MCR3U - Practice Mastery Test #6

Transcription

1 Name: Class: Date: MCRU - Practice Mastery Test #6 Multiple Choice Identify the choice that best completes the statement or answers the question.. Factor completely: 4x 2 2x + 9 a. (2x ) 2 b. (4x )(x ) c. (4x 9)(x ) d. (2x 9)(2x ) 2. A parabola has zeros at - and 7. The equation of the axis of symmetry is a. x = b. x = 6 c. x = 4 d. x=.5. Which line is the line of best fit for the data shown below? a. line b. line 2 c. line d. line 4 4. A function is defined by y =. Its domain is x a. {x x 0, x ò} c. {x x, x ò} b. {x x > 0, x ò} d. {x x, x ò} 5. The input/output diagram illustrates a number of transformations to y=f(x). The equation for this new function is... a. y = 4f(2x + ) c. y = 4(f(2x + ) ) b. y = 4f(2(x + )) d. y = 4(f(2(x + )) ) 6. The equation of the image of y = 2x 2 after a reflection in the x-axis is a. y = 2x 2 + b. y = 2x 2 + c. y = 2x 2 d. y = 2( x) Simplify 4x 2 2x 2x a. 2x c. 4x 2 b. 4x 2 - d. 2x -

2 Name: 8. Simplify 2x 8 4 x a. x - 2 b. -2 c. 2 d. x The I/O diagram for a function is... The equation for the inverse is a. y = x + 2 b. y = x 2 c. y = x d. y = x Evaluate 5 2 a. 25 b. 25 c. 25 d Evaluate 6 2 a. 4 b. 4 c. -4 d Evaluate 27 a. 4 4 b. 4 9 c. 6 d Evaluate 27 a. 9 b. 2 9 c. 6 d Simplify x y 2. Assume x,y 0 x 6 4 y a. x 9 y 6 b. x 2 y 2 c. x y 2 d. x 9 y 2 5. Simplify Ê x y ˆ Ê x y 4 ˆ. Assume x,y 0 a. y b. x 6 y c. x 9 y 2 d. x 6 y 7 2

3 Name: 6. The equation of the function shown below could be... a. y = 5 x 2 + c. y = 5 x b. y = 5 x + 2 d. y = 5 x After the graph of y = 2 x has been translated horizontally units to the left, and translated vertically units up, the resulting equation of the function would be... a. y = 2 x + + b. y = 2 x + c. y = 2 x + d. y = 2 x 8. The function defined by y = x + 2 has range... a. Ì y y >,y ò Ó c. Ì y y > 2,y ò Ó b. Ì y y <,y ò Ó d. Ì y y < 2,y ò Ó 9. The function defined by y = 5 x 2 has range... a. Ì y y > 2,y ò Ó c. Ì y y >,y ò Ó b. Ì y y < 2,y ò Ó d. Ì y y <,y ò Ó 20. In the diagram, A is closest to a. 67 b. 70 c. 5 d. 6

4 Name: 2. In the diagram, C is closest to a. 67 b. c. 27 d If θ is an angle in a triangle, and cos θ = , then θ is approximately a b c d Which trigonometric ratio of angle S is easiest to determine x in the following diagram? a. sin S b. cos S c. tan S 24. Which equation would be used to determine x in the following diagram? a. a 2 = b 2 + c 2 2bc cosa c. c 2 = a 2 + b 2 2ab cos C b. b 2 = a 2 + c 2 2ac cosb d. a sina = b sinb = c sinc 25. If cos θ = where -80 θ 80, then 2 a. θ = -50 or θ = 50 c. θ = 20 or θ = -20 b. θ = -20 or θ = -60 d. θ = -0 or θ = If cos θ = 2 where 0 θ 60, then a. θ = -45 or θ = 5 c. θ = 225 or θ = 5 b. θ = 5 or θ = 5 d. θ = 5 or θ = 225 4

5 Name: 27. Complete the identity sin 2 θ = a. cos 2 θ b. cos 2 θ c. cosθ d. cos θ 28. Complete the identity 2cos 2 θ + sin 2 θ = a. 2 b. cos 2 θ c. sin 2 θ d. cos 2 θ Complete the identity tanx = a. sin x b. c. cos 2 x d. cos 2 x 0. Complete the identity tanx = a. b. c. sin 2 x d. cos 2 x 5

6 MCRU - Practice Mastery Test #6 Answer Section MULTIPLE CHOICE. ANS: A Arrange the tiles into a rectangle: So 4x 2 2x + 9 = (2x ) 2 PTS: 2. ANS: A The axis will be mid-way between - and = PTS:. ANS: D If you imagine each of the points being little magnets and the line of best fit being a needle, hopefully you can imagine it moving into the spot that line 4 occupies. It is 'closer' to more points than any other line and shows the general trend of the data. PTS: 4. ANS: B In the real numbers, x is defined only if the expression under the square root symbol is greater or equal to 0, so x 0. However, if x was equal to 0, we would get y = 0 = 0 which would be undefined, so x must be just greater than 0 PTS:

7 5. ANS: B If we apply the sequences of operations listed in the input/output diagram to x, we get... PTS: 6. ANS: A The original equation defines a parabola with vertex at (0,-) that opens up. When it is reflected in the x-axis, it will have vertex at (0,), and will open down, so its equation will be y = 2x 2 +. We could also do this problem with an input/output diagram, but this is a parabola, so it should be easy for you to picture. PTS: 7. ANS: D 4x 2 2x 2x Factor the numerator 2x(2x ) = 2x Divide out identical factors = 2x - if x?0 PTS: 8. ANS: B Factor the numerator. Note that factoring out -, makes the other factor the same as the denominator. 2x 8 4 x 2( 4 + x) = Divide out identical factors x 4 = -2 if x?4 PTS: 2

8 9. ANS: A The I/O diagram for the inverse will be... Once the diagram is drawn, just follow the expression through the diagram to get the equation y = x + 2 Alternatively, we could use the calculator to graph each one and compare to the graph of the original. PTS: 0. ANS: D 5 2 Note that order of operations requires you to apply the exponent to the 5 BEFORE the negative is = 25 applied. PTS:. ANS: C 6 = 4 2 PTS: 2. ANS: D 27 Ê 4 = 27 = ( ) 4 = 8 ˆ An exponent of 2 4 means that you take the square root of the base. We are using the fact that Ê x a b ˆ = x ab to split up the power. An exponent of means that you take the rd root of the base. I.e., = 27 so 27 =. PTS:

9 . ANS: A 2 27 Ê = 27 = ( ) 2 ˆ 2 We are using the fact that Ê x a b ˆ = x ab to split up the power. An exponent of means that you take the rd root of the base. I.e., = 27 so 27 =. = 2 = 9 PTS: 4. ANS: C x y 2 x 6 y 4 = x ( 6) 2 ( 4) y To divide powers with the same base, subtract the exponents (beware... you may be subtracting a negative) = x + 6 y = x y 2 Or... x y 2 x 6 y 4 = x y 2 x 6 y 4 x 6 y 4 x 6 y 4 Multiply the numerator and denominator by something that will get rid of the denominator (make the exponents in the denominator 0 because x 0 = ) = x y 2 = x y 2 PTS: 5. ANS: Ê A x y ˆ Ê x y 4 ˆ = x + ( ) y + 4 = x 0 y = y To multiply powers with the same base, add the exponents Note that x 0 is equal to (as long as x is not 0) PTS: 4

10 6. ANS: A First note (from the choices available that the base function should be y = 5 x. This graph looks like this... Because of the direction of the given graph, we can see that it is NOT reflected in the x-axis. This immediately rules out y = 5 x + 2 and y = 5 x Also, it is not reflected in the y-axis This means that y = 5 x is eliminated as a choice. Note also... With a horizontal asymptote at y =, the base function of y = 5 x has to be translated up (down if negative) units, so the last part of the equation must be +. 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. PTS: 7. ANS: A The I/O diagram will become... Once the diagram is drawn, just follow the expression through the diagram to get the equation y = 2 x + + Alternatively, we could use the calculator to graph each one and compare to the graph of y = 2 x PTS: 5

11 8. ANS: A The I/O diagram will become... The base function, y = x looks like.... It has range Ì y y > 0,y ò Ó Once the diagram is drawn, just follow the expression through the diagram to get the equation y = x + 2 Alternatively, we could use the calculator to graph each one and compare to the graph of y = x PTS: 9. ANS: A The I/O diagram will become... The base function, y = 5 x looks like.... It has range Ì y y > 0,y ò Ó Once the diagram is drawn, just follow the expression through the diagram to get the equation y = 5 x 2 Alternatively, we could use the calculator to graph each one and compare to the graph of y = 5 x PTS: 6

12 20. ANS: A Sin A = opp hyp (any trig ratio could be used) = 2 Ê sin 2ˆ 67û Since A is acute, it is approximately 67 PTS: 2. ANS: D Sin C = opp hyp (any trig ratio could be used) = 5 Ê sin 5 ˆ 2û Since C is acute, it is approximately 2 PTS: 22. ANS: D Make sure the calculator is set to degrees (see MODE), and get cos - (-0.46) which is approximately 0.2 PTS: 2. ANS: B x is the adjacent side and 7 is the hypotenuse, so we have to use cos S = adj hyp PTS: 24. ANS: B We know all sides and we want to find angle B, so use the cosine law as follows: 2 = (8)(0) cos(x û ) PTS: to find x 7

13 25. ANS: A The unit circle shows us the coordinates of all points with principal angles 0, 45 and 60. It shows that cos θ = 2 at points H and J. These points are 0 away from the x axis, so the angles are -50 and 50 PTS: 8

14 26. ANS: D The unit circle shows us the coordinates of all points with principal angles 0, 45 and 60. It shows that cos θ = 2 at points G and K. These points are 45 away from the x axis, so the angles are 5 and 225. PTS: 27. ANS: B We know that cos 2 θ + sin 2 θ = for all θ. If we rearrange it (by subtracting sin 2 θ from both sides) we get cos 2 θ = sin 2 θ. Another method is to substitute = cos 2 θ + sin 2 θ into the given expression to get: sin 2 θ Ê = cos 2 θ + sin 2 ˆ θ sin 2 θ = cos 2 θ + sin 2 θ sin 2 θ = cos 2 θ PTS: 9

15 28. ANS: D We know that cos 2 θ + sin 2 θ = for all θ. If we separate the 2cos 2 θ, we get 2cos 2 θ + sin 2 θ = cos 2 θ + cos 2 θ + sin 2 θ = cos 2 θ + Another method is to substitute sin 2 θ = cos 2 θ into the given expression to get: 2cos 2 θ + sin 2 θ = 2cos 2 Ê θ + cos 2 ˆ θ = + cos 2 θ PTS: 29. ANS: C We know that tanx = tanx = = = for all x. Substituting this in, we get = cos 2 x PTS: 0

16 0. ANS: D We know that tanx = tanx = = = = cos 2 x for all x. Substituting this in, we get Another method is to rearrange tanx = (by multiplying both sides by ) tanx = (tanx) = PTS: tanx cosx = cos 2 x Ê ˆ