MHF4U: Practice Mastery Test #3

Transcription

1 MHF4U: Practice Mastery Test #3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Simplify a. b. 4x c. d. 2. Expand and simplify (2x + 1)(x - 3) a. b. c. d. 3. Factor completely: a. b. c. d. 4. If then a. x = -3 or x = 2 b. x = 3 or x = -2 c. x = 6 or x = -1 d. x = -6 or x = 1 5. Complete the square: is equal to a. b. c. d. 6. Simplify a. b. c. d. 7. A relation is defined by. Which of the following points belong to this relation? A(1,-3) B(-1,-3) C(-2,-1) a. A b. B c. C d. B & C 8. A relation is defined by. Which of the following points belong to this relation? A(0,-1) B(-1,2) C(-2,1) a. A b. B c. C d. B & C 9. If - 5x + 8 then... a. (x - 2) (x - 3) + 2 c. (x - 2) (x + 3) - 2 b. (x + 2) (x - 3) - 2 d. (x + 2) (x + 3) A function is defined by. Its range is a. {y y 0, y } c. {y y 3, y } b. {y y -3, y } d. {y y -3, y } 11. A function g is defined in terms of another function f as shown in the diagram below. x add 1 f divide by 2 g(x) The point is on the graph of f. Which of the following is the corresponding point on the graph of g? a. c. b. d.

2 12. A function g is defined in terms of another function f as shown in the diagram below. x multiply add 1 f by 2 g(x) What transformations must be applied to the graph of f to get the graph of g? a. A horizontal translation of 1 unit(s) to the right, and a vertical stretch of factor 2 c. A horizontal translation of 1 unit(s) to the left, and a vertical stretch of factor 2 b. A horizontal translation of 1 unit(s) to the d. A horizontal translation of 1 unit(s) to the right, and a vertical stretch of factor 1 2 left, and a vertical stretch of factor is a point on the unit circle and is on the terminal arm of angle θ. Which of the following is true? a. cos θ = and tan θ = c. cos θ = and tan θ = b. sin θ = and tan θ = d. sin θ = and tan θ = 14. If and then θ could be approximately a. b. c. d. 15. The input/output diagram illustrates a number of transformations to y=sin(x). divide by 3 subtract 6 sin multiply by 4 subtract 5 The correct sequence of horizontal transformations applied to y=sin(x) is... a. a horizontal translation of 6 units right, followed by a hor. stretch of factor 3 b. a horizontal stretch of factor 3, followed by a hor. translation of 6 units right c. a horizontal translation of 6 units left, followed by a hor. stretch of factor 1 3 d. a horizontal stretch of factor 1, followed by a hor. translation of 6 units left Which trigonometric ratio of angle S is easiest to determine x in the following diagram? S x R 3 Q a. tan S b. sin S c. cos S

3 17. Which equation would be used to determine x in the following diagram? A 8 11 B x? 10 C a. c. b. d. 18. Complete the identity a. b. c. d. 19. Complete the identity a. cos x b. c. d. 20. What is the next number in the sequence? a b. 0 c d Simplify. Assume a. b. c. d. 22. Simplify. Assume a. b. c. d. 23. Simplify. Assume a. b. c. d. 24. Simplify. Assume a. b. c. d.

4 25. The equation of the function shown below could be... y x a. c. b. d. 26. The equation of the function shown below could be... y x a. c. b. d.

5 27. The tables below shows the concentration of CO 2 in the air in a room over time. The best estimate of the average rate of change of the CO 2 level over the first 3 seconds is... a b c d. 0 e Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. The average rate of change of y with respect to x between x=0 and x=6 is... a. 2 b. 0.5 c. 3 d. 1.5

6 29. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. Which of the following is most likely to be true? a. The instantaneous rate of change of f at x=-1 does not exist b. The instantaneous rate of change of f at x=1 is c. The instantaneous rate of change of f at x=4 is 0 d. The instantaneous rate of change of f at x=6 is 3

7 30. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. Which of the following is most likely to be true? a. The instantaneous rate of change of f at x=-1 is 0 b. The instantaneous rate of change of f at x=1 is c. The instantaneous rate of change of f at x=6 is 3 d. The instantaneous rate of change of f at x=4 is 1

8 31. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. Which of the following is most likely to be true? a. The instantaneous rate of change at x=-1 does not exist b. The average rate of change between x= -1 and x= 2 is -1 c. The instantaneous rate of change at x=2 does not exist d. The average rate of change between x= 2 and x=4 is A man kept track of his weight and daily food consumption every day for several months. He drew a graph with weight on the vertical axis and time on the horizontal axis. What units could be used for instantaneous rate of change on this graph? a. kg/day c. cal/day b. years/kg d. kg/cal 33. The graph illustrates the motion of a person walking toward or s(t) away from a CBR. Distance from the CBR is recorded on the vertical axis in metres. Elapsed time since the CBR was started up is recorded on the horizontal axis in seconds. Which of the following best describes the motion? The person is... t a. walking away from the CBR and speeding c. walking away from the CBR and slowing up. down b. walking toward the CBR and speeding up. d. walking toward the CBR and slowing down.

9 34. The tile chart shown at right indicates that... i) ii) is a factor of 35. iii) If, then a. i only c. iii only e. all three b. ii only d. i and ii only The tile chart shown at right indicates that... i) ii) is a factor of 36. iii) If, then a. i only c. iii only e. all three b. ii only d. i and ii only x + 2 is a factor of i) ii) a. i only b. ii only c. i and ii d. neither 37. The remainder when is divided by x - 1 is 38. a. 3 b. 3 c. 7 d. 7 Factor fully a. c. b. d. 39. The number of distinct real roots for the equatiion is a. 2 b. 3 c. 4 d The number of distinct real roots for the equatiion is a. 2 b. 3 c. 4 d. 5

10 41. If then a. c. b. d. 42. Which of the graphs shown at right is most likely to be the graph of? a. A b. B c. C d. D

11 MHF4U: Practice Mastery Test #3 Answer Section MULTIPLE CHOICE 1. ANS: A We can think of as. These are 'like terms', so they can be 'combined' to get. You may have not looked at diagrams for yet - but it is possible to illustrate this question as follows: Two of the positive solids eliminate the two negative solids. The answer is. 2. ANS: A We can think of this multiplication using tiles... but it is not necessary to draw the tiles every time. The following diagram illustrates the multiplication (as long as you understand that 2x multiplied by x is 2x.) or ANS: A Arrange the tiles into a rectangle: So = (x+2)(2x+1). 4. ANS: C If then. x - 6 = 0 or x + 1 = 0

12 x = 6 or x = ANS: B. Arrange the tiles into a square... we need to add in 16 1-tiles to complete the square, and another tiles to keep the expression equivalent to the original. We get 6. ANS: C Multiply each term in the 1st bracket by every term in the 2nd. = = OR use a diagram... and then collect like terms 7. ANS: D If we let x = 1 in the equation, we get y = 5. If x=-1, y=-3, so B is a winner, and if x=-2, y=-1, so B and C are on the parabola. 8. ANS: C If we let x = 0 in the equation, we get y = 1. If x=-1, y=-2,, and if x=-2, y=1, so C is on the parabola. 9. ANS: A One way to do this problem is probably to just sub x=2,or x= 2 and then x=3 or x= 3 into the original equation (and each of the options) to find out which ones work.

13 or partially factor the expression... so... (x - 2) (x - 3) ANS: B In the real numbers, will always give an answer which is greater or equal to 0 (the square root symbol actually means the principal or positive square root). I.e. 0 Therefore (subtracting 3 from both sides) so y ANS: A Probably the easiest way to do this question is to just try 9 and 7 for x and see what happens as they go through the I/O diagram. If we plug in 9, we get... 9 add 1 10 f divide by 2 g(x) because the flow is left to right so 1 is added to 9 to get 10. We know this is wrong because the value of x that must be going into f is 8. If we plug in 7, we get... 7 add f divide by 2 1 because the flow is left to right so 1 is added to 7 to get 8. When 8 is fed into f, out comes 2. Then, 2 is divided by 2 to get 1. This is correct, so is the correct point on g. 12. ANS: A Probably the easiest way to do this question is to pick an ordered pair that might belong to f, say (2,1), and then... see what happens when you work away from f in the I/O diagram. So plug in 2 to the left of f, and 1 to its right as shown below. 2 1 multiply?? add 1 f??... 3 add 1 by f multiply by 2 2

14 The flow is left to right so we note that if we add 1 to the new x coordinate, we get 2. This means that the new x coordinate is 1 more than the old one... so we must be moving all the points right 1 unit(s). When 2 is fed into f, out comes 1. Then, 1 is multiplied by 2 to get 2, so the new y-coordinates are 2 times bigger than the old y-coordinates... so it is a vertical stretch of factor ANS: A The x coordinate of the point is cos θ, so cos θ =. The y coordinate of the point is sin θ, so sin θ = and tan θ 14. ANS: A Since and, then θ is in the 4th quadrant (the coordinates of the point are ) 2nd quadrant y 1st quadrant 0.45 (0.89, 0.45) (0.89, 0.45) x 3rd quadrant 4th quadrant To find the related acute angle, use and and are both approximately (make sure your calculator is set to degrees) However, the angle is in the 4th quadrant. So... if we fit the right triangle shown into the right spot, we can see that the actual angle could be or 15. ANS: A

15 The input/output diagram for the function is shown below. divide by 3 subtract 6 sin multiply by 4 subtract 5 Horizontally, we have to work backwards from the base function, so the first operation that we have to undo is subtract 6, so the first horizontal transformation is a translation of 6 units right. The second operation we have to undo, working away from the base function, is divide by 3, so the second horizontal transformation is a horizontal stretch of factor ANS: A 3 is the opposite side & x is the adjacent side, so we have to use tan S = to find x 17. ANS: B We know all 3 sides and we want to find angle B, so use the cosine law as follows: 18. ANS: B We know that for all θ. If we rearrange it (by subtracting from both sides) we get. Another method is to substitute into the given expression to get: 19. ANS: A We know that for all x. Substituting this in, we get

16 Another method is to rearrange to get: 20. ANS: A Each term is 0.5 times the previous term (check by dividing consecutive terms), so if we take 0.25 and multiply it by 0.5, we get ANS: C To divide powers with the same base, subtract the exponents (beware... you may be subtracting a negative) Or... Multiply the numerator and denominator by something that will get rid of the denominator (make the exponents in the

17 denominator 0 because ) 22. ANS: A To multiply powers with the same base, add the exponents. The common denominator is 6.. Reduce. 23. ANS: B To multiply powers with the same base, add the exponents. The common denominator is 12. Reduce. 24. ANS: D

18 To divide powers with the same base, subtract the exponents. The common denominator is ANS: A First note (from the choices available that the base function should be. This graph looks like this y x Because of the direction of the given graph, we can see that a reflection in the x-axis is required. This immediately rules out and. Also, a reflection in the y-axis is needed. This means that is eliminated as a choice. Note also... With a horizontal asymptote at, the base function of has to be translated up (down if negative) 2 units, so the last part of the equation must be - 2. It also appears that a horizontal translation is required, but we don t need to know by how much to answer the question. Alternatively, we could use the calculator to graph each one and compare, or draw the I/O diagram corresponding to the graph and the required translations and then get the equation. 26. ANS: A First note (from the choices available that the base function should be. This graph looks like this...

19 y x Because of the direction of the given graph, we can see that it is NOT reflected in the x-axis. This immediately rules out and. Also, it is not reflected in the y-axis This means that is eliminated as a choice. Note also... With a horizontal asymptote at, the base function of has to be translated up (down if negative) 2 units, so the last part of the equation must be + 2. It also appears that a horizontal translation is required, but we don t need to know by how much to answer the question. Alternatively, we could use the calculator to graph each one and compare, or draw the I/O diagram corresponding to the graph and the required translations and then get the equation. 27. ANS: B For average rate of change, we want the slope of the line segment from 0 to 3. slope 28. ANS: B The instantaneous rate of change is the slope of the line segment from (0,0) to (6,3) which is ANS: A From the graph, the instantaneous rate of change of f at x=-1 does not exist because there is a corner. On one side, the slope is 0, but on the other side it is Also the instantaneous rate of change of f at x=2 does not exist because there is a corner. On the left side, the slope is but on the other side it is approaching 0.

20 At x=-2, the the instantaneous rate of change of f is 0 because the slope is 0. At any point between (-1,1) and (2,-1), (but not including those points), the slope is, so the instantaneous rate of change is. At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4) At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9) 30. ANS: D From the graph, the instantaneous rate of change of f at x=-1 does not exist because there is a corner. On one side, the slope is 0, but on the other side it is Also the instantaneous rate of change of f at x=2 does not exist because there is a corner. On the left side, the slope is but on the other side it is approaching 0. At x=-2, the the instantaneous rate of change of f is 0 because the slope is 0. At any point between (-1,1) and (2,-1), (but not including those points), the slope is rate of change is., so the instantaneous At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4) At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9) 31. ANS: A To the left of x=-1, the instantaneous rate of change is the same as the average rate of change anywhere along it because it is a straight line. This rate is 2 (from (-2,0) to (-1,2). The instantaneous rate of change of f at x=-1 does not exist because the graph is discontinuous there. The actual function value is -1 when x is -1. Anywhere between x=-1 and x=2, the slope is 0, so both instantaneous and average rate of change is 0. If we approach x=2 along the curve from the right, the slope approaches 0, so there is no corner there. The instantaneous rate of change at x=2 is 0. At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4) At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9) The average rate of change between... (-2,0) and (-1,-1) is -1 (rise of -1, run of 1) between (-1,1) and (2,-1) is 0; between (2,-1) and (4.0) is 0.5 (rise of 1, run of 2); between (4,0) and (6,3) it is ANS: A

21 slope would be, so the units could be kg/day 33. ANS: B With a negative slope, we know the distance from the CBR is decreasing, so the person is walking towards the CBR. The slope is becoming more and more negative, so the speed is also increasing. 34. ANS: A i) By adding up the interior of the rectangle with the rejects, we get the equation One way to get something that looks like the equation in i) is to divide both sides by x + 1, to get which simplifies to Alternatively (probably a better way in the long run), we could push the rejects (4) into an extra column in the chart... x?? is the expression that multiplies by x + 1 to get 4. This would have to be. I.e., so... x The area (the stuff inside) divided by one dimension is the other, Therefore,... ii) The remainder is the stuff that doesn t fit nicely into the chart without fractions. i.e., it is the reject pile. In this case this is 4.

22 The remainder when is divided by is 4, and iii) if, then (from the chart) and when x= 1, so, 35. ANS: A i) The area (the stuff inside) divided by one dimension is the other, Therefore,... ii) The chart shows us that is a factor of because there are no rejects. iii) if, then (from the chart) and when x= 1, so, 36. ANS: B if, then So...x + 2 is not a factor of If, then So...x + 2 is a factor of

23 37. ANS: A if, then OR we could do the division and find the remainder 38. ANS: A First, we can see that x + 2 is a factor of, because if f(x) =,. Go to the chart: Therefore = 39. ANS: A The real roots are -5 and -3. has no real roots. 40. ANS: D The real roots are -5,, 2, and ANS: B So... if 42. ANS: C There are many ways to decide which one is most appropriate - note that all four graphs have the same zeros - at 0.5, 2, and 3.

24 Graph A bounces off the x-axis at 2, so its equation probably includes the term or so that to the left and right of 2, the sign of the expression is the same. Graph B bounces off the x-axis at 3, so its equation probably includes the term or so that to the left and right of 3 the sign of the expression is the same. Graph C passes through the x-axis at all of its zeroes, so its equation cannot include an even power of a factor. Graph D bounces off the x-axis at 0.5, so its equation probably includes the term or so that to the left and right of 0.5 the sign of the expression is the same. In this case, the best choice is graph C.