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Columbia University Chapter 7 Systems and Inequalities Questions Exercises

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(Exercises for Chapter 7: Systems and Inequalities) E.7.1 CHAPTER 7: Systems and Inequalities (A) means “refer to Part A,” (B) means “refer to Part B,” etc. (Calculator) means “use a calculator.” Otherwise, do not use a calculator. SECTIONS 7.1-7.3: SYSTEMS OF EQUATIONS When solving a system, only give solutions in  2 , the set of ordered pairs of real numbers. All such solutions correspond to intersection points of the graphs of the given equations. If there are no such solutions, write ∅ , the empty set or null set. ( ) Write solutions in a solution set as ordered pairs of the form x, y . Unless otherwise specified, do not rely on graphing or “trial-and-error point-plotting.” ⎧ x+y=5 1) Consider the system ⎨ . (A-E) ⎩5x − 3y = − 23 a) The graphs of the equations in the system are distinct lines in the xy-plane that are not parallel. How many solutions does this system have? b) Solve the system using the Substitution Method. c) Solve the system using the Addition / Elimination Method. ⎧ x 2 + y2 = 2 2) Consider the system ⎨ . (A-D) y = x + 2 ⎩ a) Find the solution set of the system. b) Use the solution set from a) to graph the equations in the system in the usual xy-plane. 3 ⎧ 2 2 ⎪x + y = 2 . (A-D) 3) Consider the system ⎨ ⎪ y 2 = x2 ⎩ a) What are the graphs of the equations in the system in the usual xy-plane? b) How many solutions does the system have? c) Find the solution set of the system. (Exercises for Chapter 7: Systems and Inequalities) E.7.2 ⎧⎪ x = y 2 4) Consider the system ⎨ . (A-D; Section 1.8) 2 ⎪⎩ x = 4 − y a) What are the graphs of the equations in the system in the usual xy-plane? b) How many solutions does the system have? c) Find the solution set of the system. ⎧⎪ x 2 + y = 0 5) Consider the system ⎨ . (A-D, F) 2 ⎪⎩ y − x = 1 a) Sketch graphs of the equations in the system in the usual xy-plane. b) Based on your graphs in a), find the solution set of the system. c) Verify the solution set by using the Substitution Method or the Addition / Elimination Method to solve the system. 6) Solve the following systems. (A-D, F) ⎧⎪ y = 3x 2 − x a) ⎨ 2 ⎪⎩ y = 2x − 3x + 8 ⎧ x 2 + 4y 2 = 2 b) ⎨ ⎩ 3x − 2y = − 4 ⎧⎪ x 2 + y 2 = 1 c) ⎨ 2 2 ⎪⎩ x − y = 4 ⎧0 = 0 7) ADDITIONAL PROBLEM. Solve the system ⎨ . (A-D, F) 0 = 1 ⎩ (Exercises for Chapter 7: Systems and Inequalities) E.7.3. SECTION 7.4: PARTIAL FRACTIONS 1) Write the PFD (Partial Fraction Decomposition) Form for the following. Do not find the unknowns (A, B, etc.). (A-C) 1 a) ( x + 4 ) ( x − 3) x 2 + 1 ( b) c) x+5 ) (x ( x3 x − 1 2 2 ) ) +3 2 3t 2 + 2t − 2 ( )( ) t 2 ( 2t + 5 ) 2t 2 + 5 t 2 + t + 1 3 2) Write the PFD (Partial Fraction Decomposition) for the following. (A-G) a) 3x − 5 x 2 − 5x + 6 2x 2 − 3x + 19 b) 3 . (Hint: Use the Rational Zero Test and Synthetic Division.) x + 4x 2 − 7x − 10 9x 2 + 14x + 6 c) 2x 3 + x 2 d) x +1 x 2 − 8x + 16 e) 8x 2 + 7x + 12 ( x + 2) x2 + 1 ( ) 5x 2 − 5x + 12 f) 3 . (Hint: Use Factoring by Grouping.) x − 5x 2 + 3x − 15 − 5x 2 − 8x − 3 g) x3 + x2 + x h) 5t 3 − t 2 + 20t − 8 (t 2 +4 ) 2 x4 A B = + 3) A student writes: . Is this appropriate? Why or why not? ( x + 3) ( x + 5 ) x + 3 x + 5 (Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means “refer to Part A,” (B) means “refer to Part B,” etc. Most of these exercises can be done without a calculator, though one may be permitted. SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS 1) Give the size and the number of entries (or elements) for each matrix below. (A) ⎡ 14 1 / 5 ⎤ ⎢ ⎥ 3 ⎥ a) ⎢ − 2 ⎢ π − 4.7 ⎥ ⎣ ⎦ 1 0 0 5⎤ ⎡ 3 ⎢− 4 2 / 3 9 0 12 ⎥ ⎢ ⎥ b) ⎢ 13 0 −1 / 2 −11 14 ⎥ ⎢ ⎥ 0 0 0 0⎦ ⎣ 2 ⎡1 2 3⎤ c) ⎢⎢ 0 1 4 ⎥⎥ ⎢⎣ 0 0 1 ⎥⎦ ⎧ 3x − y = 18 2) Consider the system ⎨ . (A-D) x + 2y = −1 ⎩ a) Write the augmented matrix for the given system. b) What size is the coefficient matrix? c) What size is the right-hand side (RHS)? d) Switch Row 1 and Row 2 of the augmented matrix. Write the new matrix. e) Take the matrix from d) and add ( − 3) times Row 1 to Row 2. Write the new matrix. f) Take the matrix from e) and divide Row 2 by ( − 7 ) ; that is, multiply Row 2 by ⎛ 1⎞ ⎜⎝ − ⎟⎠ . Write the new matrix, which will be in Row-Echelon Form (Part F). 7 g) Write the system corresponding to the matrix from f). h) Solve the system from g) using Back-Substitution, and write the solution set. i) Check your solution in the original system. (This is typically an optional step.) (Exercises for Chapter 8: Matrices and Determinants) E.8.2 In Exercises 3-12, use matrices and Gaussian Elimination with Back-Substitution. ⎧ 4x + 2y = − 3 3) Solve the system ⎨ . (A-D) x + y = − 2 ⎩ ⎧ 3x1 − 9x2 = 57 4) Solve the system ⎨ . (A-D) − 5x + 4x = −18 1 2 ⎩ 16 ⎧ ⎪⎪− 2a + 3b = − 3 5) ADDITIONAL PROBLEM: Solve the system ⎨ . (A-D) 1 17 ⎪ a − 4b = ⎪⎩ 2 3 ⎧ x + 3y = 6 6) Solve the system ⎨ . (A-E) − 2x − 6y = − 9 ⎩ ⎧ x + 3y = 6 7) Consider the system ⎨ . How many solutions does the system ⎩− 2x − 6y = −12 have? (A-E) ⎧ y − 2z = 14 ⎪ 8) Solve the system ⎨ x + z = − 3 . Begin by rewriting the system. (A-D) ⎪ 4x + 6z = − 22 ⎩ ⎧ 3x − 10y + 9z = 50 ⎪ 9) Solve the system ⎨− 2x + 6y − z = − 27 . (A-D) ⎪ x − 2y − z = 10 ⎩ ⎧ a − 4b − 3c = − 5 ⎪ 10) Solve the system ⎨ a − 4b − c = − 2 . (A-D) ⎪2a − 7b − 4c = − 7 ⎩ ⎧− 2x1 + 8x2 − 10x3 = 20 ⎪ 11) Solve the system ⎨ 3x1 + 5x2 + x3 = − 5 . (A-D) ⎪ − 4x − x + 3x = −12 1 2 3 ⎩ ⎧ x1 − 2x2 − x3 = 3 ⎪ 12) Solve the system ⎨5x1 − 10x2 − 5x3 = 11 . (A-E) ⎪ 4x + 3x + 2x = −18 2 3 ⎩ 1 (Exercises for Chapter 8: Matrices and Determinants) E.8.3 ⎡1 0 − 7 0 1 ⎤ ⎢ ⎥ 13) Consider the augmented matrix ⎢ 0 1 3 0 − 2 ⎥ . (F, G) ⎢⎣ 0 0 0 1 4 ⎥⎦ a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? ⎡1 ⎢ 0 14) Consider the augmented matrix ⎢ ⎢0 ⎢ ⎣0 4 0 0 0 1 1 0 0 0 0 1 0 2⎤ ⎥ −1⎥ . (F, G) 0⎥ ⎥ 0⎦ a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? ⎡1 1 1 1 ⎤ ⎢ ⎥ 15) Consider the augmented matrix ⎢ 0 2 4 3 ⎥ . (F, G) ⎢⎣ 0 0 3 6 ⎥⎦ a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? ⎡ 1 0 2 −1⎤ ⎢ ⎥ 0 0 1 2 ⎥ . (F, G) 16) Consider the augmented matrix ⎢ ⎢0 0 1 3 ⎥ ⎢ ⎥ 0 0 0 0 ⎣ ⎦ a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? 17) ADDITIONAL PROBLEM: Solve the systems below using Gauss-Jordan Elimination. (A-H) ⎧x − z = 3 ⎧ 4x1 + 4x2 − 3x3 = 7 ⎧2x + 6y = − 6 ⎪ ⎪ a) ⎨ ; b) ⎨2y = 16 (Rewrite first); c) ⎨ 5x1 + 7x2 − 13x3 = − 9 ⎩ 4x + 13y = −14 ⎪ 3x + z = 13 ⎪ x + 2x − 5x = − 6 2 3 ⎩ ⎩ 1 (Exercises for Chapter 8: Matrices and Determinants) E.8.4 SECTION 8.2: OPERATIONS WITH MATRICES Assume that all entries (i.e., elements) of all matrices discussed here are real numbers. ⎡ 7 ⎡ −1 3 ⎤ ⎢ 1) Let A = ⎢⎢ 2 4 ⎥⎥ and B = ⎢ 8 ⎢− 9 ⎢⎣ 7 π ⎥⎦ ⎣ 0 ⎤ ⎥ − 3 ⎥ . (C) 5 ⎥⎦ a) Find A + B . b) Find 3A − 4B . ⎡4 ⎤ 2) Let A = [ 3 2 ] and B = ⎢ ⎥ . Find AB . (D) ⎣5 ⎦ ⎡ 4 −1⎤ ⎡ 1 0⎤ 3) Let A = ⎢ and B = ⎥ ⎢ − 3 2 ⎥ . (B, D, E) ⎣2 3 ⎦ ⎣ ⎦ a) Find AB. b) Find BA. c) Yes or No: Is AB = BA here? 6 4⎤ ⎡1 ⎡2 0 7⎤ ⎢ 4) Let C = ⎢ and D = ⎢ − 2 3 1 ⎥⎥ . Find the indicated matrix products. ⎥ ⎣ −1 4 − 3⎦ ⎢⎣ 2 −1 0 ⎥⎦ If the matrix product is undefined, write “Undefined.” (D, E) a) CD b) DC c) D 2 , which is defined to be DD 5) Assume that A is an 8 × 10 matrix and B is a 10 × 7 matrix. (D, E) a) What is the size of the matrix AB? b) Let C = AB . Explain how to obtain the matrix element c56 . 6) If A is an m × n matrix and B is a p × q matrix, under what conditions are both AB and BA defined? (E) ⎡2 0 0 ⎤ 7) Let D = ⎢⎢ 0 3 0 ⎥⎥ . This is called a diagonal matrix. (D, E) ⎢⎣ 0 0 4 ⎥⎦ a) Find D 2 . b) Based on a), conjecture (guess) what D10 is. (Calculator) (Exercises for Chapter 8: Matrices and Determinants) E.8.5 8) Assume that A is a 3 × 4 matrix, B is a 3 × 4 matrix, and C is a 4 × 7 matrix. Find the sizes of the indicated matrices. If the matrix expression is undefined, write “Undefined.” (C-E) a) A + 4B ; b) A − C ; c) AB ; d) AC ; e) AC + BC ; f) ( A + B ) C 9) Write the identity matrix I 4 . (F) 10) ADDITIONAL PROBLEM: In Section 8.1, Exercise 17c, you solved the system ⎧ 4x1 + 4x2 − 3x3 = 7 ⎪ ⎨ 5x1 + 7x2 − 13x3 = − 9 . This system can be written in the form AX = B . (G) ⎪ x + 2x − 5x = − 6 2 3 ⎩ 1 a) Identify A. b) Identify X (in the given system, before it is solved). c) Identify B. d) When you solved this system using Gauss-Jordan Elimination, what was the coefficient matrix of the final augmented matrix in Reduced Row-Echelon (RRE) Form? What was the right-hand side (RHS) of that matrix? SECTION 8.3: THE INVERSE OF A SQUARE MATRIX (THESE ARE ALL ADDITIONAL PROBLEMS) Assume that all entries (i.e., elements) of all matrices discussed here are real numbers. 1) If A is an invertible 2 × 2 matrix, what is AA −1 ? (B) 2) Find the indicated inverse matrices using Gauss-Jordan Elimination. If the inverse matrix does not exist, write “A is noninvertible.” (C) ⎡ 3 −1⎤ a) A −1 , where A = ⎢ . (Also check by finding AA −1 .) ⎥ ⎣1 2 ⎦ ⎡2 4 ⎤ b) A −1 , where A = ⎢ ⎥. ⎣ 6 12 ⎦ ⎡ 0 4 8⎤ c) A −1 , where A = ⎢⎢ 0 0 3 ⎥⎥ ⎢⎣ − 5 0 0 ⎥⎦ ⎡ 2 13 − 5 ⎤ d) A , where A = ⎢⎢ 1 5 2 ⎥⎥ ⎢⎣ −1 − 2 − 8 ⎥⎦ −1 (Exercises for Chapter 8: Matrices and Determinants) E.8.6 ⎧ 3x1 − x2 = 15 3) Use Exercise 2a to solve the system ⎨ . (D) x + 2x = − 2 2 ⎩ 1 ⎧ 2x1 +13x2 − 5x3 = 7 ⎪ 4) Use Exercise 2d to solve the system ⎨ x1 + 5x2 + 2x3 = −1 . (D) ⎪− x − 2x − 8x = 7 2 3 ⎩ 1 ⎡ 3 −1⎤ 5) Let A = ⎢ ⎥ , as in Exercise 2a. (E, F) 1 2 ⎣ ⎦ a) Find det ( A ) . b) Find A −1 using the shortcut from Part F. Compare with your answer to Exercise 2a. ⎡2 4 ⎤ 6) Let A = ⎢ ⎥ , as in Exercise 2b. (E, F) ⎣ 6 12 ⎦ a) Find det ( A ) . b) What do we then know about A −1 ? Compare with your answer to Exercise 2b. 7) Assume that A and B are invertible n × n matrices. Prove that ( AB ) = B −1 A −1 . (B) −1 8) Verify that the shortcut formula for A −1 given in Part F does, in fact, give the inverse ⎡a b ⎤ of a matrix A, where A = ⎢ ⎥ and det ( A ) ≠ 0 . (E, F) c d ⎣ ⎦ SECTION 8.4: THE DETERMINANT OF A SQUARE MATRIX Assume that all entries (i.e., elements) of all matrices discussed here are real numbers, unless otherwise indicated. 1) Find the indicated determinants. (B) a) Let A = [ − 4 ] . Find det ( A ) , or A . ⎡3 b) Let B = ⎢ ⎣5 ⎡5 c) Let C = ⎢ ⎣3 ⎡ 30 d) Let D = ⎢ ⎣5 2⎤ . Find det ( B ) , or B . 4 ⎥⎦ 4⎤ . Find det ( C ) , or C . Compare with b). 2 ⎥⎦ 20 ⎤ . Find det ( D ) , or D . Compare with b). 4 ⎥⎦ (Exercises for Chapter 8: Matrices and Determinants) E.8.7 ⎡3 5⎤ e) Let E = ⎢ ⎥ . Find det ( E ) , or E . Compare with b). 2 4 ⎣ ⎦ T Note: E = B , the transpose of B. Rows become columns, and vice-versa. f) Find 2 −4 . −3 −7 g) Find 4 5 . 0 0 h) Find 4 5 . 40 50 i) Find a b . Compare with g) and h). ca cb 1 ex j) Find . Here, the entries correspond to functions. x e2 x k) Find sin θ − cos θ cos θ . Here, the entries correspond to functions. sin θ 4 3⎤ ⎡2 ⎢ 2) Let A = ⎢ −1 − 3 3 ⎥⎥ . We will find det ( A ) , or A , in three different ways. (B, C) ⎢⎣ 5 1 2 ⎥⎦ a) Use Sarrus’s Rule, the shortcut for finding the determinant of a 3 × 3 matrix. b) Expand by cofactors along the first row. c) Expand by cofactors along the second column. 5 − 2⎤ ⎡1 ⎢ 3) Let B = ⎢ 3 4 0 ⎥⎥ . We will find det ( B ) , or B , in three different ways. (B, C) ⎢⎣ −1 − 4 3 ⎥⎦ a) Use Sarrus’s Rule, the shortcut for finding the determinant of a 3 × 3 matrix. b) Expand by cofactors along the second row. c) Expand by cofactors along the third column. 1 2 3 4) Find 10 20 30 . Based on this exercise and Exercise 1i, make a conjecture (guess) 4 5 6 about determinants. (B, C) (Exercises for Chapter 8: Matrices and Determinants) E.8.8. ⎡10 400 500 ⎤ 5) Let C = ⎢⎢ 0 20 600 ⎥⎥ . C is called an upper triangular matrix. Find det ( C ) , or ⎢⎣ 0 0 30 ⎥⎦ C . Based on this exercise, make a conjecture (guess) about determinants of upper triangular matrices such as C. What about lower triangular matrices? (B, C) −1 4 3 2 6) Find −1 3 0 −2 7) Find 13 92 2 e 0 0 −2 0 1 . (B, C) 0 −4 2 1 42 5 −π − 3.2 π 0 267 0 3 5 e 9876 0 . (C) 4 − λ −2 = 0 for λ (lambda). In doing so, you are finding the 1 1− λ ⎡4 − 2⎤ eigenvalues of the matrix ⎢ ⎥. ⎣1 1 ⎦ 8) Solve the equation SECTION 8.5: APPLICATIONS OF DETERMINANTS (THESE ARE ALL ADDITIONAL PROBLEMS) ⎧ 4x + 2y = − 3 1) Use Cramer’s Rule to solve the system ⎨ in Section 8.1, Exercise 3. (A) x + y = − 2 ⎩ ⎧ 3x1 − x2 = 15 2) Use Cramer’s Rule to solve the system ⎨ in Sec. 8.3, Exercise 3. (A) x + 2x = − 2 2 ⎩ 1 3) Find the area of the parallelogram determined by each of the following pairs of position vectors in the xy-plane. Distances and lengths are measured in meters. (B) a) 4, 0 and 0, 5 b) 2, 5 and 7, 3 c) 1, 4 and 3, 12 . (What does the result imply about the vectors?) 4) Find the area of the triangle with vertices ( − 2, −1) , ( 3, 1) , and (1, 5 ) in the xy-plane. Distances and lengths are measured in meters. (B) (Exercises for Chapter 9: Discrete Mathematics) E.9.1 CHAPTER 9: Discrete Mathematics (A) means “refer to Part A,” (B) means “refer to Part B,” etc. Most of these exercises can be done without a calculator, though one may be permitted. SECTION 9.1: SEQUENCES AND SERIES, and SECTION 9.6: COUNTING PRINCIPLES 1) Let an = n 2 + n . Write a1 , a2 , and a3 . (A) 2) Let an = ( −1) ( 2n ) . Write a1 , a2 , a3 , and a4 . (A-C) n 3) Let an = ( −1) n−1 ( 2n − 1) . Write a1 , a2 , a3 , and a4 . (A-C) 4) Evaluate 6!. (D) 5) How many ways are there to order five tasks on a “To Do” list? (D, E) 6) Seven of your friends are sitting in a room. You have two identical plane tickets that you will give to two of them. In how many ways can this be done? (D, E) 7) Ten basketball players are to be divided into two teams of five people each. One team will be called “Team A,” and the other team will be called “Team 1.” In how many ways can the players be assigned to the teams? We do not yet care about positions on the teams. (D, E) 8) Simplify the expressions. Assume that n is an integer such that n ≥ 2 . (D, E) ( n + 2 )! ( n − 1)! ( 3n − 2 )! a) ( n + 1) ( n!) ; b) ; c) ; d) (do not multiply out) n! ( n + 1)! ( 3n + 3)! ⎧⎪ a1 = 4 9) Consider the sequence recursively defined by: ⎨ . a = a + 10 k ∈, k ≥ 1 ⎪⎩ k +1 k Find a1 , a2 , a3 , and a4 . This sequence is an arithmetic sequence, which we will discuss further in Section 9.2. (F) ( ) ⎧ a1 = 2 ⎪ 10) Consider the sequence recursively defined by: ⎨ . 1 ⎪ ak +1 = ak k ∈, k ≥ 1 2 ⎩ Find a1 , a2 , a3 , and a4 . This sequence is a geometric sequence, which we will discuss further in Section 9.3. (F) ( ) (Exercises for Chapter 9: Discrete Mathematics) E.9.2 ⎧⎪ a1 = −1 11) Consider the sequence recursively defined by: ⎨ ⎪⎩ ak +1 = 3ak − 2 Find a1 , a2 , a3 , and a4 . (F) ⎧ a =2 ⎪⎪ 1 12) Consider the sequence recursively defined by: ⎨ a2 = 3 ⎪ ⎪⎩ ak +2 = ak +1 ak Find a1 , a2 , a3 , a4 , and a5 . (F) ( ) k ∈, k ≥ 1 ( k ∈, ) . . k ≥1 4 13) Evaluate ∑ k . (G) k =1 ∑( j 7 14) Evaluate 2 j=3 4 15) Evaluate ∑ i=2 ) + 1 . (G) ( −1)i . (i is not the imaginary unit here.) (G) i ∞ 16) Find S4 , the fourth partial sum of the series ∑a k =1 k , where ak = 3k . (G, H) 17) Write a nonrecursive expression (formula) for the apparent general n th term, an , for each of the following sequences. Let a1 be the initial term; i.e., assume that n begins with 1. (A-D) a) 7, 8, 9, 10, 11, … b) 5, 10, 15, 20, 25, … c) 4, 7, 10, 13, 16, … 1 1 1 1 d) 1, , , , ,… 2 6 24 120 7 7 7 7 e) 7, , , , ,… 4 9 16 25 2 4 6 8 10 f) ,− , ,− , ,… 3 5 7 9 11 g) − 2 , 4, − 8 , 16, − 32 , … 18) Express the apparent series using summation notation. Use k as the index of summation. (A-D, G, H) a) 3 + 6 + 9 + 12 + 15 + 18 ; this is a finite series. 1 1 1 1 1 − + − + − ... ; this is an infinite series. b) 4 16 64 256 1024 (Exercises for Chapter 9: Discrete Mathematics) E.9.3 SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS 1) Consider the arithmetic sequence: − 5 , 1, 7, 13, 19, …. (A) a) What is the initial term, a? b) What is the common difference, d? 2) Consider the arithmetic sequence: 3 1 1 , 1, , 0, − , …. (A) 2 2 2 a) What is the initial term, a? b) What is the common difference, d? 3) Consider the arithmetic sequence with initial term 7 and common difference 3. Assume that the initial term is a1 . (A, B) a) Write the first four terms of this sequence. b) Find S4 , the fourth partial sum. c) Find a60 , the 60th term of this sequence. 4) Consider the arithmetic sequence: 2 , − 3 , − 8 , −13 , −18 , …. (A, B) a) Write a simplified, nonrecursive expression (formula) for the general n th term, an , for this sequence. Let a1 be the initial term. b) Use a) to find a387 . (Calculator) 5) An arithmetic sequence has a1 = 6 and a200 = 1399 . Find a123 . (Calculator) SECTION 9.3: GEOMETRIC SEQUENCES, PARTIAL SUMS, and SERIES 1) Consider the geometric sequence: 4, 20, 100, 500, 2500, …. (A) a) What is the initial term, a? b) What is the common ratio, r? 2) Consider the geometric sequence: a) What is the initial term, a? b) What is the common ratio, r? 6 2 2 2 2 ,− , ,− , …. (A) 7 7 21 63 189 (Exercises for Chapter 9: Discrete Mathematics) E.9.4 3) Consider the geometric sequence with initial term 5 and common ratio − 4 . Assume that the initial term is a1 . (A, B) a) Write the first four terms of this sequence. b) Find S4 , the fourth partial sum. c) Find a12 , the 12th term of this sequence. (Calculator) d) As n → ∞ , do the terms of the sequence approach a real number? If so, what number? 2 1 1 3 9 , , , , , … . (A, B) 9 3 2 4 8 a) Write a simplified, nonrecursive expression (formula) for the general n th term, an , for this sequence. Let a1 be the initial term. 4) Consider the geometric sequence: b) Use a) to find a10 . (Calculator) c) Verify your answer to b) by recursively using the common ratio to find a6 , a7 , a8 , a9 , and a10 . d) As n → ∞ , do the terms of the sequence approach a real number? If so, what number? 5) A geometric sequence has a1 = 3 and a4 = − 24 . (A, B) 125 a) Find a2 and a3 . b) As n → ∞ , do the terms of the sequence approach a real number? If so, what number? 6) Let {an } be the sequence from Exercise 3. Does the series ∞ ∑a n n =1 converge or diverge? If the series converges, find its sum. (D) 7) Let {an } be the sequence from Exercise 4. Does the series ∞ ∑a n n =1 converge or diverge? If the series converges, find its sum. (D) 8) Let {an } be the sequence from Exercise 5. Does the series ∞ ∑a n =1 diverge? If the series converges, find its sum. (D) ∞ 9) For what values of x does the serie ...
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Final Answer

attached are the answers.u should give me tip becuase this was very exhaustive work. i could have done it easily with pen and paper.

Section 7.1- 7.3
Problem 1 –
𝑥+𝑦 =5
5𝑥 − 3𝑦 = −23
a) As the graphs are distinct nonparallel lines. Hence there is one solution of the system.
b) Substitution method
𝑦 = 5−𝑥
Substituting y in second equation, we get
5𝑥 − 3(5 − 𝑥) = −23
Solving this equation, we get,
𝑥 = −1
Putting x in the previous equation we get,
𝑦 = 5−𝑥
𝑦=6

c) Addition/Elimination Method
Multiplying the first equation by 3 on both sides, we get
3𝑥 + 3𝑦 = 15
5𝑥 − 3𝑦 = −23
Adding these equations, we get
8𝑥 = −8

𝑥 = −1
Back substituting the value of x in the equation,
𝑦=6
Problem 2 –
𝑥2 + 𝑦2 = 2
𝑦 =𝑥+2
a) Substituting 𝑦 = 𝑥 + 2 in first equation
𝑥 2 + (𝑥 + 2)2 = 2
Solving (𝑥 + 2)2
𝑥 2 + (𝑥 2 + 22 + (2 × 𝑥 × 2)) = 2
Solving
𝑥 2 + (𝑥 2 + 22 + 4𝑥) = 2
2𝑥 2 + 4𝑥 + 4 = 2
2(𝑥 2 + 2𝑥 + 2) = 2
𝑥 2 + 2𝑥 + 2 = 1
𝑥 2 + 2𝑥 + 1 = 0
(𝑥 + 1)(𝑥 + 1) = 0
𝑥+1=0
𝑥 = −1
𝑃𝑢𝑡𝑡𝑖𝑛𝑔 𝑥 = −1 𝑖𝑛𝑡𝑜 𝑠𝑒𝑐𝑜𝑛𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑦 = (−1) + 2

𝑦=1
Hence the solution set is (-1,1)
b)

Problem 3 –
𝑥2 + 𝑦2 =

3
2

𝑦√2 = 𝑥 2
a)
b) given,
𝑦√2 = 𝑥 2
Substituting the value of 𝑥 2 into the first equation
𝑦√2 + 𝑦 2 =

3
2

2𝑦√2 + 2𝑦 2 = 3
2𝑦√2 + 2𝑦 2 − 3 = 0
2𝑦 2 + 2𝑦√2 − 3 = 0

Using Quadratic formula 𝑥 =

−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎

Putting values for 𝑥, 𝑎 𝑎𝑛𝑑 𝑏
−2√2 ± √(2√2)2 − (4 × 2 × −3)
𝑦=

2×2

Solving above equation

𝑦=

−2√2 ± √8 − (−24)
2×2
𝑦=

−2√2 ± 4√2
4

𝑦=

√2(−2 ± 4)
4

𝑦=

√2(−2 ± 4)
4

𝑦=

√2(−2 ± 4)
4

Acc to graph, removing negative,

𝑦=

√2
2

Putting the value of y into second equation
√2
× √2 = 𝑥 2
2
Solving
𝑥 = √1
Hence there are 2 set of solutions
c) (1,

√2
),
2

(-1,

√2
)
2

Problem 4 –
𝑥 = 𝑦2
𝑥 = 4 − 𝑦2
a) –
b) Given
𝑥 = 4 − 𝑦2
Putting the value of 𝑦 2 in equation 1
𝑥 = −(𝑥 − 4)
𝑥 = −𝑥 + 4
Hence
𝑥=2
Putting this value in equation 2
2 = 4 − 𝑦2

𝑦2 = 2
Hence y = (±√2)
So the system have 2 possible solution sets
c) (2, √2), (2, −√2)

Problem 5 –
𝑥2 + 𝑦 = 0
𝑦 − 𝑥2 = 1
a)


b) As these two graphs do not intersect each other hence there is no solution
c) Given

𝑦 = 1 + 𝑥2
Putting it in first equation
𝑥2 + 1 + 𝑥2 = 0
2𝑥 2 = −1
𝑥2 = −

1
2

It has a non-real solution for x hence there is no solution
Problem 6 –
a)
𝑦 = 3𝑥 2 − 𝑥
𝑦 = 2𝑥 2 − 3𝑥 + 8
Putting the value of y from first equation to second
3𝑥 2 − 𝑥 = 2𝑥 2 − 3𝑥 + 8
Solving
3𝑥 2 − 2𝑥 2 + 3𝑥 − 𝑥 − 8 = 0
𝑥 2 + 2𝑥 − 8 = 0
(𝑥 + 4)(𝑥 − 2) = 0
Hence
𝑥 = (−4,2)
Putting the value 𝑥,

𝑦 = 3𝑥 2 − 𝑥

When 𝑥 = −4
𝑦 = 3(−4)2 − (−4)
𝑦 = 52
When 𝑥 = 2
𝑦 = 3(2)2 − (2)
𝑦 = 10
Solution sets ( -4,52), (2,10)
b)
𝑥 2 + 4𝑦 2 = 2
3𝑥 − 2𝑦 = −4
From second equation
𝑦 = −(−4 − 3𝑥)
𝑦=

3𝑥 + 4
2

Putting the value of y in first equation
𝑥 2 + 4(
𝑥 2 + 4(

3𝑥 + 4 2
) =2
2

9𝑥 2 16 24𝑥
+
+
)=2
4
4
4

10𝑥 2 + 24𝑥 + 14 = 0
5𝑥 2 + 12𝑥 + 7 = 0
Factoring above,
7
(𝑥 + ) (𝑥 + 1) = 0
5

7

Hence x = -5 𝑎𝑛𝑑 − 1
7

When x = -5
7
3( − ) + 4
5
𝑦=
2
𝑦=−

1
10

When x= -1
𝑦=

3( −1) + 4
2
𝑦=

1
2

c)
𝑥2 + 𝑦2 = 1
𝑥2 − 𝑦2 = 4
Adding these two equations,
𝑥2 + 𝑦2 + 𝑥2 − 𝑦2 = 5
2𝑥 2 = 5
Hence
𝑥2 =

5
2

Putting this value into equation gives no real value for y. Hence there is no possible
solution

Section 7.4
Problem 1 –

a)

1
(𝑥+4)(𝑥−3)(𝑥 2 +1)

Partial Fraction Decomposition for this will be
𝐴

𝐵

𝐶𝑥+𝐷

+
+ (𝑥 2
(𝑥+4) (𝑥−3)

+1)

A, B, C, D ∈ 𝑹
b)

𝑥+5
𝑥 3 (𝑥−1)2 (𝑥 2 +3)2

Partial Fraction Decomposition for this will be
𝐴
𝑥

+

𝐵
𝑥2

+

𝐶
𝑥3

+

𝐷
(𝑥−1)

+

𝐸
(𝑥−1)2

+

𝐹𝑥+𝐺
𝑥 2 +3

𝐻𝑥+𝐼

+ (𝑥 2

+3)2

A, B, C, D, E, F, G, H, I ∈ 𝑹
c)

3𝑡 2 +2𝑡−2
𝑡 2 (2𝑡+5)3 (2𝑡 2 +5)(𝑡 2 +𝑡+1)

Partial Fraction Decomposition for this will be
𝐴
𝑡

+

𝐵
𝑡2

+

𝐶
2𝑡+5

𝐷

+ (2𝑡+5)2 +

𝐸
(2𝑡+5)3

+

𝐹𝑡+𝐺
2𝑡 2 +5

A, B, C, D, E, F, G, H, I ∈ 𝑹

Question 2 –

a)

3𝑥−5
𝑥 2 −5𝑥+6

Partial Fraction Decomposition for this will be

+

𝐻𝑡+𝐼
𝑡 2 +𝑡+1

𝐴
(𝑥−3)

𝐵

+ (𝑥−2)

Comparing the equations, we get
3𝑥 − 5 = 𝐴(𝑥 − 2) + 𝐵(𝑥 − 3)
𝑇ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ℎ𝑜𝑙𝑑𝑠 𝑔𝑜𝑜𝑑 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥. Hence substituting x as 2, we get
3 ∗ 2 − 5 = 𝐴(2 − 2) + 𝐵(2 − 3)
𝐵 = −1
Similarly substituting x as 3, we get
3 ∗ 3 − 5 = 𝐴(3 − 2) + 𝐵(3 − 3)
𝐴=4
Hence, the partial fraction decomposition is,
4

1

(𝑥−3) (𝑥−2)
b)

2𝑥 2 −3𝑥+19
𝑥 3 +4𝑥 2 −7𝑥−10

Partial Fraction Decomposition for this will be
𝐴

𝐵

𝐶

+
+ (𝑥−2)
(𝑥+5) (𝑥+1)
Comparing the equations, we get
2𝑥 2 − 3𝑥 + 19 = 𝐴(𝑥 + 1)(𝑥 − 2) + 𝐵(𝑥 + 5)(𝑥 − 2) + 𝐶(𝑥 + 5)(𝑥 + 1)
𝑇ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ℎ𝑜𝑙𝑑𝑠 𝑔𝑜𝑜𝑑 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥. Hence substituting x as 2, we get
2 ∗ 2 ∗ 2 − 3 ∗ 2 + 19 = 𝐶(2 + 5)(2 + 1)
𝑆𝑜𝑙𝑣𝑖𝑛𝑔 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑤𝑒 𝑔𝑒𝑡 𝐶 = 1
Similarly substituting x as -1 we get,
2 ∗ (−1) ∗ (−1) − 3 ∗ (−1) + 19 = 𝐵(−1 + 5)(−1 + 1)

Solving the equation, we get B = -2
Similarly substituting x as -5, we get
2 ∗ (−5) ∗ (−5) − 3 ∗ (−5) + 19 = 𝐴(−5 + 1)(−5 − 2)
𝑆𝑜𝑙𝑣𝑖𝑛𝑔 𝑡ℎ𝑒 𝑒...

manankul (1815)
Boston College

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