midyear exam review - wordpress.com · midyear exam review 1. if (x 2) is a factor of 3 7 2 4x (3...
Post on 26-Jul-2018
217 Views
Preview:
TRANSCRIPT
Mathematics 3200
Midyear Exam Review
1. If )2( x is a factor of )3(47 23 kxxx , what is the value of k?
(A) -31 (B) -28 (C) 28 (D) 31
2. Which graph below represents the graph of an even degree function?
(A) (B)
(C) (D)
3. Which of the equations below is best represented by the given graph?
(A) y = –x4 + 4x3
(B) y = x4 – 4x3
(C) y = x3 – 4x2
(D) y = –x3 + 4x2
4. Which represents the value of k if the remainder is 5 for
(2x3 + 4x2 + kx – 3) ÷ (x + 1) ?
(A) –6 (B) –2 (C) 2 (D) 6
x-4 -2 2 4
y
-6
-4
-2
2
4
6
x-4 -2 2 4
y
-10
-8
-6
-4
-2
2
4
6
x-4 -2 2 4
y
-10
-8
-6
-4
-2
2
4
6
x-4 -2 2 4
y
-6
-4
-2
2
4
6
x- 2 2 4 6
y
- 8- 6- 4- 2
2468
10121416182022242628
5. Which function has each of the characteristics:
an even function
end behavior in the third and fourth quadrants
y – intercept is –6
(A) P(x) = x4 – 5x2 – 6 (B) P(x) = –x4 + 3x3 + 6
(C) P(x) = –(x + 2)(x + 3) (D) P(x) = – x3+ x – 6
6. Which statement is true for a polynomial function?
(A) All even degree polynomial functions have at least one x-intercept.
(B) Some odd degree polynomial functions have no x-intercepts.
(C) Even degree polynomial functions always have an even number
of x-intercepts.
(D) All odd degree polynomials have at least one x-intercept.
7. What are the x-intercepts of P(x) = 4x3 – 12x2 + 8x ?
(A) x = –4, –2, –1 (B) x = –2, –1, 0
(C) x = 0, 1, 2 (D) x = 1, 2, 4
8. List all possible integral zeros for P(x) = x4 + 3x3 – 2x2 – 12x – 8.
(A) ±1, ±8 (B) ±1, ±2, ±3, ±4, ±6, ±12
(C) ±1, ±2, ±4, ±8 (D) ±2, ±4
9. The volume of a rectangular prism is V= 2x3 – 5x2 – x + 6. If two of
the dimensions are x – 2 and x + 1, what is an expression for the other
dimension?
(A) 𝑥 − 6 (B) 𝑥 − 6 (C) 2𝑥 − 3 (D) 2𝑥 + 3
10. What are the x -intercepts of f (x) = x 2 (x + 3)(x − 2)?
(A) −3 and 2 (B) 3 and −2
(C) 0, −3, and 2 (D) 0, 3, and −2
11. What is the quotient and remainder for (2 x 3 − x 2 + 2 x + 4) ÷ ( x − 3)?
(A) The quotient is 2x 2 + x + 5, and the remainder is 19.
(B) The quotient is 2x 2 + 5x + 7, and the remainder is 29.
(C) The quotient is 2x 2 + 5x + 17, and the remainder is 55.
(D) The quotient is 2x 2 + x + 3, and the remainder is 7.
12. Which sketch best represents the graph of edxcxbxaxy 234
if 0a and 0e ?
(A) (B)
(C) (D)
13. How many x-intercepts are possible for the polynomial
function P(x) = ax5 + bx4 + cx3 ?
(A) 1 (B) 3 (C) 4 (D) 5
14. The function 𝑦 = 𝑓(𝑥) is stretched horizontally by a factor of 4 and
is reflected in the x-axis. What is the equation of the transformed function?
(A) −𝑦 = 𝑓 (1
4𝑥) (B) 𝑦 = 𝑓 (−
1
4𝑥)
(C) –y = f(4x) (D) y = f(–4x)
15. The graph of 𝑦 = 𝑓(𝑥) contains the point A(–6, 4). What are
The coordinates of the image point, 𝐴′, on the function 1
2(𝑦 − 2) = 𝑓(−3(𝑥 + 1))?
(A) (1, 10) (B) (3, 10) (C) (1, 0) (D) (3, 0)
16. The graph of 𝑦 = 𝑓(𝑥) is transformed according to the mapping rule
(𝑥, 𝑦) → (2𝑥 − 3, −1
4𝑦). What is the equation of the resulting function?
(A) 𝑦 = −1
4𝑓(2𝑥 − 3) (B) 𝑦 = −
1
4𝑓 (
1
2𝑥 + 3)
(C) 𝑦 = −1
4𝑓(2(𝑥 − 3)) (D) 𝑦 = −
1
4𝑓 (
1
2(𝑥 + 3))
17. Given that 𝑦 = 𝑓(𝑥) contains the point (𝑚, 𝑛), which of the following
points must lie on the graph of 1
3𝑦 = 𝑓(𝑥 + 𝑚) ?
(A) (2m, 3n) (B) (2𝑚,1
3𝑛)
(C) (0, 3n) (D) (0,1
3𝑛)
18. The function 𝑦 = 𝑓(𝑥) is transformed to 𝑦 = 3𝑓(4𝑥). Which describes how
𝑦 = 𝑓(𝑥) is transformed?
Horizontal Stretch Vertical Stretch
A 1
4
1
3
B 1
4 3
C 4
1
3
D 4 3
19. The graph of 𝑦 = 𝑓(𝑥) is transformed to produce the graph of
𝑦 = −5𝑓(2𝑥 − 6) − 1 . Which describes the horizontal translation?
(A) 6 units left (B) 6 units right
(C) 3 units left (D) 3 units right
20. Which function would produce a graph with the
same y-intercept as the graph of 𝑦 = 𝑓(𝑥)?
(A) y = f(x – 4) (B) y = f(x) + 4
(C) y = 3f(x) (D) y = f(3x)
21. The graph of 𝑦 = 𝑓(𝑥), shown below, is transformed to produce the
graph of −2𝑦 = 𝑓(𝑥 + 3). Which of the labelled points (A, B, and C)
would be invariant?
(A) A and B only
(B) C only
(C) A, B, and C
(D) None of them would be invariant
x
y
A B
C
22. When compared to 𝑦 = 𝑓(𝑥), what is the vertical stretch factor
of −2(𝑦 + 1) = 𝑓(1
3𝑥)?
(A) –2 (B) 2 (C) −1
2 (D)
1
2
23. What transformations of the graph of 𝑦 = 2𝑓(𝑥 + 3) are required
to produce the graph of 𝑦 = 6𝑓(𝑥 + 1)?
(A) Vertical Stretch of 4 and a horizontal translation of 2 units left
(B) Vertical Stretch of 3 and a horizontal translation of 2 units left
(C) Vertical Stretch of 4 and a horizontal translation of 2 units right
(D) Vertical Stretch of 3 and a horizontal translation of 2 units right
24. The function 𝑦 = 𝑓(𝑥) is stretched vertically by a factor of 3, and
translated 5 units right and 1 unit down. What is the equation of the
resulting function?
(A) 𝑦 =1
3𝑓(𝑥 + 5) − 1 (B) 𝑦 =
1
3𝑓(𝑥 − 5) + 1
(C) 𝑦 = 3𝑓(𝑥 + 5) − 1 (D) 𝑦 = 3𝑓(𝑥 − 5) − 1
25. The mapping rule (𝑥, 𝑦) → (5𝑥 + 2, 𝑦 − 3) is applied to 𝑦 = 𝑓(𝑥) to
produce a function of the form 𝑦 = 𝑎𝑓(𝑏(𝑥 − ℎ)) + 𝑘. Which values
are correct for 𝑎, 𝑏, ℎ, and 𝑘?
(A) 𝑎 = 1, 𝑏 = 1
5, ℎ = 2, 𝑘 = −3 (B) 𝑎 = 1, 𝑏 = 5, ℎ = −2, 𝑘 = 3
(C) 𝑎 = 1, 𝑏 = 1
5, ℎ = −2, 𝑘 = 3 (D) 𝑎 = 1, 𝑏 = 5, ℎ = 2, 𝑘 = −3
26. Which combination of transformations is required to map 𝑦 = 𝑓(𝑥)
onto 𝑦 =1
2𝑓(−𝑥)?
(A) Reflection in the x-axis, Stretched vertically by a factor of 2
(B) Reflection in the x-axis, Stretched vertically by a factor of 1
2
(C) Reflection in the y-axis, Stretched vertically by a factor of 2
(D) Reflection in the y-axis, Stretched vertically by a factor of 1
2
27. The domain of 𝑦 = 𝑓(𝑥) is {𝑥/−12 ≤ 𝑥 ≤ 6, 𝑥 ∈ ℝ}. What is the
domain of ℎ(𝑥) = 3𝑓(2𝑥) ?
(A) {𝑥|−24 ≤ 𝑥 ≤ 12, 𝑥 ∈ 𝑅} (B) {𝑥|−6 ≤ 𝑥 ≤ 3, 𝑥 ∈ 𝑅}
(C) {𝑥|−4 ≤ 𝑥 ≤ 2, 𝑥 ∈ 𝑅} (D) {𝑥|−36 ≤ 𝑥 ≤ 18, 𝑥 ∈ 𝑅}
28. The function 𝑦 = 𝑓(𝑥) has zeroes 𝑥 = −4 and 𝑥 = 2. What are the
zeroes of the function 𝑔(𝑥) = 3𝑓 (−1
2𝑥)?
(A) x = 2 and x = –1 (B) x = 8 and x = –4
(C) x = 6 and x = –3 (D) x = 6 and x = –12
29. The graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) are shown below. Which
mapping rule would map 𝑦 = 𝑓(𝑥) onto 𝑦 = 𝑔(𝑥)?
(A) (x, y) → (x + 2, –2y – 1)
(B) (x, y) → (x – 2, –2y – 1)
(C) (x, y) → (x + 2, –2y + 2)
(D) (x, y) → (x – 2, –2y + 2)
30. A function is defined as f(x) = x2 – 5. What is the minimum value of the
function y = √𝑓(𝑥) ?
(A) –5 (B) 0 (C) 1 (D) 5
31. The equation of a function is given as 𝑓(𝑥) = 𝑥2 + 1. What is the domain
of 𝑦 = √𝑓(𝑥)?
(A) 𝑥 ∈ (−∞, ∞) (B) 𝑥 ∈ [1, ∞)
(C) 𝑥 ∈ (1, ∞) (D) 𝑥 ∈ (−∞, −1]⋃[1, ∞)
32. Which set of transformations would map 𝑦 = √𝑥
onto 𝑦 − 3 = −√4(𝑥 + 2)?
(A) Reflection in the x-axis, horizontal stretch by a factor of 4,
translation of 2 units right and 3 units down.
(B) Reflection in the x-axis, horizontal stretch by a factor of 1
4,
translation of 2 units left and 3 units up.
(C) Reflection in the y-axis, horizontal stretch by a factor of 4,
translation of 2 units right and 3 units down.
(D) Reflection in the y-axis, horizontal stretch by a factor of 1
4,
translation of 2 units left and 3 units up.
x- 6 - 4 - 2 2 4 6
y
- 8
- 6
- 4
- 2
2
4
6
8
y = f(x)
y = g(x)
33. Which function best represents the graph shown below?
(A) 𝑦 = √−(𝑥 − 2)
(B) 𝑦 = √−𝑥 − 2
(C) 𝑦 = −√𝑥 − 2
(D) 𝑦 = −√𝑥 − 2
34. The graph of a function 𝑦 = 𝑓(𝑥) is shown. Which statement about the
function 𝑦 = √𝑓(𝑥) is true?
(A) It has two invariant points and has range 𝑦 ≤ 1
(B) It has three invariant points and has range 𝑦 ≤ 1
(C) It has two invariant points and has range 𝑦 𝜖 [0, 1]
(D) It has three invariant points and has range 𝑦 𝜖 [0, 1]
35. Which function has domain {𝑥/𝑥 ≥ −2, 𝑥 ∈ 𝑅} and {𝑦/𝑦 ≤ 3, 𝑦 ∈ 𝑅}?
(A) 𝑦 − 3 = −√𝑥 + 2 (B) 𝑦 + 3 = −√𝑥 − 2
(C) 𝑦 − 3 = √𝑥 − 2 (D) 𝑦 + 3 = √𝑥 + 2
36. The function 𝑦 = 𝑓(𝑥) contains the point (9, 4). Which point must lie on
the graph of 𝑦 = √𝑓(𝑥)?
(A) (3, 2) (B) (9, 2) (C) (3, 4) (D) (2, 3)
37. The point (𝑛, −2) lies on the graph of the function 𝑦 = 2√𝑥 − 1 − 6 ?
Which value is correct for 𝑛?
(A) 2 (B) 3 (C) 5 (D) 50
38. Which mapping rule would map 𝑦 = √𝑥 onto −1
3(𝑦 + 2) = √2𝑥 + 6 ?
(A) (𝑥, 𝑦) → (1
2𝑥 − 6, −3𝑦 − 2) (B) (𝑥, 𝑦) → (
1
2𝑥 − 3, −3𝑦 − 2)
(C) (𝑥, 𝑦) → (2𝑥 − 6, −1
3𝑦 + 2) (D) (𝑥, 𝑦) → (2𝑥 − 3, −
1
3𝑦 + 2)
x
y
x- 4 - 3 - 2 - 1 1 2 3 4 5 6
y
- 1
1
2
3
39. The graph of the function 𝑦 = √𝑥 is reflected in the x-axis and is translated
4 units left and 2 units up. Which describes the domain and range of the
resulting function?
(A) Domain: 𝑥 ≤ −4 Range: 𝑦 ≤ 2
(B) Domain: 𝑥 ≤ −4 Range: 𝑦 ≥ 2
(C) Domain: 𝑥 ≥ −4 Range: 𝑦 ≤ 2
(D) Domain: 𝑥 ≥ −4 Range: 𝑦 ≥ 2
40. Convert 160° to radians.
(A) 9π
16 (B)
9π
8 (C)
8π
9 (D)
5π
6
41. Which best approximates the value of cot(200°) + csc(3)?
(A) 0.3273 (B) 1.7374 (C) 9.8336 (D) 21.8548
42. If 𝑐𝑜𝑡𝜃 < 0 and 𝑠𝑒𝑐𝜃 > 0, in which quadrant would the terminal arm of
𝜃 lie?
(A) I (B) II (C) III (D) IV
43. What is the length of the arc cut by a 120° sector in a circle
having radius 6 cm?
(A) 4π (B) π
9 (C)
π
3 (D) 5π
44. Which expression represents all angles co-terminal with 2𝜋
3 ?
(A) 2π
3+ kπ, k ∈ R (B)
2π
3+ kπ, k ∈ I
(C) 2π
3+ 2kπ, k ∈ R (D)
2π
3+ 2kπ, k ∈ I
45. Which represents the missing coordinate if the point 𝑃 (𝑥,3
4)
lies on the terminal arm of the unit circle in the second quadrant?
(A) −1
4 (B) −
√7
4 (C) −
5
4 (D) −
√2
2
46. Which represents a circle centered at the origin with a radius of 5√3 ?
(A) 𝑥2 + 𝑦2 = 15 (B) 𝑥2 + 𝑦2 = 5√3
(C) 𝑥2 + 𝑦2 = 75 (D) 𝑥2 + 𝑦2 = 30
47. What is the measure of the central angle if P(θ) = (√2
2, −
√2
2)?
(A) π
4 (B)
5π
4 (C)
5π
3 (D)
7π
4
48. What is the exact value of: csc4π
3 ?
(A) −√3
2 (B) −
1
2 (C) −2 (D) −
2√3
3
49. If the point P(5, –12) lies on the terminal arm of Ө, then which
represents the ratio for sec Ө ?
(A) −13
12 (B)
13
5 (C) −
12
13 (D)
5
13
50. If 𝑐𝑜𝑡 𝜃 =7
24, 𝜋 < 𝜃 <
3𝜋
2, then which ratio is true?
(A) 𝑐𝑜𝑠𝜃 =7
25 (B) 𝑐𝑠𝑐𝜃 =
25
24
(C) 𝑠𝑒𝑐𝜃 = −25
24 (D) 𝑠𝑖𝑛𝜃 = −
24
25
51. Solve for x: cos Ө = 1
(A) kπ, k ∈ I (B) 2πk, k ∈ I
(C) π
2+ kπ, k ∈ I (D)
π
2+ 2πk, k ∈ I
52. If 𝑠𝑒𝑐 𝜃 = −2.5, 0° ≤ 𝜃 ≤ 180°, what is a possible value of 𝜃?
(A) 23.6° (B) 66.4° (C) 113.6° (D) 156.4°
53. Which represents the exact value of 𝑡𝑎𝑛2 5𝜋
6+ 𝑐𝑜𝑡2 5𝜋
6 ?
(A) 3
10 (B)
10
3 (C)
4
3 (D) 1
54. Solve for θ: 2sin2θ = sin θ where 0 ≤ θ ≤ 2π.
(A) {π
6,
5π
6} (B) {
π
3,
2π
3}
(C) {0,π
6,
5π
6, π, 2π} (D) {0,
π
3,
2π
3, π, 2π}
55. What is the period of: y = 3cos [1
2x]?
(A) 𝜋
2 (B) (C) 4 (D) 8
56. Which represents the sine function with amplitude 3 and a period of 5𝜋
6 ?
(A) y = 3sin (5π
6θ) (B) y = 3sin (
6
5πθ)
(C) y = 3sin (5
12θ) (D) y = 3sin (
12
5θ)
57. For what value of k will the polynomial 3 2( ) 4 3 6P x x x kx have the
same remainder when it is divided by both 1 and 3x x ?
58. Give the function, 65632)( 234 xxxxxP , determine the other roots.
59. Given the graph, determine the equation of the polynomial in factored form.
60. An open top box is made from a 16 m by 12 m rectangular piece of sheet
metal by cutting congruent squares of length x from each corner and folding
up the sides. Identify any restrictions on x and algebraically determine what
size squares must be removed to produce a box with a volume of 192 m3.
61. Sketch the graph of the function 3 22 5 4 3y x x x and clearly label the
x-intercept(s) and the y-intercept.
62. The graph of the function y g(x) represents a transformation
of the graph of y f (x). Determine the equation of g(x) in the
form y af (b(x h)) k.
63. The graph of )(xfy is shown below. Sketch the graph of the transformed
function ))1(2(63 xfy . Identify the specific values of a, b, h and k
required and state the mapping rule.
64. Algebraically determine the inverse function for 𝑓(𝑥) = 3𝑥2 + 6𝑥 − 1,
stating any necessary restictions.
x- 10 - 5 5 10
y
- 15
- 10
- 5
5
10
y = f (x)
y = g(x)
65. Determine the domain and range of 8)1(2 2 xy .
Show algebraic workings.
66. (i) Write the mapping rule that maps xy onto
the function 1)4(23 xy .
(ii) State the domain and range of the transformed function.
(iii) Sketch the graph on the grid provided, showing the image points for
those shown on the graph of xy .
Mapping Rule: ______________
Domain: ___________________
Range: ___________________
67. Determine the approximate solution to each equation graphically
xx 392 2 . Verify your answer algebraically.
68. Determine the exact value of 𝑐𝑠𝑐𝜋
3+ 𝑐𝑜𝑡
11𝜋
4
69. Solve for x:
(a) 2𝑐𝑜𝑠𝑥 + √3 = 0, 𝑥 ∈ [0,180°)
(b) 𝑐𝑠𝑐𝑥 + 12 = 2 − 4𝑐𝑠𝑐𝑥, 𝑥 ∈ [𝜋
2,
5𝜋
2)
(c) 3 csc2x – 4 = 0 where 0 ≤ x ≤ 2π
(d) 2𝑠𝑖𝑛2𝑥 + 3𝑠𝑖𝑛𝑥 = −1, 𝑥 ∈ [−180°, 540°)
(𝑒) 3𝑡𝑎𝑛2𝑥 = 14 tan 𝑥 + 5, 𝑥 ∈ [0,3𝜋)
(𝑓) 2𝑠𝑒𝑐2(𝑥) = sec(𝑥) + 15, where 𝑥 ∈ [−2𝜋, 2𝜋)
70. Determine the length of 𝐴�̂� (in cm), to the nearest tenth of a unit.
ANSWERS:
1.D 2. B 3. A 4. A 5. C 6. D 7. C
8. C 9. C 10. C 11. C 12. D 13. B 14.A
15. A 16. D 17. C 18. B 19. D 20. D 21.D
22. D 23. D 24. D 25. A 26. D 27. B 28.B
29. A 30. B 31. A 32. B 33. A 34. D 35.A
36. B 37. C 38. B 39. C 40. C 41. C 42.D
43. A 44. D 45. B 46. C 47. D 48. D 49.B
50. D 51. B 52. C 53. B 54. C 55. C 56.D
57. k = –34 58. −1, −1, 2,3
2 59. 𝑦 = −
1
2𝑥(𝑥 + 2)2(𝑥 − 3)
60. 0 < x < 6, a 2m x 2m square must be removed from each corner
61. 62. y = –2f(x + 3)
63.
64. Restriction: x ≥ –1 𝑦 = √𝑥+4
3− 1
65. Domain: –1 ≤ x ≤ 3 Range: 0 ≤ y ≤ √8
B (-15,8)
A (8,15)
x- 6 - 4 - 2 2 4 6
y
- 8
- 6
- 4
- 2
2
4
6
x
y
A'
B' C' D'
66. (i) (𝑥, 𝑦) → (−1
2𝑥 + 4, 3𝑦 + 1)
(ii) Domain: x ≤ 4 Range: y ≥ 1
(iii)
67.
68. 2√3−3
3
69.(a) 150ᵒ (b) 7π
6,
11π
6 (c)
π
3,
2π
3,
4π
3,
5π
3
(d) −30°, −90°, −150°, 210°, 270°, 330°
(e) 1.4 𝑟𝑎𝑑, 2.8 𝑟𝑎𝑑, 4.5 𝑟𝑎𝑑, 5.98 𝑟𝑎𝑑, 7.7 𝑟𝑎𝑑
(f)
−5.05 𝑟𝑎𝑑, −4.3 𝑟𝑎𝑑, −1.98 𝑟𝑎𝑑, −1.23 𝑟𝑎𝑑, 1.23 𝑟𝑎𝑑, 1.98 𝑟𝑎𝑑, 4.3 𝑟𝑎𝑑, 5.05 𝑟𝑎𝑑
70. 26.7 cm
x-6 -4 -2 2 4 6
y
-2
2
4
6
8
10
x-4 -2 2 4
y
-2
2
4
Solutions:
{𝑥 = 0 𝑎𝑛𝑑 𝑥 = 2}
top related