This Is a List of Major Topics Covered This Year
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Math 10 Exam Topics June 2012
**This is a list of major topics covered this year. ** **You are responsible for all material covered – refer to your notes to check for additional topics. **
Chapter 3 – Patterns, Relations, Equations, and Scatter plots Predictions Regression analysis and correlation coefficient You should be able to do most of the above with and without a Graphing Equations graphing calculator! o Table of values o y = mx + b (using slope and y-intercept) o x and y intercept Chapter 5 – How Far? How tall? How steep? Determining equations o From a graph o Given a point and the slope Similar triangles Congruent triangles o Given 2 points Pythagorean theorem Parallel and perpendicular lines Labelling sides and angles in a triangle – adjacent, Domain and range opposite, hypotenuse Polynomials Trig ratios – sine, cosine, tangent Multiplying (distributive property) o Solving for sides using trig ratios o Factoring Solving for angles using trig ratios (using “2nd” or “shift” . Greatest common factor key on your calculator) . Guess and check Angles of elevation and depression . Decomposition method Word problems involving the above . Difference of squares Coordinate geometry- midpt, & distance. o Expanding and then simplifying (FOIL) Radicals o Rational Expressions o Simplifying – No perfect squares underneath . Simplifying radical sign!! . Mult., div., adding and subtracting o Adding, subtracting, multiplying, dividing Reading/interpreting graphs o Rationalizing the denominator Dependent and independent variables Describing patterns Discrete and continuous data Chapter 7 – Linear Programming
Chapter 4 – Modeling Functional Relationships Solving inequalities Graphing and finding equations of inequalities Solving systems of equations (looking for a point of Transformations of absolute and quadratic functions intersection between 2 lines) Horizontal and vertical translation o o Graphically Reflection (in the x-axis) o o Substitution Vertical and horizontal stretch o o Elimination Mapping rule Linear programming (optimization) problem Solving quadratics for roots, zeros and x-int. by factoring and solving non-factorable eq. on TI83. Graphing quadratics Getting equations (quadratic functions) from graph Functions and function notation – f(x)
Chapter 6 – Geometry
Be able to name various geometric shapes Compute perimeter and area Compute volume Determine Surface area Calculate economy rate Determine the packaging cost
Chapter 1 – Data Management
Independent and dependent variables Cause and effect relationships Mean, mode, median, standard deviation Histogram – frequency table, bin, frequency Stem and leaf plot Box and whisker plots – 5 key points Normal distribution – 68%, 95%, 99.7% Math 10 – Exam Review
1. Graph each of the following: a. y = -3 x + 10 b. y = 5 c. x = -2 4 2. Rearrange the equations into y = mx + b form and graph: a. x – 2y + 1 = 8 b. 4x – 6y = 0
3. Find the x and y intercepts for: a. 3x – 5y = 12 b. y = -2 x + 7 3 4. Write the equations for the following linear graphs: a. b.
5. Julie rents a jet ski and is charged an initial flat fee of $5 plus $2 for every litre of fuel consumed. a. Set up a table of values for the situation described b. Using the table of values created, draw a graph for this relationship c. Write an equation for this relationship
6. On weekends, Gina operates a window cleaning business. The table shows her fee depending on hours worked. Hours Worked Charge ($) 1 15 3 35 5 55 7 75
a. What is the hourly rate? b. What is her initial fee? c. How much would Gina charge for a 10 hour job? d. Find the equation that describes this relationship e. Graph the relationship
7. Find the equations of the line: a. Slope = -1/2 and passing through the point (-4,5) b. x-intercept = 8 and y-intercept = -3 c. passing through points (-1,7) and (2,-2) d. parallel to 2x – 3y = -3 and passing through (-6,5) e. parallel to y-axis and passing through (-4, 2) f. perpendicular to y = 3x – 8 and with the same x-intercept as 6x – 5y = 12 2 8. Solve for the variable a. 2(6d + 3) = 18 – 3(16 – 3d) b. (2x – 1)(x + 7) – 3(x + 1) = -2x(5 – x)
c. (x – 4)(x + 5) – (x + 6)(x – 2) = 8 d. a – 1 = a + 1 3 2 4 3 e. 1(4x + 1) – 3 – 2x = 2x – 5 2 3 6
9. Find the intersection point by solving and graphing a. y = -3 x + 4 and y = 1 x – 8 2 2 b. y = 4x – 7 and y = -2x – 5
10. Are the following sets of data (shown by graphs) continuous or discrete? A. B. C. D.
11. State the domain and range (using proper notation) for each of the graphs in question 10.
12. Expand and simplify a. (x – 4)2 b. (a + b)2 c. 2a(2a2 – m) d. (2y – 3)(4y + 7) e. (y + 1)(y – 1)
13. Factor Completely a. a2 – 25 b. 72 – 8x2 c. y2 – 5y + 4 d. 3a2 + 4a + 1 e. 2x2 + 4x – 16 f. 16m2 + 4n2 g. 3n2 + 24n + 48 h. 6a2 – a – 2 i. 5a2 + 48a – 20 j. 4a2 + 27a – 7 k. x2 + x + 1 l. 4x2 – 9
14. Simplify a. 3x – 12 b. a 3 b – a 2 b 2 c. x 2 + 4x – 21 12x – 48 a2b – ab2 x2 – 4x + 3
15. Multiply/Divide/Add/Subtract x 2 15 4x 24 x 2 6x 27 x 2 10x 9 6x 3 9x 2 a. b. c. 2x 12 x 2 3x 40 2x 2 11x 15 16x 40 4x 2 1 6x 3 3 16. Kendra is travelling at a constant speed along the highway and is stopped by a police officer. The officer gives her a ticket, and then she continues on her way at a slower speed. Draw a graph that represents this scenario with time on the x axis and speed on the y axis.
17. Use the graph below and answer the questions that follow.
a. What is the difference between the sections with positive slopes and the section with a negative slope? b. What do the flat regions on the graph show? c. Write a story to describe this person’s trip to the supermarket.
18. Which of the following relations are functions? a. (1, 4), (-1, 5), (2, -4), (-2, 5) b. b. (3, 6), (4, -4), (8, 9), (, 3-1) c. x y e. f. 1 6 2 6 3 6 4 6
.
19. Given: f(x) = -8x – 23 g(x) = x2 + 2x – 7 h(x) = -2 j(x) = 7x – 20 5 Find each of the following: a. f(-5) b. g(-1) c. h(4) d. j(3) e. g(2m) f. find x if f(x) = 1 g. find x if j(x) = 3 h. find x if h(x) = f(x) i. find x if g(x) = f(x)
20. For the following quadratic and absolute value equations, state the: i. transformations on y = x2 or y = |x| iv. mapping rule ii. vertex v. graph the function iii. axis of symmetry 4 a. y – 6 = (x + 1)2 b. –(y – 3) = (x – 5)2 c. 1(y + 4) = (x + 7)2 d. –2(y + 1) = x2 3 e. y = |x + 4| f. –4(y – 2) = |x + 1| g. –y = |x – 2| h. –3y = (x – 2)2
5 15 y
10
5
x
-15 -10 -5 5 10 15
-5
-10
-15
21. State the equations or each of the following graphs.
22. Solve for x. Note: Some may require the graphing calculator. a. x2 + 4x – 32 = 0 b. x2 – 9x +14= 5 c. 2x2 + 9x – 5 = 0 d. 4x2 – 36 = 0 e. 18x2 + 27x = 0 f. 5(x²-3)=-x(3x+2)
23. Timmy plants a tree and would like it to grow very tall. His mother tells him that the more he waters the tree, the taller it will grow. What is the dependent variable? What is the independent variable?
24. Isabelle is starting a tree farm. She has gathered data about trees grown by 2 different companies. She wants to grow the biggest fir tree in the shortest period of time.
Firs ‘r’ Us (diameter in cm of 1-yr old trees at a height of 1m) 2.2 2.1 2.5 1.8 1.6 1.4 1.7 1.9 2.3 2.8 2.4 2.7
Balsam City (diameter in cm of 1-yr old trees at a height of 1m) 1.6 1.7 1.9 2.9 1.4 1.8 1.6 1.4 2.1 2.6 2.4 2.4
a. Draw box and whisker plots for each of the two sets of data in order to compare them. b. Which company should she choose?
25. When the reading level of 1200 junior high students was tested, the results were normally distributed with a mean grade level of 7.6 and a standard deviation of 1.8.
a. Draw a diagram and indicate the key values. b. What percent of the students were reading at a level between 4 and 11.2? c. What percent of the students were reading at a level above 11.2? d. How many students were reading above a grade level of 5.8? e. How many students were within one standard deviation of the mean? f. Give an example of a reading level that would be considered an outlier.
26. The diameter of trees a Christmas tree farm are found to be normally distributed with a mean of 5cm and a standard deviation of 1.5cm. 6 a. What is the range of trees within 1 standard deviation of the mean? b. 95% of the trees on the farm have a diameter within what range? c. Suppose that there are 900 trees. Approximately how many would have the diameter more than 8cm? d. What percent of the trees had diameters more than 2cm and less than 6.5cm?
27. After 1 week, the heights of 30 radish seed plants were measured. They are shown in the table below.
5 36 32 48 15 21 36 39 7 17 10 22 33 25 25 24 29 38 18 37 25 11 10 16 22 26 45 37 26 13 a. Use bin size 10 starting at 0 to make a frequency table. b. Draw a histogram to represent the data. 28. Boards from a rail shipment of wood are selected at random and measured. The following lengths, in meters, are obtained. 4.01 3.96 4.10 3.98 3.92 3.95 4.08 4.03 4.03 3.98 a. Calculate the mean. b. Calculate the standard deviation. c. Does the shipment pass inspection?
29. The following data gives the blood pressure of 30 adults.
151 123 120 123 110 115 117 112 120 122 121 117 110 136 108 124 110 115 155 130 116 121 120 117 99 137 123 127 131 110 a. Construct a box and whisker plot for the data. Indicate the key values clearly on your plot. b. Make a frequency table and construct a histogram using a bin width of 10. c. State each of the following: i. Range ii. Mean iii. Mode iv. Median v. Standard deviation
30. For each of the following sets of data calculate the correlation coefficients (linear, quadratic and exponential). Then, decide whether the data are best modeled using a liner, quadratic or exponential function. a. x 5 10 15 20 25 30 35 40 y 600 400 200 100 20 0 30 120 7 b. x 8 10 12 14 16 18 20 22 y 3.3 7.0 10.5 13.9 17.0 20.0 23.0 25.5
c. x 10 20 30 40 50 60 70 80 y 0.20 0.25 0.40 0.70 1.0 1.7 2.8 4.5 31. Regression analysis was performed on a set of data and following correlation coefficients were obtained: Linear r2 = 0.0017 Quadratic r2 = 0.6921 Exponential r2 = 0.7822
a. Which model best fits the data? Why? b. Would you say that the model you chose is a good fit? Why or why not?
32. Another set of data was analyzed using regression analysis and the following information was found:
Function Equation r2 Linear y = 0.7355x – 54.6819 0.8359 Quadratic y = 0.0022x2 + 0.2294x – 27.3712 0.8384 Exponential y = (0.5339)(1.034)x 0.8084
a. Which model best fits the data? b. Suppose the data represented the number of goals you make vs. slap shot speed. How many goals would you expect to get if your slap shot speed is 129km/hr?
33. Solve for the missing variables a. b. Find g.
c. d.
8 e. Given: BD = 10; B C D 34. Which of the following are Pythagorean triples? a. 10-24-26 b. 7-24-25 c. 10-12-15 35. Solve ΔMNP for all missing information if given: m = 8, n = 12, 36. Determine whether or not the following is a right-angled triangle. 37. For maximum safety, a ladder should make an angle of 72° with the ground. If a ladder is 4 meters long, how far will it reach up the wall? 38. Find the angle of elevation to the top of a radar tower 112m high from an observation point 245m from the base of the tower. 39. Three towns (A, B, and C) are located so that A is 30km due north of C, and B is 45km due east of C. Calculate the distance from A to B. 40. From the second floor of QEH, the angle of elevation to the top of a nearby building is 31° and the angle of depression to the base of the building is 48°. If the distance between QEH and the nearby building is 15m, calculate the height of the building. 41. When standing 30m away from a flagpole, the angle of elevation is 35°. What is the height of the flagpole? 42. An airplane is flying at an altitude of 140 000m. The pilot spots a forest fire at an angle of depression of 48°. What is the distance from the pilot to the fire? 43. Which of the following are similar triangles? (not drawn to scale!) A. B. 6 C. 3.6 D. 18 E. 8 0.9 1.5 15 10 7.2 4.8 24 30 10 1.2 7.5 9 44. Simplify each of the following radicals, leaving your answer in radical form. a. b. c. 1 2 d. e. f. 500 g. 5 3 45. Solve each of the inequalities below: a. 6 + 11x > -60 b. 32 – 15x < 2 46. Graph the regions described by the following inequalities: a. y > -2x + 1 b. –2x < 2y + 5 c. 2x + 2y < -6 d. –4y ≤ x + 12 47. Solve the following systems graphically: a. x + 2y = -4 b. y = ½ x – 3 4y = 3x + 12 y = ¼ x – 1 48. Solve by substitution or elimination: a. x + y = 8 b. x + 2y = 5 c. 5x – y = 13 d. 3x – 6y = 24 x – y = 4 x + y = 3 2x + 3y = 12 5x + 4y = 12 e. 2x – 3y = -1 f. 2x – 4y = -2 g. 3x + 3y = 3 h. 2x + 3y = 5 10x + 10y = 5 3x + 2y = 3 4x + 2y = 2 4x + 6y = 6 i. 3x = 24 j. x + 3y = 5 x + 2y = 0 2x – 3y = -8 49. A sail maker makes 2 kinds of sails: jibs and spinnakers. Each kind must be processed by 2 different machines: a cutter and a sewer. To complete 1 jib, the cutter must work 1 hour and the sewer must work 2.5 hours. To complete one spinnaker, the cutter and sewer must work 4 hours and 2 hours respectively. The cutter may not be operated more than 8 hours per day and the sewer not more than 12 hours per day. If the profit on a jib is $100 and profit on a spinnaker is $150, how many sails of each kind should the manufacturer make per day to maximize profit? 50. A furniture manufacturer makes 2 different types of ping pong tables: the Bounce Easy Model and the Catch-a-Spin Model. Each model must be processed by 2 machines, Alpha and Beta. Machine Alpha can be operated a maximum of 18 hours per day and Beta a maximum of 21 hours. To make the Bounce Easy model, 2 hours is needed on Alpha and 5 hours on Beta. For one Catch-a-Spin model, 4 hours is needed on Alpha and 2 hours on Beta. The manufacturer can sell all the tables on both types. His profit on each Bounce Easy is $30 and on each Catch-a-Spin is $50. How many of each type should he produce daily to maximize his profit? 51. Mike’s Famous Toy Trucks manufactures 2 kinds of toy trucks – a standard model and a deluxe model. In the manufacturing process, each standard model requires 2 hours of grinding and 2 hours of finishing, and each deluxe model needs 2 hours of grinding and 4 hours of finishing. The company has 2 grinders and 3 finishers, each of whom work 40 hours per week. Each standard model toy truck brings a profit of $3 and each deluxe model a profit of $4. Assuming that every truck made will be sold, how many of each should be made to maximize profits? 10 52. Nutt’s Nuts has 75 pounds of cashews and 120 pounds of peanuts. These are to be mixed in 1 pound packages as follows: a low-grade mixture that contains 0.25 pounds of cashews and 0.75 pounds of peanuts, and a high-grade mixture that contains 0.5 pounds of cashews and 0.5 pounds of peanuts. Suppose that in addition to the information given above, we also know what profit will be on each type of mixture. For example, suppose the profit is $0.25 on each package of the low-grade mixtures and is $0.45 on each package of the high-grade mixture. The question of importance to the manager is “How many packages of each type of mixture should be prepared to maximize profit?” (assuming that all packages are sold) 53. Find the Volume, Surface area and Economy Rate of the following objects: a) b) c) 54. Paul wants to paint the exterior walls of his house and the roof. The paint costs $34.95 per 4 L can. Each 4 L can covers 30m2. How much will it cost Paul to paint his house? 3m 2m 4m 10m 7m 55. A golfer carries 8 golf balls in his bag. Each golf ball has a diameter of 42.7 mm. How much volume do the balls take up in golfer’s bag? 11 Math 10 Answers to review questions for final exam June 2012 1. Graphs 1 7 2 2. (a) y x (b) y x 2 2 3 12 21 3. (a) x = 4 ; y = or –2.4 (b) x = or 10.5 ; y = 7 5 2 1 4. (a) y = – x (b) y x 1 2 5. (c) y = 2x + 5 6. (a) $10/h (b) $5 (c) $105 (d) y = 10x + 5 1 3 7. (a) y x 3 (b) y x 3 2 8 2 (c) y 3x 4 (d) y x 9 3 2 2 (e) x = – 4 (f) y x 3 3 1 16 1 8. (a) –12 (b) (c) or – 5 2 3 3 1 (d) 10 (e) 2 1 17 9. (a) (6, – 5) (b) , 3 3 10. (a) (b) (c) continuous (d) discrete 11. (a) Domain: {x | x 3, x R} Range: {y | y 3, y R} (b) Domain: {x | 3 x 3, x R} Range: {y | 1 y 3, y R} (c) Domain {x | x 3, x R} Range: {y | 3 y 3, y R} 12. (a) x 2 8x 16 (b) a 2 2ab b 2 (c) 4a 3 2am (d) 8y 2 2y 21 13. (a) a 5a 5 (b) 83 x3 x (c) y 1y 4 (d) 3a 1a 1 (e) 2x 4x 2 (f) 44m2 n 2 (g) 3n 42 (h)3a 22a 1 (i) 5a 2a 10 (j) 4a 1a 7 (k) Can’t factor (l) 2x 32x 3 12 1 x 7 14. (a) (b) a (c) 4 x 1 2x 5 x 6 8 15. (a) (b) (c) x 8 2x 13x 1 x 1 17. (a) Positive slopes – moving away from starting position Negative slope – returning to starting position (b) Stationary (no movement) (c) Answers will vary! 16.Graph 18.(a) (c) (e) (f) Yes (b) (d) No 1 19. (a) 17 (b) – 8 (c) – 2 (d) or 0.2 5 21 (e) 4m 2 4m 7 (f) – 3 (g) 5 (h) or – 2.625 8 (i) – 2 , – 8 20. Transformation Vertex Axis of Mapping rule symmetry (a) vt of 6, ht of –1 (–1, 6) x = – 1 x, y x 1, y 6 (b) Rx, vt of 3, ht of 5 (5, 3) x = 5 x, y x 5,y 3 (c) vs of 3, vt of –4, ht of – 7 (– 7, – 4) x = – 7 x, y x 7,3y 7 (d) 1 1 Rx, vs of , vt of – 1 x, y x, y 1 2 (0, –1) x = 0 2 (e) ht of –4 (–4, 0) x = –4 x, y x 4, y (f) 1 1 Rx, vs of , vt of 2, ht of –1 x, y x 1, y 2 4 (–1, 2) x= –1 4 (g) Rx, ht of 2 (2, 0) x = 2 x, y x 2,y (h) 1 1 Rx, vs of , ht of 2 x, y x 2, y 3 (2, 0) x = 2 3 -1 21. A. (y–2)=(x+3)² B. (y+6)=–|x–2| C. (y - 8) 2(x - 7)² D. 4(y+5)=|x+10| 2 1 22. (a) – 8, 4 (b) 2, 7 (c) –5, 2 6 (d) –3, 3 (e) (f) 0 2 13 23. Dependent variable – height of tree (over time) Independent variable – volume of water added 24. (a) (Some values rounded to nearest tenth) Tree farm Lower Lower Median Upper Upper extreme Quartile Quartile extreme Firs ‘r’ Us 1.4 1.8 2.2 2.5 2.8 Balsam City 1.4 1.6 1.9 2.4 2.9 (b) Firs ‘r’ Us 26. (a) 5.8 – 9.4 (b) 95 % (c) 2.5 % (d) 84 % of 1200 = 1008 (e) 68% of 1200 = 816 (f) < 2 or > 3 27. (a) 3.5 cm – 6.5 cm (b) 2 cm to 8 cm (c) 2.5 % of 900 (22 or 23) (d) 81.5 % 28. (a) bin frequency 0 - 10 2 10 - 20 8 20 - 30 10 30 - 40 8 40 - 50 2 29. (a) Mean = 4.0 (b) standard deviation = 0.054 (c) Shipment will likely pass inspection since the standard deviation is very low indicating very little variation from the mean. 30. (a) Lower Lower Median Upper Upper extreme Quartile Quartile extreme Blood pressure 99 115 120 124 155 (b) bin frequency 90 – 100 1 100 – 110 1 110 – 120 11 120 – 130 11 130 – 140 4 140 – 150 0 150 – 160 2 14 (c) (i) Range 56 (99 – 155) (ii) Mean 121.3 (iii) Mode 110 (iv) Median 120 (v) Standard deviation 11.7 31. (a) Model r r2 R2 Linear -.8089 .6543 na Quadratic na na .9982 Exponential -.4241 .1799 na Note: Exponential regression can only be performed with positive y-values. (30, 0) was replaced with (30, 0.001) to obtain the r and r2 values for the exponential model above! The data is best modeled by the quadratic regression. (R2 value closest to 1) 31. (b) Model r r2 R2 Linear .9985 . 9970 na Quadratic na na .9999 Exponential .9447 .8924 na The data is best modeled by the quadratic regression. (R2 value closest to 1) (c) Model r r2 R2 Linear .9059 . 8206 na Quadratic na na .9864 Exponential .9974 .9949 na The data is best modeled by the exponential regression. (R2 value closest to 1) 32. (a) Exponential. (r2 value closest to 1) (b) It is the best fit of the three but since the r2 value is not that close to 1 there is indication that there might be a lot of scatter in the data or that some other function might fit it better. (Really need to see the scatterplot!) 33. (a) Quadratic. (R2 value closest to 1, but not that much different from linear.) (b) Quadratic model would predict 39 goals. (Linear predicts 40) 34. (a) 39° (b) 36.8 cm (c) 5.3 (d) 13.3 (e) A 300 CB = 17.3 CD = 20.0 AC = 34.6 15 35. (a) (b) Yes (c) No 36. p = 14.4, M 340 N 560 37. It is a right-angled triangle 38. 3.8 m 39. 25° 40. 54.1 km (54 to nearest km) 41. 25.7 m 42. 21 m 43. If distance on line of sight, approximately 190 000 m. If horizontal (land) distance, approximately 130 000 m. (Rounded off) 44. A, B and D are similar to each other. C and E are similar to each other. 45. (a) 65 3 (b) 20 5 (c) 864 46. (a) x > –6 (b) x > 2 47. Graphs 48. (a) (–4, 0) (b) (8, 1) 49. (a) (6, 2) (b) (1, 2) (c) (3, 2) (d) (4, –2) 1 2 1 3 (e) , (f) , (g) (0, 1) (h) no solution 10 5 2 4 (i) (8, –4) (j) (–1, 2) 50. 4 jibs, 1 spinnaker, $550 51. 3 easy bounce, 3 catch a spin, $240 52. 20 standard, 20 deluxe, $140 52. 90 low grade, 105 high grade, $69.75 16