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30s 2012 review Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Determine the first 4 terms of an arithmetic sequence, given the first term, difference, . A. –4, –6, –8, –10 C. –2, 2, 6, 10 B. –4, –2, 0, 2 D. –2, –6, –10, –14 ____ 2. Determine t11 of this arithmetic sequence: –20, –35, –50, –65, ... A. C. B. D. ____ 3. The sum of the first 15 terms of an arithmetic series is –210. The sum of the first 16 terms is –243.2. The common difference is –2.4. Determine the first 4 terms of the series. A. C. B. D. ____ 4. Which sequence could be geometric? A. 2 2 6, 2, , , ... 3 9 B. 9 18 6, , , 3, ... 2 5 C. 2 2 6, 2, , , ... 3 9 D. 3 6, 3, 2, , ... 2 ____ 5. Determine the 6th term of this geometric sequence: –9, 27, –81, 243, ... A. C. B. D. ____ 6. In a finite geometric sequence, A. B. ____ 7. Determine the 5th term of this geometric sequence: –16, 8, 4, 2, ... A. 2 C. 1 2 B. 1 D. 1 2 ____ 8. The sum of the first 12 terms of which geometric series is 5460? A. C. B. D. ____ 9. Which geometric series could this graph represent? and . Determine t10. C. D. , and the common Geom etric Series 320 280 240 Partial sums 200 160 120 80 40 0 1 2 3 4 5 6 7 Num ber of term s A. 1 1 1 + + + ... 12 24 48 B. 6 + 12 + 24 + 48 + ... 6+ C. D. 6 + 18 + 54 + 162 + ... ____ 10. The common ratio of a geometric sequence is 1.5. Which graph could represent this geometric sequence? A. C. Geom etric Sequence Geom etric Sequence 120 Term value Term value 80 40 0 –40 1 2 3 4 5 6 7 Term num ber 80 40 0 1 2 3 4 5 6 7 Term num ber –80 B. D. Geom etric Sequence Geom etric Sequence 120 Term value Term value 80 40 0 –40 –80 1 2 3 4 5 6 7 Term num ber 80 40 0 1 2 3 4 5 6 7 Term num ber ____ 11. Determine the first 4 terms of an infinite geometric series with and . A. t1 = –3; t2 = 15; t3 = 75; t4 = 375 B. C. 3 3 3 t1 = –3; t2 = ; t3 = ; t4 = 5 25 125 D. 3 3 3 t1 = –3; t2 = ; t3 = ; t4 = 5 25 125 1 1 1 t1 = –3; t2 = ; t3 = ; t4 = 5 75 1125 ____ 12. An infinite geometric series has and . Determine . C. This series does not have a finite sum. D. A. B. ____ 13. Use an infinite geometric series to express A. as a fraction. C. B. D. ____ 14. Which point on the number line has an absolute value of 2.5? W –10 A. Y ____ 15. Evaluate A. 7252 –8 –6 –4 –2 X Y 0 2 4 B. X when B. 7259 Z 6 8 10 C. Z D. W C. 7189 D. 7217 C. 945 D. –17 C. D. C. D. C. D. C. D. . ____ 16. Evaluate: A. 15 B. 47 ____ 17. Write this mixed radical as an entire radical: A. B. ____ 18. Write this mixed radical as an entire radical: A. B. ____ 19. Write this entire radical as a mixed radical: A. B. ____ 20. Write this entire radical as a mixed radical: A. B. ____ 21. For which values of the variable, x, is this radical defined? A. C. B. D. ____ 22. Simplify by adding or subtracting like terms: A. B. C. D. ____ 23. Expand and simplify this expression: A. B. C. D. ____ 24. Rationalize the denominator: A. B. C. D. ____ 25. Expand and simplify this expression: A. B. C. D. ____ 26. Expand and simplify this expression: A. B. ____ 27. Solve this equation: A. 9 x= 4 C. D. B. x= 4 9 C. x= 16 81 D. x= ____ 28. Which statement is true for the equation ? A. 7 and –1 are roots. B. 7 is a root of the original equation and –1 is an extraneous root. C. 1 is a root of the original equation and –7 is an extraneous root. D. 7 and 1 are both extraneous roots. ____ 29. Solve by factoring: A. B. C. D. ____ 30. Without solving, determine the number of real roots of this equation: A. 2 B. 0 C. 1 ____ 31. Without solving, determine the number of real roots of this equation: A. 0 B. 2 C. 1 ____ 32. Calculate the value of the discriminant for this equation: A. 20 B. 24 C. 5 D. –6 ____ 33. Which statement below is NOT true for the graph of a quadratic function? A. The vertex of a parabola is its highest or lowest point. 81 16 B. When the coefficient of is positive, the vertex of the parabola is a minimum point. C. The axis of symmetry intersects the parabola at the vertex. D. The parabola is symmetrical about the y-axis. ____ 34. Identify the quadratic function that this table of values represents, then determine the value of y when x = 7. x y –1 –9 0 –3 3 –33 A. B. ; –185 ; –182 C. D. ; 185 ; 185 ____ 35. A rectangular dog pen is to be enclosed with 20 m of fencing. The area of the dog pen, A square metres, is modelled by the function , where x is the width, in metres. What is the width that gives maximum area? Write the answer to the nearest tenth, if necessary. A. 5 m C. 20 m B. 25 m D. 10 m ____ 36. Use the graph of to determine the roots of . y 12 8 4 –6 –4 –2 0 2 4 6 x –4 –8 –12 A. B. and and C. D. and and ____ 37. Use graphing technology to approximate the solution of this equation: Write the roots to 1 decimal place. and . A. The roots are approximately and . B. The roots are approximately and . C. The roots are approximately and . D. The roots are approximately ____ 38. Identify the coordinates of the vertex of the graph of this quadratic function: B. (–4, –4) A. (4, 4) ____ 39. Match the quadratic function y A. to a graph below. C. 6 –3 –2 –1 4 2 2 0 1 2 –1 –2 –1 0 –2 –4 –4 D. y = h(x) 2 2 2 3 x 3 x 2 3 x y 4 1 2 y = g(x) 6 4 0 1 –6 y = f(x) y –2 –3 3 x –2 6 –3 y 6 4 –6 B. D. (4, –4) C. (–4, 4) –3 –2 –1 0 –2 –2 –4 –4 –6 –6 y = k(x) 1 ____ 40. A sports equipment company sells skates for $65 a pair. At this price, the company sells approximately 200 pairs a week. For every increase in price of x dollars, the company will sell 40x fewer pairs. Determine the equation that should be used to maximize the revenue, R dollars. A. C. B. D. ____ 41. Two numbers have a difference of 10. The sum of their squares is a minimum. Determine the numbers. A. 5 and 15 B. –2 and 8 C. 0 and 10 D. –5 and 5 ____ 42. Does the quadratic function value? A. minimum value; 484 B. minimum value; 340 have a maximum value or a minimum value? What is that C. maximum value; 484 D. maximum value; 340 ____ 43. A rectangular dog pen is to be fenced with 24 m of fencing. Determine the maximum area and the width of this rectangle. A. C. ; ; B. D. ; ; ____ 44. Three rectangular areas are to be fenced with 78 m of fencing. Which equation could be used to determine the dimensions that enclose the maximum area? l w w w w l A. C. B. D. ____ 45. Represent the solution of this quadratic inequality on a number line: A. –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 B. C. D. ____ 46. Which coordinates are a solution of the inequality A. (2, 8) B. (0, 1) C. (3, 7) ? D. (5, 8) ____ 47. Match the inequality –x + 2y 8 to its graph. A. C. y –6 –4 –2 y 6 6 4 4 2 2 0 2 4 6 x –6 –4 –2 0 –2 –2 –4 –4 –6 –6 2 4 6 x B. D. y 6 –6 –4 –2 y 6 4 4 2 2 0 2 4 –6 6 x –4 –2 –2 –2 –4 –4 –6 –6 ____ 48. Which ordered pair is a solution of the quadratic inequality A. (0, 5) B. (3, 12) C. (2, 4) ____ 50. Which graph represents the inequality –2 B. ? D. (2, 19) y 6 4 4 2 2 0 2 4 6 x D. (1, 5) C. y –4 4 ? 6 –6 2 ? ____ 49. Which ordered pair is a solution of the quadratic inequality A. (3, 36) B. (1, 1) C. (–1, 2) A. 0 –6 6 x –4 –2 0 –2 –2 –4 –4 –6 –6 2 4 6 x 2 4 6 x D. y 6 y 6 4 4 2 2 –6 –4 –2 0 –2 2 4 6 x –6 –4 –2 0 –2 –4 –4 –6 –6 ____ 51. Write an inequality to describe this graph. y 6 4 2 –3 –2 –1 0 1 2 3 x –2 –4 –6 A. B. C. D. ____ 52. Use a graphing calculator. Which graph represents this system of equations? A. C. y 6 –3 –2 –1 y 6 4 4 2 2 0 1 2 3 x –3 –2 –1 0 –2 –2 –4 –4 –6 –6 1 2 3 x B. D. y 6 –3 –2 –1 y 6 4 4 2 2 0 1 2 –3 3 x –2 0 –1 –2 –2 –4 –4 –6 –6 1 2 3 x ____ 53. Two numbers are related: The sum of twice the square of the first number plus the second number is 9. The difference between twice the first number and the second number is 15. Which system of equations represents this relationship? A. C. B. D. ____ 54. Solve this linear-quadratic system algebraically. A. (5, 2) B. (2, 5) C. (–2, 1) D. (2, 33) ____ 55. Solve this linear-quadratic system algebraically. A. (1, –2) and (–1, 6) B. (1, 6) and (–1, –2) C. (6, 1) and (–2, –1) D. (–2, 1) and (6, –1) ____ 56. A ladder leans against a wall. The base of the ladder is on level ground 1.4 m from the wall. The angle between the ladder and the ground is 74. To the nearest tenth of a metre, how far up the wall does the ladder reach? A. 5.1 m B. 4.9 m C. 1.5 m D. 0.4 m ____ 57. An angle has its terminal arm in Quadrant 2. Which primary trigonometric ratio is greater than 0? A. cos B. tan C. sin D. all 3 ratios ____ 58. Determine the reference angle for the angle 290° in standard position. A. 290° B. 20° C. 110° D. 70° ____ 59. Determine the exact value of tan 210. A. B. D. C. ____ 60. Point P(5, 0) is a terminal point of an angle in standard position. Determine the value of cos . A. undefined B. 0 C. 5 D. 1 ____ 61. In XYZ, XY = 6 cm and X = 33°. For which value of YZ is no triangle possible? A. 6 cm B. 3 cm C. 4 cm D. 12 cm ____ 62. In ABC, AB = 9.1 m and A = 41°. For which value of BC can one scalene triangle be drawn? A. 9.1 m B. 11.4 m C. 6 m D. 7.3 m ____ 63. In DEF, DE = 11 cm and EF = 9 cm. For which measure of D is it possible to draw two scalene triangles? A. 54° B. 65° C. 58° D. 90° ____ 64. For DEF, determine the measure of E to the nearest degree. E 7.4 cm 54° D F 4.2 cm A. 155° B. 27° C. 25° D. 58° ____ 65. In PQR, determine the measure of Q to the nearest degree. P R 44° 5.5 cm 3.5 cm Q A. 70° B. 110° C. 136° D. 154° B. C. D. ____ 66. Solve. A. ____ 67. Solve. A. B. C. D. no solution ____ 68. This is the graph of the absolute value of a linear function. y 4 2 –6 –4 –2 0 2 4 6 x –2 –4 Which is the graph of the linear function? A. C. y y 4 4 2 2 –6 –6 –4 –2 0 2 4 6 –4 –2 0 2 4 6 x 2 4 6 x –2 x –2 –4 –4 B. D. y y 4 4 2 2 –6 –6 –4 –2 0 2 4 6 –4 –2 0 –2 x –2 –4 –4 ____ 69. Sketch the graph of the absolute value function . A. 12 –8 –4 B. –4 12 8 8 4 4 0 4 8 –8 x –4 –4 –8 –8 D. y 12 8 8 4 4 0 4 8 –8 x –4 –4 –8 –8 C. D. 4 8 x 4 8 x y 0 –4 ____ 70. Solve this equation: and A. and B. y 0 –4 12 –8 C. y and and ____ 71. The graph of the reciprocal of a linear function has this vertical asymptote. What is the linear function? y 6 4 2 –6 –4 –2 0 2 4 6 x –2 –4 –6 A. B. ____ 72. Which graph represents the function C. D. and its reciprocal function ? A. C. y –4 –2 4 4 2 2 0 2 4 –2 0 –2 –4 –4 D. y –2 –4 x –2 B. –4 y 4 2 2 2 4 4 x 2 4 x y 4 0 2 –4 x –2 0 –2 –2 –4 –4 ____ 73. A linear function has a positive slope. Which graph below is a possible graph of its reciprocal function? y y A. C. –4 –2 4 4 2 2 0 2 4 0 –2 –4 –4 D. y –2 –2 –2 B. –4 –4 x 4 2 2 2 4 –4 x –2 0 –2 –2 –4 –4 ____ 74. The graphs of the reciprocal functions the coordinates of any points of intersection? 4 x 2 4 x y 4 0 2 and are plotted on the same grid. What are C. (–1, –6) A. 1 (1, ) 6 B. 1 ( , –1) 6 D. 1 ( , 1 ) 6 ____ 75. This is the graph of the reciprocal of a linear function. Which graph below represents the linear function? y 6 4 2 –6 –4 –2 0 2 4 6 x –2 –4 –6 A. C. y –8 –4 8 8 4 4 0 4 8 –4 0 –4 –8 –8 D. y –4 –8 x –4 B. –8 y 8 4 4 4 8 x –8 –4 0 –4 –4 –8 –8 Short Answer 1. Could this sequence be arithmetic? –1, –5, –10, –15, ... 8 x 4 8 x y 8 0 4 2. The sum of the first n terms of an arithmetic series is: Determine the first 4 terms of the series. 3. Determine r, t5, and t6 of this geometric sequence: 3125, 625, 125, 25, ... 4. Evaluate each expression, then order the values of the expressions from greatest to least. i) ii) iii) iv) 5. Evaluate each expression, then order the values of the expressions from least to greatest. i) ii) iii) iv) 6. Determine the root of each equation. a) b) c) d) 7. The total area of the large rectangle below is 24 m2. Determine the value of x. x m 2m 2m x m x m 8. A baseball is hit upward. The approximate height of the baseball, h metres, after t seconds is modelled by this formula: When is the baseball 11 m high? 9. When 3 times a number is added to the square of the number, the result is 40. Determine the number. 10. a) Calculate the value of the discriminant for the equation b) Are the roots rational or irrational? = 0. 11. Use a graphing calculator to graph the quadratic function Determine: a) the intercepts b) the coordinates of the vertex c) the equation of the axis of symmetry d) the domain of the function e) the range of the function Round the answers to the nearest hundredth, if necessary. . 12. The weekly profit, P hundred dollars, of a company is modelled by the equation , where x is the number of units produced per week, in thousands. a) Use a graphing calculator to determine the number of units the company should produce per week to earn the maximum weekly profit. b) What is the maximum weekly profit? 13. The graph of a quadratic function is shown. What can you say about the discriminant of the corresponding quadratic equation? y 12 8 4 –6 –4 –2 0 2 4 6 x –4 –8 –12 14. Determine an equation of a quadratic function with the given characteristics of its graph. coordinates of the vertex: V(–4, 1); y-intercept 33 15. Use a graphing calculator or graphing software. Graph each quadratic function. How many x-intercepts does the graph have? a) b) c) 16. Use a graphing calculator or graphing software. Graph each quadratic function. How many x-intercepts does the graph have? a) b) c) 17. The equation of the axis of symmetry of the graph of a quadratic function is . The graph passes through the points R(3, 7) and T(–1, –25). Determine an equation of the function. 18. Does the quadratic function have a maximum value or a minimum value? What is that value? 19. A coffee shop sells coffee for $1.40 a cup. At this price, the store sells approximately 800 cups per day. Research indicates that for every $0.05 increase in price, the store will sell 40 fewer cups. Determine the price of a cup of coffee that will maximize the revenue. 20. Represent the solution of this quadratic inequality on a number line: 21. a) Determine the critical values of this quadratic inequality: b) Complete the table. Interval Sign of Value of x c) What is the solution of the inequality? 22. a) Graph the inequality: b) Write the coordinates of 2 points that satisfy the inequality. 23. Use technology to graph the inequality . Sketch the graph. 24. Use a graphing calculator. Graph this system of equations. a) Sketch the graphs on the same grid. b) Write the coordinates of the point of intersection. 25. Use a graphing calculator. Graph this system of equations. a) Sketch the graphs on the same grid. b) Write the coordinates of the points of intersection. 26. Use a graphing calculator. Graph this system of equations. a) Sketch the graphs on the same grid. b) Write the coordinates of the point of intersection. 27. How is a linear-quadratic system different from a quadratic-quadratic system in terms of the number of solutions the system may have? 28. Solve this quadratic-quadratic system algebraically. 29. Solve this quadratic-quadratic system algebraically. 30. A baseball is thrown upward at an initial speed of 9.5 m/s. The approximate height of the ball, h metres, after t seconds is given by the equation . At the same time, a golf ball is hit upward at an initial speed of 12 m/s. The approximate height of the ball, h metres, after t seconds is given by the equation . a) When are both balls at the same height? b) What is this height? Give your answer to the nearest tenth of a metre. 31. Solve this quadratic-quadratic system algebraically. Give the solutions to the nearest tenth, when necessary. 32. Solve this quadratic-quadratic system algebraically. Give the solutions to the nearest tenth, when necessary. 33. A tree is supported by a guy wire. The guy wire is anchored to the ground 5.0 m from the base of the tree. The angle between the wire and the level ground is 60. To the nearest tenth of a metre, how far up the tree does the wire reach? 34. A flagpole is 11.0 m high. At a certain point, the angle between the ground and Jon’s line of sight to the top of the flagpole is 61. To the nearest tenth of a metre, how far is Jon from the flagpole? 35. A ladder is 5 m long. It leans against a house. The base of the ladder is 1.4 m from the house. What is the angle of inclination of the ladder to the nearest tenth of a degree? 36. A ski jump is 116 m long. It has a vertical rise of 54 m. What is the angle of inclination of the jump to the nearest tenth of a degree? 37. A 2.8-m cable is attached to a sign. The cable is anchored to the ground 1.8 m from the base of the sign. What is the angle of inclination of the cable to the nearest tenth of a degree? 38. To the nearest degree, which values of satisfy this equation for ? 39. If AB = 7 cm, BC = 3 cm, and A = 70°, is it possible to construct ABC? 40. For JKL, can the Sine Law be used to determine the length of JL? If your answer is yes, determine the length of JL to the nearest tenth of a centimetre. If your answer is no, explain why. L J 7 cm 3 cm 101° K 41. Given the following information about ABC, determine how many triangles can be constructed. a = 5.6 cm, c = 7.8 cm, A = 38° 42. In ABC, AB = 3.9 cm, C = 63°, and BC = 5.2 cm. a) Determine how many triangles can be constructed. Justify your answer. b) Sketch a diagram to show any possible triangles. 43. Complete this table of values. x –2 –1 0 1 2 3 –4 –3 –2 –1 0 1 44. Complete this table of values. x –2 –1 0 1 2 3 –16 –14 –8 2 16 34 45. Write this absolute value function in piecewise notation. y 8 4 –4 –2 0 2 4 x –4 –8 46. Write the absolute value function in piecewise notation. y 24 16 8 –8 –6 –4 0 –2 2 4 6 8 x –8 –16 –24 47. Write this absolute value function in piecewise notation. y 24 16 8 –12 –8 –4 0 4 8 12 x –8 –16 48. Solve this equation: y 20 16 12 8 4 –8 –6 –4 –2 0 –4 2 4 6 8 x 49. A student used this graph to solve an absolute value equation. What might the equation have been? y 8 6 4 2 –8 –6 –4 –2 0 2 4 6 8 x –2 –4 50. A student used this graph to solve an absolute value equation. What might the equation have been? y 8 6 4 2 –8 –6 –4 –2 0 2 4 6 8 x –2 –4 –6 –8 Problem 1. In this arithmetic sequence, k is a natural number: a) Determine t6. b) Write an expression for tn. c) Suppose ; determine the value of k. ... 2. a) What is the sum of the first 124 multiples of 4? b) What is the value of t124? 3. a) A geometric sequence has these terms: 1 t4 = 8, t5 = 2, t6 = 2 State the common ratio, then write the first 3 terms of the sequence. b) Identify the sequence as convergent or divergent. Explain. 4. Show how you can use geometric series to determine this sum: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 + 64 + 96 + 128 + 192 + 256 + 384 + 512 1 the length of the preceding one. 4 A frog is programmed to jump to a position 12 m away in 12 jumps. Determine the horizontal distance of each of the first 3 jumps. Give your answers to the nearest hundredth of a metre. 5. A robotic frog is programmed to make a sequence of jumps, each jump 6. Joel worked at Series Inc. for the summer. He earned $56 on the first day. His pay was increased at a rate of 2.1% with each subsequent day. Determine the total amount Joel earned after 16 days. 7. Suppose the lengths of a frog’s jumps form a geometric series. After 12 jumps, the frog has travelled a 7 horizontal distance of approximately 6265.56 cm. The common ratio is . 5 Determine the length of the frog’s first jump. 8. Create the first 6 terms of a geometric sequence for this description of a graph: the term values and increase in numerical value as more points are plotted. 9. a) Without graphing, describe the graph of this geometric sequence: 3, 9, 27, 81, 243, 729, ... b) Without graphing, describe the graph of the partial sums of this geometric series: ... c) Verify your descriptions by graphing. Sketch and label each graph on a grid below. 10. An infinite geometric series with is represented by this equation: a) Determine the first 4 terms of the series. b) Determine whether the series diverges or converges. c) If the series has a finite sum, determine the sum. 11. Three congruent rectangles are joined as shown. The length of each rectangle is 3 times its width. The area of each rectangle is 39 square units. In simplest form, write a radical expression for the perimeter of the shape formed. Show your work. 3x A = 39 units 2 I x I I I 12. Expand and simplify this expression: Show your work. 13. The side length of a square is units. Write, then simplify, a radical expression for the area of the square. Show your work. 14. Write this expression in simplest form: Describe your strategy. 15. Determine whether the given value of x is a root of this equation. Justify your answer. ; 16. The volume of water in Lake Ontario is about 1710 km3. a) To the nearest kilometre, determine the edge length of a cube with the same volume. b) To the nearest kilometre, determine the radius of a sphere with the same volume. 17. Determine the lengths of the legs in this right triangle. Explain your strategy. 29 cm x cm x + 1 cm 18. Determine the values of k for which the equation equation. has two real roots, then write a possible 19. Determine the values of k for which the equation equation. has no real roots, then write a possible 20. Create two different quadratic equations whose discriminant is 64. Explain your strategy. 21. Use a graphing calculator to graph the quadratic function Determine: a) the intercepts b) the coordinates of the vertex c) the equation of the axis of symmetry d) the domain of the function e) the range of the function Write the answers to the nearest hundredth, if necessary. . 22. a) Identify the coordinates of the vertex, the domain, the range, the direction of opening, the equation of the axis of symmetry, and the intercepts for this quadratic function: b) Sketch a graph. 23. The arch of a bridge forms a parabola. The arch is 72 m wide and its maximum height is 27 m above its base. Determine an equation to model this parabola. Explain your strategy. 24. A hospital sells raffle tickets to raise funds for new medical equipment. Last year, 2000 tickets were sold for $24 each. The fund-raising coordinator estimates that for every $1 decrease in price, 125 more tickets will be sold. a) What decrease in price will maximize the revenue? b) What is the price of a ticket that will maximize the revenue? c) What is the maximum revenue? Show and explain your work. 25. Consider this inequality: a) Solve the inequality by factoring. b) Illustrate the solution on a number line. c) What do you notice about the solution of the inequality. Explain why. 26. is a quadratic equation. For which values of b does the quadratic equation have: a) 2 real roots? b) exactly 1 root? c) no real roots? Show your work. 27. The length of a rectangular garden is 6 m greater than its width. The area of the garden is at least 20 m2. What are possible dimensions of the garden, to the nearest tenth of a metre? Show your work. Verify the solution. 28. Graph each inequality for the given restrictions on the variables. Explain your strategy and describe the solution. a) b) ; for ; for 29. For A(a, 3) to be a solution of Show and explain your work. , what must be true about a? 30. Use a graphing calculator. Graph this system of equations. a) Sketch the graphs on the same grid. b) Write the coordinates of the points of intersection. 31. Write the system of equations represented by this graph, then solve the system. Explain your work. y 12 10 8 6 4 2 –12 –10 –8 –6 –4 –2 0 2 4 6 8 10 x –2 –4 –6 32. Two numbers are related in this way: Twice the square of the first number minus the second number is 3. The square of the sum of the first number and 5 is equal to the second number minus 2. a) Create a system of equations to represent this relationship. b) Solve the system to determine the numbers. Explain the strategy you used. 33. A football is kicked upward at an initial speed of 10 m/s. The approximate height of the ball, h metres, after t seconds is modelled by the equation . A soccer ball is kicked upward with an initial speed of 12 m/s. The approximate height of the ball, h metres, after t seconds is modelled by the equation . a) The soccer ball is kicked 1 s after the football is kicked. Determine when both balls reached the same height. b) What is this height? Explain your strategy. Give your answers to the nearest tenth of a unit. 34. Determine the measures of A and B to the nearest tenth of a degree. Explain your strategy. B 21 11 A C 35. a) In BCD, identify the side opposite angle and the side adjacent to angle . b) Determine sin to the nearest tenth. Describe what the value of sin indicates. c) Determine the measure of to the nearest tenth of a degree. Show your work. A B 8 cm D 14 cm C 36. A coast guard patrol boat is due west of the Carmanah lighthouse. An overturned fishing boat is due north of the lighthouse. The patrol boat travels 9.1 km directly to the fishing boat. The angle between due east and the patrol boat’s path is 54°. To the nearest tenth of a kilometre, determine the distance between the fishing boat and the lighthouse. Explain your work. 37. In , AB = 35 cm, BC = 19 cm, and . a) Explain why it is possible to draw two different triangles with these measures. b) Sketch a diagram to show both triangles. 38. In ABC, AB = 6 cm and Length of BC (cm) Value of . Complete the chart below for your own values of BC. How does compare with sin A? Description of possible triangles No triangles are possible. 1 isosceles triangle 1 scalene triangle 2 scalene triangles 39. A pair of campers paddle a canoe 3.4 km [W24°N], then 2.4 km [S28°E]. To the nearest tenth of a kilometre, what is the straight-line distance from their start point to their end point? To the nearest degree, what is the bearing of the end point from the start point? Show your work. 40. Solve ABC. Give angle measures to the nearest degree. 10 cm C A 9 cm 5 cm B 41. Without solving the equation, explain how you know that this equation has no solution. 42. Explain how you would solve this rational equation: 43. Complete this table of values, then sketch the graphs of x –2 –1 –6 –4 0 2 1 0 and 2 3 2 4 on the same grid. y 8 6 4 2 –8 –6 –4 0 –2 2 4 6 8 x –2 –4 –6 –8 44. Write an equation for the absolute value function. Explain your strategy. 8 y 6 4 2 –8 –6 –4 –2 0 2 4 6 8 x –2 –4 –6 –8 45. Write an equation for this absolute value function. Explain your strategy. y 8 6 4 2 –8 –6 –4 –2 0 2 4 6 8 x –2 –4 –6 –8 46. Solve the equation . Show your work. 47. A motorcycle is travelling toward the Ontario-Quebec border. The motorcycle is 255 km from the border, and is travelling at an average speed of 95 km/h. a) Write an absolute value equation to represent the distance, d kilometres, of the motorcycle from the border after t hours. b) Determine how far the motorcycle is from the border after 1.5 h. Give distance to the nearest kilometre. 48. The function is linear. The line line intersects the graph of the function . intersects the graph of at x = 2 and x = 6. The at x = 3 and x = 5. Use a graph to determine the equation for y 20 16 12 8 4 –8 –6 –4 –2 0 –4 2 4 6 8 x 49. Use the graph of to sketch a graph of . Write the equation of the linear and reciprocal functions. Show your work. y 8 6 4 2 –6 –4 –2 0 2 4 6 x –2 –4 –6 –8 50. a) Write a reciprocal function that describes the time, t hours, it takes an athlete to run 1 km, as a function of her speed, s kilometres per hour. b) What are the domain and range of the reciprocal function in part a? c) Sketch a graph of the reciprocal function in part a. t 10 8 6 4 2 0 2 4 6 8 10 s 30s 2012 review Answer Section MULTIPLE CHOICE 1. ANS: LOC: 2. ANS: LOC: 3. ANS: LOC: 4. ANS: LOC: 5. ANS: LOC: 6. ANS: LOC: KEY: 7. ANS: LOC: 8. ANS: LOC: 9. ANS: REF: TOP: 10. ANS: REF: TOP: 11. ANS: LOC: 12. ANS: LOC: 13. ANS: LOC: 14. ANS: REF: TOP: 15. ANS: REF: TOP: 16. ANS: REF: TOP: 17. ANS: REF: TOP: 18. ANS: A PTS: 1 DIF: Easy REF: 1.1 Arithmetic Sequences 11.RF9 TOP: Relations and Functions KEY: Procedural Knowledge B PTS: 1 DIF: Easy REF: 1.1 Arithmetic Sequences 11.RF9 TOP: Relations and Functions KEY: Procedural Knowledge A PTS: 1 DIF: Difficult REF: 1.2 Arithmetic Series 11.RF9 TOP: Relations and Functions KEY: Procedural Knowledge C PTS: 1 DIF: Easy REF: 1.3 Geometric Sequences 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge C PTS: 1 DIF: Easy REF: 1.3 Geometric Sequences 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge D PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences 11.RF10 TOP: Relations and Functions Conceptual Understanding | Procedural Knowledge B PTS: 1 DIF: Easy REF: 1.3 Geometric Sequences 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge D PTS: 1 DIF: Moderate REF: 1.4 Geometric Series 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge B PTS: 1 DIF: Easy 1.5 Graphing Geometric Sequences and Series LOC: 11.RF9 | 11.RF10 Relations and Functions KEY: Procedural Knowledge C PTS: 1 DIF: Moderate 1.5 Graphing Geometric Sequences and Series LOC: 11.RF9 | 11.RF10 Relations and Functions KEY: Conceptual Understanding C PTS: 1 DIF: Easy REF: 1.6 Infinite Geometric Series 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge B PTS: 1 DIF: Easy REF: 1.6 Infinite Geometric Series 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge B PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series 11.RF10 TOP: Relations and Functions KEY: Procedural Knowledge A PTS: 0 DIF: Easy 2.1 Absolute Value of a Real Number LOC: 11.AN1 Relations and Functions KEY: Conceptual Understanding B PTS: 0 DIF: Moderate 2.1 Absolute Value of a Real Number LOC: 11.AN1 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Moderate 2.1 Absolute Value of a Real Number LOC: 11.AN1 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge C PTS: 0 DIF: Easy 2.2 Simplifying Radical Expressions LOC: 11.AN2 Relations and Functions KEY: Procedural Knowledge A PTS: 0 DIF: Moderate 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. REF: 2.2 Simplifying Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge ANS: A PTS: 0 DIF: Moderate REF: 2.2 Simplifying Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge ANS: D PTS: 0 DIF: Moderate REF: 2.2 Simplifying Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge ANS: A PTS: 0 DIF: Moderate REF: 2.2 Simplifying Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge ANS: C PTS: 0 DIF: Moderate REF: 2.3 Adding and Subtracting Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge ANS: D PTS: 0 DIF: Easy REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge ANS: A PTS: 0 DIF: Easy REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge ANS: C PTS: 0 DIF: Moderate REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge ANS: A PTS: 0 DIF: Moderate REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge ANS: C PTS: 0 DIF: Easy REF: 2.5 Solving Radical Equations LOC: 11.AN3 TOP: Relations and Functions KEY: Procedural Knowledge ANS: B PTS: 0 DIF: Moderate REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.AN3 TOP: Algebra and Number KEY: Conceptual Understanding ANS: B PTS: 0 DIF: Moderate REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.RF5 TOP: Relations and Functions KEY: Procedural Knowledge ANS: B PTS: 0 DIF: Easy REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Conceptual Understanding ANS: C PTS: 0 DIF: Easy REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Conceptual Understanding ANS: A PTS: 0 DIF: Easy REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Procedural Knowledge ANS: D PTS: 0 DIF: Easy REF: 4.1 Properties of a Quadratic Function LOC: 11.RF4 TOP: Relations and Functions KEY: Conceptual Understanding ANS: C PTS: 0 DIF: Moderate REF: 4.1 Properties of a Quadratic Function LOC: 11.RF4 TOP: Relations and Functions KEY: Procedural Knowledge 35. ANS: REF: TOP: 36. ANS: REF: TOP: 37. ANS: REF: TOP: 38. ANS: REF: LOC: 39. ANS: REF: LOC: 40. ANS: REF: LOC: 41. ANS: REF: LOC: 42. ANS: REF: LOC: 43. ANS: REF: LOC: 44. ANS: REF: LOC: 45. ANS: REF: TOP: 46. ANS: REF: TOP: 47. ANS: REF: TOP: 48. ANS: REF: TOP: 49. ANS: REF: TOP: 50. ANS: REF: TOP: 51. ANS: A PTS: 0 DIF: Moderate 4.1 Properties of a Quadratic Function LOC: 11.RF4 Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge B PTS: 0 DIF: Easy 4.2 Solving a Quadratic Equation Graphically LOC: 11.RF5 Relations and Functions KEY: Conceptual Understanding B PTS: 0 DIF: Easy 4.2 Solving a Quadratic Equation Graphically LOC: 11.RF5 Relations and Functions KEY: Procedural Knowledge D PTS: 0 DIF: Easy 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q 11.RF3 TOP: Relations and Functions KEY: Conceptual Understanding D PTS: 0 DIF: Easy 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q 11.RF3 TOP: Relations and Functions KEY: Conceptual Understanding B PTS: 0 DIF: Easy 4.7 Modelling and Solving Problems with Quadratic Functions 11.RF4 TOP: Relations and Functions KEY: Conceptual Understanding D PTS: 0 DIF: Moderate 4.7 Modelling and Solving Problems with Quadratic Functions 11.RF4 TOP: Relations and Functions KEY: Procedural Knowledge C PTS: 0 DIF: Easy 4.7 Modelling and Solving Problems with Quadratic Functions 11.RF4 TOP: Relations and Functions KEY: Procedural Knowledge D PTS: 0 DIF: Moderate 4.7 Modelling and Solving Problems with Quadratic Functions 11.RF4 TOP: Relations and Functions KEY: Procedural Knowledge B PTS: 0 DIF: Moderate 4.7 Modelling and Solving Problems with Quadratic Functions 11.RF4 TOP: Relations and Functions KEY: Procedural Knowledge A PTS: 0 DIF: Easy 5.1 Solving Quadratic Inequalities in One Variable LOC: 11.RF8 Relations and Functions KEY: Procedural Knowledge A PTS: 0 DIF: Easy 5.2 Graphing Linear Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Procedural Knowledge D PTS: 0 DIF: Moderate 5.2 Graphing Linear Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge C PTS: 0 DIF: Easy 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Procedural Knowledge D PTS: 0 DIF: Easy 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Procedural Knowledge B PTS: 0 DIF: Easy 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge D PTS: 0 DIF: Moderate 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: REF: TOP: ANS: LOC: ANS: LOC: ANS: LOC: ANS: LOC: ANS: LOC: KEY: ANS: LOC: ANS: LOC: ANS: LOC: KEY: ANS: LOC: 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Easy 5.4 Solving Systems of Equations Graphically LOC: 11.RF6 Relations and Functions KEY: Procedural Knowledge B PTS: 0 DIF: Easy 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 Relations and Functions KEY: Conceptual Understanding C PTS: 0 DIF: Easy 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 Relations and Functions KEY: Procedural Knowledge A PTS: 0 DIF: Easy 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 Relations and Functions KEY: Procedural Knowledge B PTS: 0 DIF: Moderate 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills C PTS: 0 DIF: Easy 6.2 Angles in Standard Position in All Quadrants LOC: 11.T1 Trigonometry KEY: Conceptual Understanding D PTS: 0 DIF: Easy 6.2 Angles in Standard Position in All Quadrants LOC: 11.T1 Trigonometry KEY: Conceptual Understanding | Procedural Knowledge B PTS: 0 DIF: Moderate 6.2 Angles in Standard Position in All Quadrants LOC: 11.T2 Trigonometry KEY: Conceptual Understanding | Procedural Knowledge D PTS: 1 DIF: Moderate 6.2 Angles in Standard Position in All Quadrants LOC: 11.T2 Trigonometry KEY: Conceptual Understanding | Procedural Knowledge B PTS: 0 DIF: Easy REF: 6.3 Constructing Triangles 11.T3 TOP: Trigonometry KEY: Procedural Knowledge B PTS: 0 DIF: Easy REF: 6.3 Constructing Triangles 11.T3 TOP: Trigonometry KEY: Procedural Knowledge A PTS: 0 DIF: Moderate REF: 6.3 Constructing Triangles 11.T3 TOP: Trigonometry KEY: Procedural Knowledge B PTS: 0 DIF: Easy REF: 6.4 The Sine Law 11.T3 TOP: Trigonometry KEY: Procedural Knowledge B PTS: 0 DIF: Moderate REF: 6.4 The Sine Law 11.T3 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge D PTS: 0 DIF: Easy REF: 7.5 Solving Rational Equations 11.AN6 TOP: Algebra and Number KEY: Procedural Knowledge A PTS: 0 DIF: Moderate REF: 7.5 Solving Rational Equations 11.AN6 TOP: Algebra and Number KEY: Procedural Knowledge D PTS: 0 DIF: Easy REF: 8.1 Absolute Value Functions 11.RF2 TOP: Relations and Functions Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Moderate REF: 8.1 Absolute Value Functions 11.RF2 TOP: Relations and Functions KEY: 70. ANS: REF: TOP: 71. ANS: REF: TOP: 72. ANS: REF: TOP: 73. ANS: REF: TOP: 74. ANS: REF: TOP: 75. ANS: REF: TOP: Procedural Knowledge | Communication C PTS: 0 DIF: Easy 8.2 Solving Absolute Value Equations LOC: 11.RF2 Relations and Functions KEY: Procedural Knowledge C PTS: 0 DIF: Moderate 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Moderate 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Moderate 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge A PTS: 0 DIF: Difficult 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge C PTS: 0 DIF: Moderate 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge SHORT ANSWER 1. ANS: No, this could not be an arithmetic sequence because the difference between consecutive terms is not constant. PTS: 1 LOC: 11.RF9 2. ANS: DIF: Easy REF: 1.1 Arithmetic Sequences TOP: Relations and Functions KEY: Conceptual Understanding PTS: 1 LOC: 11.RF9 3. ANS: 1 r = , t5 = 5, t6 = 1 5 DIF: Difficult REF: 1.2 Arithmetic Series TOP: Relations and Functions KEY: Conceptual Understanding PTS: 1 LOC: 11.RF10 4. ANS: i) ii) 1 iii) 4 5 iv) DIF: Moderate REF: 1.3 Geometric Sequences TOP: Relations and Functions KEY: Procedural Knowledge 1 The values of the expressions from greatest to least are: 49, 37, 4 , 4 5 PTS: 0 DIF: Moderate REF: 2.1 Absolute Value of a Real Number LOC: 11.AN1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 5. ANS: i) ii) iii) iv) The values of the expressions from least to greatest are: –8, 8, 10, 26 PTS: 0 DIF: Moderate REF: 2.1 Absolute Value of a Real Number LOC: 11.AN1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 6. ANS: a) b) c) The equation has no real root. d) PTS: 0 DIF: Moderate REF: 2.5 Solving Radical Equations LOC: 11.AN3 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 7. ANS: PTS: 0 DIF: Moderate REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.RF5 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 8. ANS: The baseball is 11 m high after 1 s and after 2 s. PTS: 0 DIF: Moderate REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.RF5 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 9. ANS: There are 2 numbers: 5 and –8 PTS: 0 DIF: Moderate REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.RF5 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 10. ANS: a) b) The square root of the discriminant is rational, so the roots are rational. PTS: 0 DIF: Moderate REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 11. ANS: a) x-intercepts: none y-intercept: 9 b) vertex: (–2, 3) c) axis of symmetry: d) domain: e) range: , PTS: 0 DIF: Moderate REF: 4.1 Properties of a Quadratic Function LOC: 11.RF4 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 12. ANS: a) The company should produce 1000 units per week to earn the maximum weekly profit. b) The maximum weekly profit is $600. PTS: 0 DIF: Moderate REF: 4.1 Properties of a Quadratic Function LOC: 11.RF4 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 13. ANS: The graph intersects the x-axis at 2 points, so the related quadratic equation has 2 real roots. This means that the discriminant is greater than 0. PTS: 0 DIF: Moderate REF: 4.2 Solving a Quadratic Equation Graphically LOC: 11.RF5 TOP: Relations and Functions KEY: Communication | Conceptual Understanding 14. ANS: PTS: 0 DIF: Moderate REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q LOC: 11.RF3 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 15. ANS: a) The graph has 1 x-intercept. b) The graph has 0 x-intercepts. c) The graph has 2 x-intercepts. PTS: 0 DIF: Moderate REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q LOC: 11.RF3 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 16. ANS: a) The graph has 2 x-intercepts. b) The graph has 1 x-intercept. c) The graph has 0 x-intercepts. PTS: REF: LOC: KEY: 17. ANS: 0 DIF: Moderate 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q 11.RF3 TOP: Relations and Functions Conceptual Understanding | Procedural Knowledge PTS: 0 DIF: Difficult REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q LOC: 11.RF3 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 18. ANS: maximum value; –2 PTS: 0 DIF: Easy REF: 4.7 Modelling and Solving Problems with Quadratic Functions LOC: 11.RF4 TOP: Relations and Functions KEY: Conceptual Understanding 19. ANS: $1.83 PTS: 0 DIF: Moderate REF: 4.7 Modelling and Solving Problems with Quadratic Functions LOC: 11.RF4 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 20. ANS: The solution is: , 1.5 –9 –8 –7 –6 –5 –4 –3 –2 –1 PTS: 0 LOC: 11.RF8 21. ANS: 0 1 2 3 4 5 DIF: Moderate REF: 5.1 Solving Quadratic Inequalities in One Variable TOP: Relations and Functions KEY: Procedural Knowledge a) The critical values are and . b) Interval Sign of Value of x positive negative positive c) The solution is: or , PTS: 0 DIF: Moderate REF: 5.1 Solving Quadratic Inequalities in One Variable LOC: 11.RF8 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 22. ANS: a) y 6 4 2 –6 –4 0 –2 2 4 6 x –2 –4 –6 b) Sample response: Two points that satisfy the inequality have coordinates: (0, 3), (–2, –1) PTS: 0 DIF: Moderate REF: 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 23. ANS: y 6 4 2 –6 –4 –2 0 2 4 6 x –2 –4 –6 PTS: 0 LOC: 11.RF7 24. ANS: a) and b) DIF: Moderate REF: 5.3 Graphing Quadratic Inequalities in Two Variables TOP: Relations and Functions KEY: Procedural Knowledge y 8 6 4 2 –3 (–1, 0) –2 –1 0 1 2 3 x –2 –4 PTS: 0 LOC: 11.RF6 25. ANS: DIF: Moderate REF: 5.4 Solving Systems of Equations Graphically TOP: Relations and Functions KEY: Procedural Knowledge y 20 (–3, 16) 16 12 8 4 –10 –8 (–7, 0) –6 –4 –2 0 2 4 6 8 x –4 –8 PTS: 0 LOC: 11.RF6 26. ANS: DIF: Moderate REF: 5.4 Solving Systems of Equations Graphically TOP: Relations and Functions KEY: Procedural Knowledge y 20 16 12 8 (2, 9) 4 –8 –6 –4 –2 0 2 4 6 8 x –4 –8 PTS: 0 DIF: Moderate REF: 5.4 Solving Systems of Equations Graphically LOC: 11.RF6 TOP: Relations and Functions KEY: Procedural Knowledge 27. ANS: A linear-quadratic system may have 2 solutions, 1 solution, or no solution. A quadratic-quadratic system may have 2 solutions, 1 solution, no solution, or infinite solutions. PTS: 0 DIF: Easy REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Communication | Conceptual Understanding 28. ANS: The solutions are: (2, 0) and (0, 4) PTS: 0 DIF: Moderate REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Procedural Knowledge 29. ANS: The solutions are: (0, 2) and (–1, –5) PTS: 0 DIF: Moderate REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Procedural Knowledge 30. ANS: a) After 0.4 s b) Approximately 4.0 m PTS: 0 DIF: Moderate REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 31. ANS: The solutions are approximately: (1.7, –7.6) and (–0.4, 0.1) PTS: 0 LOC: 11.RF6 32. ANS: The solutions are: DIF: Moderate REF: 5.5 Solving Systems of Equations Algebraically TOP: Relations and Functions KEY: Procedural Knowledge and PTS: 0 DIF: Moderate REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Procedural Knowledge 33. ANS: approximately 8.7 m PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills 34. ANS: approximately 6.1 m PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills 35. ANS: approximately 73.7 PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills 36. ANS: approximately 27.7 PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills 37. ANS: approximately 50.0 PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Problem-Solving Skills 38. ANS: and PTS: 0 DIF: Moderate REF: 6.2 Angles in Standard Position in All Quadrants LOC: 11.T2 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 39. ANS: It is not possible to construct ABC. PTS: 0 DIF: Moderate REF: 6.3 Constructing Triangles LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 40. ANS: No, the Sine Law cannot be used because only one angle measure is given and the angle is contained between the two given sides. PTS: 0 DIF: Easy REF: 6.4 The Sine Law LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Communication 41. ANS: Two triangles can be constructed. PTS: 0 DIF: Easy REF: 6.4 The Sine Law LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 42. ANS: a) ... Since , no triangle can be constructed. b) No triangle can be constructed. PTS: 0 DIF: Moderate REF: 6.4 The Sine Law LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Communication 43. ANS: x –2 –1 0 1 2 3 –4 –3 –2 –1 0 1 4 3 2 1 0 1 PTS: 0 DIF: Easy REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 44. ANS: x –2 –1 0 1 2 3 –16 –14 –8 2 16 34 16 14 8 2 16 34 PTS: 0 DIF: Easy REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 45. ANS: PTS: 0 DIF: Moderate REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 46. ANS: PTS: 0 DIF: Moderate REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 47. ANS: PTS: 0 DIF: Moderate REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 48. ANS: The solutions are and . PTS: 0 DIF: Easy REF: 8.2 Solving Absolute Value Equations LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding 49. ANS: The student might have used the graph to solve the equation: PTS: 0 DIF: Moderate REF: 8.2 Solving Absolute Value Equations LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge 50. ANS: The student might have used the graph to solve the equation: PTS: 0 DIF: Difficult REF: 8.2 Solving Absolute Value Equations LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge PROBLEM 1. ANS: a) The common difference, d, is: b) c) PTS: 1 LOC: 11.RF9 2. ANS: DIF: Difficult REF: 1.1 Arithmetic Sequences TOP: Relations and Functions KEY: Problem-Solving Skills a) b) PTS: 1 DIF: Moderate REF: 1.2 Arithmetic Series LOC: 11.RF9 TOP: Relations and Functions KEY: Conceptual Understanding | Problem-Solving Skills 3. ANS: a) The first 3 terms are: b) The sequence is convergent because the terms approach a constant value of 0. PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences LOC: 11.RF10 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Problem-Solving Skills 4. ANS: Sample response: This sum comprises 3 geometric series: 1 + 4 + 16 + 64 + 256 3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 2 + 8 + 32 + 128 + 512 For the first series: For the second series: For the third series: PTS: 1 DIF: Difficult REF: 1.4 Geometric Series LOC: 11.RF10 TOP: Relations and Functions KEY: Conceptual Understanding | Problem-Solving Skills 5. ANS: To determine t1, use: The 1st jump is approximately 9.00 m. To determine t2, use: The 2nd jump is approximately 2.25 m. To determine t3, use: The 3rd jump is approximately 0.56 m. PTS: 1 DIF: Difficult REF: 1.4 Geometric Series LOC: 11.RF10 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Problem-Solving Skills 6. ANS: The payments form a geometric series, with first term 56, common ratio 1.021, and number of terms 16. Joel earned approximately $1051.94 after 16 days. PTS: 1 DIF: Moderate REF: 1.4 Geometric Series LOC: 11.RF10 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Problem-Solving Skills 7. ANS: The length of the frog’s first jump was approximately 45 cm. PTS: 1 DIF: Difficult REF: 1.4 Geometric Series LOC: 11.RF10 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 8. ANS: Sample response: For the term values to alternate between positive and negative,and increase in numerical value, the 25 125 common ratio must be ; for example, with , a possible sequence is: –4, 5, , , 4 16 625 3125 , , ... 64 256 PTS: 1 DIF: Moderate REF: 1.5 Graphing Geometric Sequences and Series LOC: 11.RF9 | 11.RF10 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Procedural Knowledge 9. ANS: a) b) c) Geom etric Sequence 1000 800 600 Term value 400 200 0 –200 1 2 3 4 5 6 7 Term num ber –400 –600 –800 –1000 To graph the geometric series, determine the partial sums. S1 3 S2 S3 S4 S5 S6 1600 Geom etric Series 1200 Partial sums 800 400 0 –400 1 2 3 4 5 6 7 Num ber of term s –800 –1200 –1600 PTS: 1 DIF: Moderate REF: 1.5 Graphing Geometric Sequences and Series LOC: 11.RF9 | 11.RF10 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Procedural Knowledge 10. ANS: a) t2 is: –5 t3 is: 5 9 t4 is: b) 5 9 5 81 5 81 5 729 , so the series converges. c) PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series LOC: 11.RF10 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 11. ANS: Use the formula for the area, A, of a rectangle: Substitute: and Perimeter of shape formed An expression for the perimeter of the shape formed is: PTS: 0 DIF: Moderate REF: 2.3 Adding and Subtracting Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication | Problem-Solving Skills 12. ANS: PTS: 0 DIF: Easy REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication 13. ANS: Use the formula for the area, A, of a square: PTS: 0 DIF: Moderate REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication 14. ANS: Simplify the denominators. Rationalize the denominators. PTS: 0 DIF: Difficult REF: 2.4 Multiplying and Dividing Radical Expressions LOC: 11.AN2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication 15. ANS: Since the left side does not equal the right side, is not a root of the equation. PTS: 0 DIF: Easy REF: 2.5 Solving Radical Equations LOC: 11.AN3 TOP: Relations and Functions KEY: Procedural Knowledge | Communication 16. ANS: a) Use the formula for the volume, V, of a cube: , where s represents the edge length of the cube. A cube with the same volume as Lake Ontario would have an edge length of about 12.0 km. b) Use the formula for the volume, V, of a sphere: , where r represents the radius of the sphere. A sphere with the same volume as Lake Ontario would have a radius of about 7.4 km. PTS: 0 DIF: Difficult REF: 2.5 Solving Radical Equations LOC: 11.AN3 TOP: Relations and Functions KEY: Procedural Knowledge | Problem-Solving Skills 17. ANS: Use the Pythagorean Theorem. Solve by factoring. Since length cannot be negative, . The length of the shorter leg is 20 cm. The length of the longer leg is: PTS: 0 DIF: Difficult REF: 3.2 Solving Quadratic Equations by Factoring LOC: 11.RF5 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 18. ANS: For an equation to have two real roots, Substitute: For to have two real roots, k must be less than 3. Sample response: A possible value of k is 2. So, an equation with two real roots is: PTS: 0 DIF: Moderate REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 19. ANS: For an equation to have no real roots, Substitute: For 9 to have no real roots, k must be greater than . 8 Sample response: A possible value of k is 3. So, an equation with no real roots is: PTS: 0 DIF: Moderate REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 20. ANS: Use guess and test to determine two values of a, b, and c so that: Substitute: This is satisfied by So, one equation is: and . and . Substitute: This is satisfied by So, another equation is: PTS: 0 DIF: Difficult REF: 3.5 Interpreting the Discriminant LOC: 11.RF5 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 21. ANS: a) x-intercepts: –1.70, 0.95 y-intercept: 3.25 b) vertex: (–0.38, 3.53) c) axis of symmetry: d) domain: e) range: , PTS: 0 DIF: Moderate REF: 4.1 Properties of a Quadratic Function LOC: 11.RF4 TOP: Relations and Functions KEY: Communication | Procedural Knowledge 22. ANS: Compare with the vertex form . a) a is positive, so the graph opens up. and , so the coordinates of the vertex are: (2, 4) The equation of the axis of symmetry is ; that is . To determine the y-intercept, substitute . The y-intercept is 6. To determine the x-intercepts, substitute . This equation has no solution, so there are no x-intercepts. The domain is: The graph opens up, so the vertex is a minimum point with y-coordinate 4. The range is: b) y 20 16 12 8 4 –12 –8 –4 0 4 8 12 x –4 PTS: 0 DIF: Moderate REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q LOC: 11.RF3 TOP: Relations and Functions KEY: Communication | Conceptual Understanding | Procedural Knowledge 23. ANS: Sketch the parabola on the coordinate plane so the line of symmetry is the y-axis. The y-intercept represents the maximum height. The vertex of the parabola is 27 m above the base, so the coordinates of the vertex are V(0, 27). Since the bridge is 72 m wide, the x-intercepts are 36 and 36. , y 30 24 18 12 6 –36 –30 –24 –18 –12 –6 0 6 12 18 24 30 36 x –6 The equation of the parabola has the form , with vertex (p, q). The coordinates of the vertex are V(0, 27), so and . So, the equation of the parabola becomes . To determine the value of a, substitute the coordinates of an x-intercept: (36, 0) An equation of the parabola is: PTS: 0 DIF: Difficult REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q LOC: 11.RF3 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 24. ANS: Determine an equation to represent the situation. For each $1 decrease in price, 125 more tickets will be sold. Let x represent the number of $1 decreases in the price of a ticket. When the price decreases by $1 x times: • the price, in dollars, of a ticket is • the number of tickets sold is • the revenue, in dollars, is . . . Let the revenue be R dollars. An equation is: Use a graphing calculator. Graph: Use the CALC function to determine the coordinates of the vertex. a) From the graph, the maximum revenue occurs when the number of $1 decreases is 4. So, the decrease in price that will maximize the revenue is $4. b) The price of a ticket that will maximize the revenue is: c) Substitute in to determine the maximum revenue. The maximum revenue is $50 000. PTS: 0 DIF: Difficult REF: 4.7 Modelling and Solving Problems with Quadratic Functions LOC: 11.RF4 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 25. ANS: a) Solve: When , such as , L.S. = 1 and R.S. = 0; so When , such as , L.S. = 1 and R.S. = 0; so The solution is: , x 2.5 b) satisfies the inequality. satisfies the inequality. 2.5 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 c) There is only one number that does not satisfy the inequality. Since the square of any non-zero number is positive, any real number but 2.5 satisfies the inequality. PTS: 0 DIF: Moderate REF: 5.1 Solving Quadratic Inequalities in One Variable LOC: 11.RF8 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 26. ANS: a) For to have 2 real roots, its discriminant is greater than 0. b) For to have exactly 1 root, its discriminant is 0. c) For to have no real roots, its discriminant is negative. , which is written PTS: 0 DIF: Difficult REF: 5.1 Solving Quadratic Inequalities in One Variable LOC: 11.RF8 TOP: Relations and Functions KEY: Communication | Conceptual Understanding 27. ANS: Write an inequality to represent the problem. Let x m represent the width of the garden. Then its length is m. And its area, in square metres, is . The area is at least 20 m2. So, an inequality is: Use a graphing calculator to graph the corresponding quadratic function: Determine the x-intercepts: and From the graph, the inequality is greater than or equal to 0 for Since the width of the garden is positive, the solution of the problem is: The width of the garden is greater than or equal to approximately 2.4 m and its length is greater than or equal to approximately , or 8.4 m. Verify the solution. The area of the garden with dimensions 2.4 m and 8.4 m is: This is greater than 20 m2, so the possible dimensions are correct. PTS: 0 DIF: Difficult REF: 5.1 Solving Quadratic Inequalities in One Variable LOC: 11.RF8 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 28. ANS: a) Since , the graph is in Quadrant 4. The graph of the related function has slope –3 and y-intercept 4. Draw a broken line to represent the related function in Quadrant 4. Shade the region below the line. The axes bounding the graph are broken lines. y 6 4 2 –6 –4 –2 0 2 4 6 x –2 –4 –6 b) Since , the graph is in Quadrant 3. Graph the related function. When , . When , . Draw a solid line to represent the related function in Quadrant 3. Use (0, 0) as a test point. L.S. = 6; R.S. = 0 Since 6 > 0, the origin is not in the shaded region. Shade the region below the line. The axes bounding the graph are solid lines. y 6 4 2 –6 –4 0 –2 2 4 6 x –2 –4 –6 PTS: 0 DIF: Difficult REF: 5.2 Graphing Linear Inequalities in Two Variables LOC: 11.RF7 TOP: Relations and Functions KEY: Communication | Procedural Knowledge 29. ANS: In , substitute: , Solve for a. Divide both sides by –3. Square both sides. That is, PTS: 0 DIF: Moderate REF: 5.3 Graphing Quadratic Inequalities in Two Variables LOC: 11.RF7 TOP: Relations and Functions KEY: Communication | Procedural Knowledge 30. ANS: 10000 y 8000 (40, 5400) (50, 5000) 6000 4000 2000 –60 –40 –20 0 –2000 20 40 60 80 100 x PTS: 0 DIF: Moderate REF: 5.4 Solving Systems of Equations Graphically LOC: 11.RF6 TOP: Relations and Functions KEY: Problem-Solving Skills | Procedural Knowledge 31. ANS: The line has slope 3 and y-intercept 4, so its equation is: The parabola has vertex (5) and is congruent to . So, its equation is: The system of equations represented by the graph is: The coordinates of the points of intersection are: (0, 4) and (5, 11) PTS: 0 DIF: Difficult REF: 5.4 Solving Systems of Equations Graphically LOC: 11.RF6 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 32. ANS: a) Let x represent the first number and let y represent the second number. The statement that twice the square of the first number minus the second number is 3 can be modelled with the equation: The statement that the square of the sum of the first number and 5 is equal to the second number minus 2 can be modelled with the equation: So, the system of equations that represents this relationship is: b) From equation , substitute in equation . So, or Substitute each value of x in equation . When : When : The numbers are: 2 and 11; 12 and 291 Verify the solutions using the statement of the problem. For and : 2 Two times (2) minus 11 is 3. The square of is equal to These numbers satisfy the problem statement. For and : 2 Two times (12) minus 291 is 3. is equal to 291 minus 2. These numbers satisfy the problem statement. So, the numbers are: 2 and 11; 12 and 291 PTS: 0 DIF: Difficult REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 33. ANS: a) Write an equation that represents the height of the soccer ball t seconds after the football is kicked. Solve the system formed by the 2 equations: From equation , substitute in equation . The football and the soccer ball reached the same height after approximately 1.5 s. b) Substitute in equation . After 1.5 s, both balls are at a height of approximately 4.9 m. PTS: 0 DIF: Difficult REF: 5.5 Solving Systems of Equations Algebraically LOC: 11.RF6 TOP: Relations and Functions KEY: Communication | Problem-Solving Skills 34. ANS: First determine the measure of B. In right ABC, In a right triangle, when one acute angle is , the other acute angle is . So, B is approximately 58.4 and A is approximately 31.6. PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Conceptual Understanding | Communication 35. ANS: a) In right BCD, BD is the hypotenuse. BC is the side opposite angle and CD is the side adjacent to angle . b) sin is approximately 0.6. This means that in any right triangle similar to BCD, the length of the side opposite angle is approximately 0.6 times the length of the hypotenuse. c) is approximately 34.8. PTS: 0 DIF: Moderate REF: 6.1 Angles in Standard Position in Quadrant 1 LOC: 11.T2 TOP: Trigonometry KEY: Procedural Knowledge | Communication 36. ANS: Draw a labelled diagram to represent the problem. In right FLP, FP is the hypotenuse and FL is the side opposite P. So, use the sine ratio to determine the length of FL. F 9.1 km Solve the equation for FL. P 54° L The distance between the fishing boat and the lighthouse is approximately 7.4 km. PTS: 0 DIF: Moderate REF: 6.2 Angles in Standard Position in All Quadrants LOC: 11.T2 TOP: Trigonometry KEY: Communication | Problem-Solving Skills 37. ANS: a) , or 0.5428... Since , there are two possible triangles with the given measures. b) B 35 cm 19 cm 19 cm 30° A C1 C2 PTS: 0 DIF: Moderate REF: 6.3 Constructing Triangles LOC: 11.T3 TOP: Trigonometry KEY: Communication | Problem-Solving Skills 38. ANS: Possible solution: Length of BC (cm) Value of How does compare with sin A? Description of possible triangles 5 0.8333... No triangles are possible. 6 1 1 isosceles triangle 7 1.1666... 1 scalene triangle 5.9 0.9833... 2 scalene triangles PTS: 0 DIF: Moderate REF: 6.3 Constructing Triangles LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Problem-Solving Skills 39. ANS: Sketch a diagram to represent their trip from A, through B, to C. Determine the measure of B in ABC. B 24° In ABC: 3.4 km 28° 2.4 km 24 C Determine the measure of angle . B 24° 3.4 km 28° Determine the angle bearing, . 2.4 km C The straight-line distance is approximately 2.1 km. The bearing of the end point from the start point is approximately 250°. PTS: 1 DIF: Difficult REF: 6.5 The Cosine Law LOC: 11.T3 TOP: Trigonometry KEY: Procedural Knowledge | Communication | Problem-Solving Skills 40. ANS: Use: 24 Use: PTS: 1 DIF: Moderate REF: 6.5 The Cosine Law LOC: 11.T3 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 41. ANS: Factor the expressions and simplify. Since the expressions have the same denominator but different numerators, the equation has no solution. PTS: 0 DIF: Moderate REF: 7.5 Solving Rational Equations LOC: 11.AN6 TOP: Algebra and Number KEY: Conceptual Understanding | Communication 42. ANS: First, identify the non-permissible values and a common denominator: The non-permissible values are: and A common denominator is: Then, multiply each term in the equation by the common denominator and simplify. Finally, solve the equation using the quadratic formula. The solutions are: and PTS: 0 DIF: Difficult REF: 7.5 Solving Rational Equations LOC: 11.AN6 TOP: Algebra and Number KEY: Procedural Knowledge | Communication | Problem-Solving Skills 43. ANS: x –2 –1 0 1 2 3 –6 –4 –2 0 2 4 6 4 2 0 2 4 y 8 6 4 y=|f(x)| 2 –8 –6 –4 –2 y=f(x) 0 2 4 6 8 x –2 –4 –6 –8 PTS: 0 DIF: Moderate REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication 44. ANS: Choose two points on the line to determine the slope of the linear function: (–5, 5) and (–1, –3) 8 y 6 4 2 y-intercept: –5 An equation for the absolute value function is: –8 –6 –4 0 –2 2 4 6 8 x –2 –4 –6 –8 PTS: 0 DIF: Difficult REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication | Problem-Solving Skills 45. ANS: Assume the middle piece of the graph was y reflected in the x-axis. So, the graph of the 8 quadratic function opens up and has vertex (4, – 5). The equation has the form 6 Use the point (5, 0) to determine a. 4 2 An equation of the absolute value function is: –8 –6 –4 –2 0 2 –2 –4 –6 –8 PTS: 0 DIF: Difficult REF: 8.1 Absolute Value Functions LOC: 11.RF2 TOP: Relations and Functions KEY: Procedural Knowledge | Communication | Problem-Solving Skills 46. ANS: Simplify the equation: The absolute value function creates two quadratic equations: 4 6 8 x and So, the solutions are , , and . PTS: 0 DIF: Moderate REF: 8.2 Solving Absolute Value Equations LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication 47. ANS: a) After 1 h, the motorcycle has travelled 95 km. After t hours, the motorcycle has travelled 95t kilometres. The distance from the border after t h can be represented by the equation: b) Substitute: t = 1.5 So, the motorcycle is approximately 113 km from the border after 1.5 h. PTS: 0 DIF: Moderate REF: 8.2 Solving Absolute Value Equations LOC: 11.RF2 TOP: Relations and Functions KEY: Conceptual Understanding | Communication | Problem-Solving Skills 48. ANS: The graph of passes through the points (2, 6), (6, 6), (3, 3), and (5, 3). Plot these points on the coordinate y grid. The points are symmetrical about the 20 line , so plot a point at (4, 0). Join the points with straight lines. 16 An equation for the right branch of the graph has the form . 12 Use the points (4, 0) and (3, 3) to 8 determine the slope, m. 4 –16 –12 –8 –4 0 4 8 12 16 x –4 An equation of the line is: Use the point (2, 6) to determine the y-intercept, b. So, an equation for the absolute value function is PTS: 0 LOC: 11.RF2 . DIF: Difficult REF: 8.2 Solving Absolute Value Equations TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication 49. ANS: An equation of the line has the form . Use the points and to determine m and b. y 8 6 4 So, an equation of the linear function is For the graph of the reciprocal function: . The equation is: 2 –6 Horizontal asymptote: x-intercept is 0, so vertical asymptote is –4 –2 0 2 4 6 x –2 . –4 –6 –8 Mark points at y = 1 and y = on the graph of . Draw a smooth curve through each point so that the curve approaches the asymptotes but never touches them. PTS: 0 DIF: Difficult REF: 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication 50. ANS: a) Use the formula for speed: Substitute: d = 1 b) Both time and speed are positive. The reciprocal function has vertical asymptote So, the domain is and the range is c) A sketch of the reciprocal function is: and horizontal asymptote . . t 10 8 6 4 2 0 2 4 6 8 10 s PTS: 0 DIF: Moderate REF: 8.3 Graphing Reciprocals of Linear Functions LOC: 11.RF11 TOP: Relations and Functions KEY: Procedural Knowledge | Communication | Problem-Solving Skills