Alg2 - CH4 Practice Test

Transcription

1 lg2 - CH4 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Evaluate + C, if possible a. Not possible c. b Find the product, if possible a b C d c. Not possible d July 2004 sales of car and car by two car dealers is shown in the first table. Use a product matrix to find the total profit from the cars for each dealer. July 2004 Car Sale Car Car ealer 6 ealer Car Profits

2 Revenue ($) ealer Cost Profit ($) ($) Car 8,700 2,00 5,800 Car 2,800 3,200,600 a. The total profit from cars and for dealer is $2,200 and for dealer 2 is $65,600. b. The total profit from cars and for dealer is $04,400 and for dealer 2 is $48,200. c. The total profit from cars and for dealer is $87,000 and for dealer 2 is $2,200. d. The total profit from cars and for dealer is $65,600 and for dealer 2 is $2, Evaluate 4 a. b , if possible. c d. Not possible What effect does multiplying the coordinates of a figure by have? 0 2 a. The figure is reduced and rotated counter-clockwise 0º. b. The figure is enlarged and reflected across the y-axis. c. The figure is reduced and reflected across the x-axis. d. The figure is enlarged and rotated clockwise 0º. 6. Use Cramer s rule to solve the system of equations. Ï 3x 4y Ì Ô ÓÔ x + 0y 3 a. (2, 0) c. (78, 72) b. (2, 3) d. (3, 2) Find the determinant of a. 2 c. 5.

3 b. 5 d nutrition specialist planned a low-calorie, low-fat diet for a patient. The patient s daily calorie intake from 540 grams of food will be 240 calories with 40% of the calories from protein. There are 4 calories per gram of protein, 4 calories per gram of carbohydrate, and calories per gram of fat. How many daily grams of protein (p), carbohydrates (c), and fat (f) will this diet include? (Hint: Use Cramer s rule to solve.) a. p 24, c 24, f 50 c. p 24, c 255, f 44 b. p 200, c 20, f 50 d. p 200, c 26, f 44. etermine whether and are inverses. a. No, they are not inverses. b. Yes, they are inverses. Ï x + y 8 0. Write the augmented matrix for the system Ì Ô z y 4. a. b ÓÔ x z + 3 c. 0 8 d. 3x + 3y Ï. Write the augmented matrix for Ì Ô, and solve. Round your answer to the nearest ÓÔ 2x + 2y 6 tenth. a. c ; x , y 2.5 ; x 2.0, y b ; x 5.0, y 2.0 d ; x 2.5, y 6.8

4 2. homeless shelter used a generous donation to purchase items worth a total of $2,200. lankets cost $5 each, a pair of boots cost $20 each, and coats cost $25 each. There are 7 blankets for every coat, and twice as many pairs of boots as coats. Solve by using row reduction on a calculator. How many of each item were purchased? a. There are 5 blankets, 70 pairs of boots, and 35 coats. b. There are 54 blankets, 44 pairs of boots, and 22 coats. c. There are 220 blankets, 5 pairs of boots, and 3 coats. d. There are 440 blankets, 0 pairs of boots, and 88 coats. Numeric Response 3. Solve for x x Find the determinant of the matrix. 8 5 Matching Match each vocabulary term with its definition. a. row operation b. coefficient matrix c. reduced row-echelon form d. square matrix e. constant matrix f. variable matrix g. row reduction h. augmented matrix 5. a matrix that consists of the coefficients and the constant terms in a system of linear equations 6. the process of performing elementary row operations on an augmented matrix to transform the matrix to reduced row-echelon form 7. an operation performed on a row of an augmented matrix that creates an equivalent matrix 8. the matrix of the variables in a linear system of equations. a form of an augmented matrix in which the coefficient columns form an identity matrix 20. the matrix of the constants in a linear system of equations Match each vocabulary term with its definition. a. reflection matrix b. rotation matrix c. translation matrix d. matrix equation

5 e. multiplicative inverse matrix f. multiplicative identity matrix g. additive inverse matrix h. additive identity matrix 2. a matrix used to rotate a figure about the origin 22. a square matrix with in every entry of the main diagonal and 0 in every other entry 23. an equation of the form X, where is the coefficient matrix, X is the variable matrix, and is the constant matrix of a system of equations 24. a matrix used to translate points on the coordinate plane 25. a matrix used to reflect a figure across a specified line of symmetry 26. a matrix where the product of it and the original matrix is the identity matrix

6 lg2 - CH4 Practice Test nswer Section MULTIPLE CHOICE. NS: Multiply each entry in by and each entry in C by, and then add the results C First, multiply each entry in matrix by the coefficient of and each entry in matrix C by the coefficient of C. Then, add the results. First, multiply each entry in the matrices by the coefficients. Then, add the results. First, multiply each entry in the matrices by the coefficient. Then, add the results. PTS: IF: asic REF: Page 248 OJ: 4-.4 Simplifying Matrix Expressions TOP: 4- Matrices and ata 2. NS: Check the dimensions. is 3 2, and is 2 3, so the product is defined and is ( 8) + 6( 2) 5( 0) + 6( 6) 5( 3) + 6( 4) 2( 8) + 3( 2) 2( 0) + 3( 6) 2( 3) + 3( 4) 4( 8) + 0( 2) 4( 0) + 0( 6) 4( 3) + 0( 4)

7 You found the product, not. The Commutative Property does not hold for multiplication of matrices. C is 3 2, and is 2 3, so the product is defined and is 3 3. The first row of should contain the results of multiplying the first row of and each column of. The second row of should contain the results of multiplying the second row of and each column of, and so on. PTS: IF: verage REF: Page 254 TOP: 4-2 Multiplying Matrices 3. NS: Use the product matrix to find the revenue, cost, and profit for each dealer. 6 8, 700 2, 00 5, , 800 3, 200, , , , , , 800 +, , , , , , , 600 ealer ealer 2 Revenue Cost Profit 67, , 200 2, , , , 600 The total profit from cars and for dealer is $2,200 and for dealer 2 is $65,600. C To find an element of the product matrix multiply the elements of a row of the first matrix by the elements of a column of the second and add the results. To find an element of the product matrix multiply the elements of a row of the first matrix by the elements of a column of the second and add the results. You reversed the values of dealer and dealer 2 profits. PTS: IF: verage REF: Page 255 TOP: 4-2 Multiplying Matrices 4. NS: ( 5) + ( 4) 5( ) + ( 2) 4( 5) + 2( 4) 4( ) + 2( 2) The ij-th element of is the sum of the products of consecutive entries in row i

8 C in matrix and column j in matrix. The ij-th element of is the sum of the products of consecutive entries in row i in matrix and column j in matrix. Square matrices can be multiplied by themselves any number of times. PTS: IF: verage REF: Page 256 OJ: Finding Powers of Square Matrices TOP: 4-2 Multiplying Matrices 5. NS: Multiply a point (x, y) 2 0 on the figure by the matrix. 0 2 x y 2 0 2x 2y 2 x y 0 2 The point is enlarged by a factor of 2 and reflected on the y-axis. The point is arbitrary. Thus, the figure is also enlarged and reflected on the y-axis. C Multiply an arbitrary point (x, y) on the figure by the matrix to determine the effect of the matrix. Multiply an arbitrary point (x, y) on the figure by the matrix to determine the effect of the matrix. Multiply an arbitrary point (x, y) on the figure by the matrix to determine the effect of the matrix. PTS: IF: dvanced TOP: 4-3 Using Matrices to Transform Geometric Figures 6. NS: Find, the determinant of the coefficient matrix 3 4 ( 3)(0) ()( 4) so the system is consistent. Solve for each variable by replacing the coefficients of that variable with the constants..

9 x y 6 The solution is (3, 2). ()(0) ( 3)( 4) ( 3)( 3) ()() To find the determinant, multiply the matrix components diagonally. Use x-coefficients to solve for y, and y-coefficients to solve for x. C ivide by the determinant to find the solution. PTS: IF: verage REF: Page 27 OJ: Using Cramer's Rule for Two Equations TOP: 4-4 eterminants and Cramer s Rule 7. NS: C The determinant of Step Multiply each down diagonal and add. 5( 0) ( 0) + ( 3) ( ) ( ) + ( 2) ( 6) ( 2) 2 Step 2 Multiply each up diagonal and add. ( ) ( 0) ( 2) + ( 2) ( ) ( 5) + ( 0) ( 6) ( 3) 30 Step 3 Find the difference of the sums C You multiplied the sums of the diagonals instead of adding the products. Subtract the sum of the "up" diagonals products from the sum of the "down" diagonals products to find the determinant. Rewrite the first two columns at the right side of the determinant. Then subtract the sum of all the "up" diagonals products from the sum of all the "down" diagonals products. PTS: IF: verage REF: Page 272 OJ: Finding the eterminant of a 3 x 3 Matrix TOP: 4-4 eterminants and Cramer s Rule 8. NS:

10 The diet will include p grams of protein, c grams of carbohydrates, and f grams of fat. 4p + 4c + f 240 Equation for total calories p + c + f 540 Total grams of food 4p + 0c + 0f 64 Calories from protein, 40%(240) Find the determinant of the coefficients. Replace the coefficients of each variable with the corresponding constant values and calculate each new determinant. Then divide by the determinant of the coefficients p p p c c c f f f 50 The diet includes 24 grams of protein, 24 grams of carbohydrates, and 50 grams of fat. C Identify three equations using the information about total grams, total calories, and percent of protein. Then use Cramer's Rule to solve. Identify three equations using the information about total grams, total calories, and percent of protein. Then use Cramer's Rule to solve. Identify three equations using the information about total grams, total calories, and percent of protein. Then use Cramer's Rule to solve. PTS: IF: verage REF: Page 273 OJ: pplication TOP: 4-4 eterminants and Cramer s Rule. NS: Multiply the matrices to see if their product is the identity matrix.

11 Two matrices and are inverses if and only if their product is the identity matrix, or x I. PTS: IF: verage REF: Page 278 OJ: 4-5. etermining Whether Two Matrices re Inverses TOP: 4-5 Matrix Inverses and Solving Systems 0. NS: Step Write each equation in x + y + Cz form. Ï x + y + 0z 8 Ì Ô 0x y + z 4 ÓÔ x + 0y z 3 Step 2 Write the augmented matrix with coefficients and constants C Write each equation in x + y + Cz form. z y y + z Write each equation in x + y + Cz form. PTS: IF: asic REF: Page 287 OJ: 4-6. Representing Systems as Matrices TOP: 4-6 Row Operations and ugmented Matrices

12 . NS: The first and second equations becomes the first and second rows, respectively, of the augmented matrix Multiply row by 2. Multiply row 2 by 3. Then subtract row 2 from row and write the result in row ivide row 2 by 2. Then multiply row 2 by 3 and subtract row 2 from row. Write the result in row. Finally, divide row by Round your answer to the nearest tenth to obtain x 5.0 and y 2.0. C The first and second equations become the first and second rows, respectively, of the augmented matrix. Write the coefficients of x and y in order to create the rows of the augmented matrix. The first and second equations become the first and second rows, respectively, of the augmented matrix. PTS: IF: verage REF: Page 288 OJ: Solving Systems with an ugmented Matrix TOP: 4-6 Row Operations and ugmented Matrices 2. NS: Use the facts to write three equations. Let k number of blankets, t number of pairs boots, and c number of coats. 5k + 20t + 25c 2, 200 k 7c 0 t 2c 0 The cost of each item times the number of items equals the total cost. There are 7 blankets for each coat. There are twice as many pairs of boots as coats. Enter the 3 4 augmented matrix as.

13 [ ]: , Press the 2nd and MTRIX buttons, select MTH,.and move down the list to :rref to find the reduced row-echelon form of the augmented matrix. There are 54 blankets, 44 pairs of boots, and 22 coats. There are 7 blankets for every coat. C There are twice as many boots as coats. The total donation is $2,200. PTS: IF: verage REF: Page 20 OJ: pplication TOP: 4-6 Row Operations and ugmented Matrices NUMERIC RESPONSE 3. NS: 2 PTS: IF: dvanced TOP: 4- Matrices and ata 4. NS: 3 PTS: IF: verage TOP: 4-4 eterminants and Cramer s Rule MTCHING 5. NS: H PTS: IF: asic REF: Page 287 TOP: 4-6 Row Operations and ugmented Matrices 6. NS: G PTS: IF: asic REF: Page 288 TOP: 4-6 Row Operations and ugmented Matrices 7. NS: PTS: IF: asic REF: Page 288 TOP: 4-6 Row Operations and ugmented Matrices 8. NS: F PTS: IF: asic REF: Page 27 TOP: 4-5 Matrix Inverses and Solving Systems. NS: C PTS: IF: asic REF: Page 288 TOP: 4-6 Row Operations and ugmented Matrices 20. NS: E PTS: IF: asic REF: Page 27 TOP: 4-5 Matrix Inverses and Solving Systems

14 2. NS: PTS: IF: asic REF: Page 264 TOP: 4-3 Using Matrices to Transform Geometric Figures 22. NS: F PTS: IF: asic REF: Page 255 TOP: 4-2 Multiplying Matrices 23. NS: PTS: IF: asic REF: Page 27 TOP: 4-5 Matrix Inverses and Solving Systems 24. NS: C PTS: IF: asic REF: Page 262 TOP: 4-3 Using Matrices to Transform Geometric Figures 25. NS: PTS: IF: asic REF: Page 263 TOP: 4-3 Using Matrices to Transform Geometric Figures 26. NS: E PTS: IF: asic REF: Page 278 TOP: 4-5 Matrix Inverses and Solving Systems