all steps. 6. Find the inverse of the matrix A given below, if it exists. Show all steps A = Page 8

9 7. Solve the system x 6y + 5z = 3 x 3y + 2z = 1 3x + 8y 4z = 2 is using the fact that the inverse of. (You must use the requested method). Page 9

10 6 Infinite Sequences and Summation Notation 1. Find the first three terms and 12th term of a sequence whose nth term is given by a n = ( 1)n 1 1 6n 2. Find the 33th term of the sequence whose nth term is defined by a n = (n + 2)! (n 1)!, n 1 3. Evaluate the following sum 8 ( 1) n (5n) n=5 4. (a) Write an expression for a n, the nth term of the sequence of even numbers 2, 4, 6, 8, 10, (b) Write an expression for b n, the nth term of the sequence of odd numbers 1, 3, 5, 7, 9, 11,... (c) Write an expression for c n, the nth term of the sequence whose first six terms are 7, 9, 11, 13, 15, 17,... (d) Write an expression for d n, the nth term of the sequence whose first six terms are 12, 14, 16, 18, 20, 22,... (e) Write an expression for e n, the nth term of the sequence whose terms are multiples of 7, so its first six terms are 7, 14, 21, 28, 35, 42,... (f) Write an expression for f n, the nth term of the sequence whose first six terms are 4, 11, 18, 25, 32, 39, Express the following sum in summation notation Page 10

11 7 The Binomial Theorem 1. Expand the binomial (2v 6 3w) 5. (Show all steps; if you use Pascal s triangle, write it as far as you use it). 2. Use a binomial expansion to simplify the complex number (2 + 5i) 4 and write the final answer as a complex number in standard form. 3. Find the fifth term of the binomial expansion of (2x y 2 ) Find the term of the binomial expansion (x + y) 8 that contains x 6 y 2 5. Find the term of the binomial expansion (x 2 2y) 15 that contains x 18 Page 11