Algebra II
Unit 7B/5A & 5B Review
1.
(7B/5A-1,6) Simplify:
(3√6)2 − (2 + √−5)(2 − √−5)
A. 45
B. 53
C. 63
D. 53 + 4𝑖√5
2. Simplify: (7√2)2 + (8 + √−2)(8 − √−2)
3
3.
(7B/5A-1,2) Given x > 0, y > 0, which expression is equivalent to:
3
A.
C.
D.
√18𝑥 6 𝑦 10
4
2 √𝑥 √3
𝑥 2 𝑦 3 √2
3
B.
4
√8𝑥 4 √243𝑦8
4
2√2 √𝑥 √3
2𝑥 2 𝑦 3
3
4
3
4
√𝑥 √3√2
𝑥2𝑦3
√𝑥 √6
𝑥2𝑦3
3
√250𝑎5 4√256𝑏 7
4. Given x>0, y>0, simplify the expression:
√243𝑎8 𝑏 12
5. (7B/5A-1,2) Rewrite this expression in simplest form. Assume variables represent positive
numbers.
A.
B.
C.
D.
6.
25𝑤 6 𝑦 5
√
72𝑥 7
5𝑤 3 𝑦 2 √𝑦
6𝑥 3 √2𝑥
5𝑤 3 𝑦 2 √𝑦
6𝑥 3 √𝑥
5𝑤 3 𝑦 2 √2𝑥𝑦
12𝑥 3
5𝑤 3 𝑦 2 √2𝑥𝑦
12𝑥 4
Rewrite this expression in simplest form. Assume variables represent positive numbers.
162𝑤 9 𝑦 4
√
242𝑥 3
3
3
7. (7B/5A-1) Variables a, b, and c are real numbers where b = c4 and a = b5. Write √𝑏𝑐 3 + √𝑎2 in
terms of c.
3
A. (2𝑐 2 ) √𝑐
3
B. 𝑐15 √𝑐
3
C. 2𝑐 √𝑐
3
D. (𝑐 2 + 𝑐13 ) √𝑐
4
4
8. Variables x, y, and z are real numbers where y = z3 and x = y2. Write √𝑦𝑧 5 + √𝑥 3 in terms of z.
9. (7B/5A-1) Given:
𝑏 = √𝑎
𝑐 = 𝑎2
𝑑 = 𝑏2 𝑐
𝑑3 𝑏
Which expression is equivalent to 𝑎𝑐 3 in terms of a?
A. 𝑎√𝑎
B. 𝑎2 √𝑎
C. 𝑎2
D. 𝑎10
𝑏 = √𝑎 ,
10. Given:
𝑏3 𝑑
𝑐 = 𝑎2 , and 𝑑 = 𝑏 2 𝑐 , write an expression that is equivalent to 𝑎𝑐 2 in terms
of a?
3
11. (7B/5A-1,2) Completely simplify
√500𝑘 145
3
√162
3
A.
5 √6𝑘𝑘 48
9
3
B.
5 √18𝑘𝑘 48
9
3
C.
5𝑘 48 √𝑘
3
D.
125𝑘 72 √𝑘
3
3
3
12. Completely simplify
√2500𝑘 153
3
√216
13. (7B/5A-2) Rationalize the denominator and simplify:
A. −(2 + √6 − √3 − 3√2 )
B. 4 + 2√6 − 2√3 − 3√2
C.
4+2√6−2√3−3√2
D. −
2
4+2√6−2√3−3√2
2
14. Rationalize the denominator and simplify:
6−√5
6+√10
2−√3
2−√6
15. (7B/5A-4) Mr. Johnson bought a conical camping tent for his 2 daughters. The radius of the
circular base of the tent measures 6.5 ft, and the tent’s lateral surface area is 240.6 ft2. Use the
formula = 𝜋𝑟√𝑟 2 + ℎ2 , where S is the lateral surface area and r is the radius, to find the height,
h, of the tent to the nearest tenth of a foot. (Note: π ≈ 3.14)
A. 9.8
B. 11.6
C. 11.7
D. 96.7
16. Mr. Johnson bought a conical camping tent for his 3 sons. The radius of the circular base of the
tent measures 10 ft, and the tent’s lateral surface area is 392.5 ft2. Use the formula
𝑆 = 𝜋𝑟√𝑟 2 + ℎ2 , where S is the lateral surface area and r is the radius, to find the height, h, of
the tent to the nearest tenth of a foot. (Note: π ≈ 3.14)
17. (7B/5A-4) Scientists use the Beaufort wind scale to approximate wind speed. The formula is
𝐵 = 1.69√𝑠 + 4.45 − 3.49 , where B is the Beaufort number and s is the wind speed in miles per
hour. To the nearest mile per hour, what is the approximate wind speed if the Beaufort number is
15 ?
A.
6
B.
8
C.
45
D. 115
18. Scientists use the Beaufort wind scale to approximate wind speed. The formula is
𝐵 = 1.69√𝑠 + 4.45 − 3.49 , where B is the Beaufort number and s is the wind speed in miles per
hour. To the nearest mile per hour, what is the approximate wind speed if the Beaufort number is
3?
𝑥
3
19. (7B/5A-4) If 𝑓(𝑥) = √𝑥 − 5𝑥 and 𝑔(𝑥) = 2−𝑥 , what is 𝑓(𝑔(𝑥)) ?
3
𝑥
3
𝑥
A. √2−𝑥 − 5𝑥
5𝑥
B. √2−𝑥 − 2−𝑥
C.
D.
√𝑥−5𝑥
2−𝑥
𝑥√𝑥−5𝑥2
2−𝑥
20. If 𝑓(𝑥) = √𝑥 − 5 and 𝑔(𝑥) =
𝑥−2
𝑥
, what is 𝑔(𝑓(𝑥)) ?
𝑥
21. (7B/5A-4) Consider the functions 𝑓(𝑥) = 4√6 and 𝑔(𝑥) = 3𝑥 2 . Find and simplify (𝑔°𝑓)(𝑥)
for x > 0.
3𝑥 2
A. 4√
6
B. 2𝑥√2
C. 8𝑥
𝑥
D. 12√6
𝑥
22. Consider the functions 𝑓(𝑥) = 4√6 and 𝑔(𝑥) = 3𝑥 2 . Find and simplify (𝑓°𝑔)(𝑥) for x > 0.
23. (7B/5A-4) What is the solution set to the compound inequality x2 > 5 and x < 7?
A. {𝑥|√5 < 𝑥 < 7}
B. {𝑥|7 < 𝑥 < √5}
C. {𝑥|𝑥 < −√5 𝑜𝑟 𝑥 > √5}
D. {𝑥|𝑥 < −√5 𝑜𝑟 √5 < 𝑥 < 7}
24. What is the solution set to the compound inequality x2 < 6 or x > 5?
25. (7B/5A-6) Rationalize
1−𝑖
1+𝑖
A. –1
B. 1
C. –i
D.
i
26. Rationalize
𝑖−3
3−𝑖
27. (7B/5A-6) What is the product of (6 + 5𝑖) and (−2 − 7𝑖) ?
A.
23 − 52𝑖
B. −23 − 52𝑖
C. −12 − 87𝑖
D. −47 − 52𝑖
28. What is the product of (3 − 7𝑖) and (4 − 5𝑖) ?
29. (7B/5A-6) If 𝑐 − 𝑑 = 3 and 𝑐 = 5 + 3𝑖 , what is d ?
A. −2 − 3𝑖
B. −2 + 3𝑖
2 − 3𝑖
C.
D. 2 + 3𝑖
30. If 𝑐 − 𝑑 = 2 and 𝑑 = 4 − 5𝑖 , what is c ?
31. (7B/5A-6) What is the sum of 3i, 4 − 5𝑖, and –8 ?
A. −4 − 2𝑖
B. −12 − 2𝑖
C. −12 − 8𝑖
D. −4 + 2𝑖
32. What is the sum of 6i + 2, –5+3𝑖, and 7 ?
33. (7B/5A-6) Write
25−𝑖 2
5+𝑖
in standard form.
A. 5 − 𝑖
B. 5 + 𝑖
C.
D.
13
2
5
4
34. Write
𝑖 2 −36
𝑖−6
in standard form.
6−4𝑖
35. (7B/5A-7) What is the first step in simplifying −2−3𝑖 ?
6+4𝑖
A. Multiply the fraction by 6+4𝑖
−2−3𝑖
B. Multiply the fraction by −2−3𝑖
−2+3𝑖
C. Multiply the fraction by −2+3𝑖
D. Multiply the fraction by
2+3𝑖
2+3𝑖
36. What is the first step in simplifying
−3+4𝑖
5+7𝑖
?
37. (7B/5A-7) What is the complex conjugate of 2 + √−45 ?
A. 2 + 9𝑖√5
B. 2 − 9𝑖√5
C. 2 + 3𝑖√5
D. 2 − 3𝑖√5
38. What is the complex conjugate of 9 + √−32 ?
39. (7B/5A-7) What is the complex conjugate of
A.
2
3
2
3
+ 3𝑖 ?
− 3𝑖
2
B. 3 − 3 𝑖
C.
D.
2
3
2
3
𝑖−3
+ 3𝑖
40. What is the complex conjugate of
1
4
− 5𝑖 ?
41. (5B-1) What are the zeros of the quadratic function 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 4 ?
A. −3 ± √5
B.
−6±√20
2
C. 3 ± √5
D.
6±√20
2
42. What are the zeros of the quadratic function 𝑓(𝑥) = 𝑥 2 + 5𝑥 + 3 ?
43. (5B-1) What is the solution set for 4𝑥 2 = 3𝑥 + 2 ?
3
1
A. {− 8 ± 8 √41}
3
1
3
𝑖
3
𝑖
B. {8 ± 2 √2}
C. {8 ± 8 √35}
D. {8 ± 8 √23}
44. What is the solution set for 3𝑥 2 = 5𝑥 + 6 ?
45. (5B-1) What are the roots of this equation? 𝑥 2 − 4𝑥 + 10 = 0
A. −4 ± 2𝑖√6
B.
2 ± 𝑖√6
C.
2 ± 2𝑖√6
D. 4 ± 2𝑖√6
46. What are the roots of this equation? 𝑥 2 − 5𝑥 + 12 = 0
1
47. (5B-1) For what values of x does the graph of 𝑓(𝑥) = 𝑥 2 − 1 2 𝑥 − 7 intersect the x-axis?
A. –1 and 7
B. 1 and –7
1
C. −3 2 and 2
1
D. 3 2 and −2
1
48. For what values of x does the graph of 𝑓(𝑥) = 𝑥 2 − 5 3 𝑥 − 4 intersect the x-axis?
49. (5B-1) What is the solution set of
A. {
1
2
𝑥2 + 5 = 𝑥 − 3 ?
4
3±𝑖√42
3
}
−3±2√42
B. {
}
3
−6±2𝑖√42
C. {
D. {
}
3
6±2𝑖√42
3
}
50. What is the solution set of
3
1
𝑥 2 + 3 = 2𝑥 − 5 ?
4
51. (5B-1) What are the solutions for 𝑥 2 − 6𝑥 + 7 = 2𝑥 − 5 ?
A. 2, 6
B. −2, −6
C. –2, 6
D. 2, −6
52. What are the solutions for 𝑥 2 − 2𝑥 + 8 = 𝑥 + 5 ?
53. (5B-1) What condition will yield real zeros of a quadratic function
A. – b < b2
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ?
B. b2 < 4ac
C. 2a < 0
D. b2 > 4ac
54. How do you know if a zero is real or non-real?
55. (5B-2) For the equation 𝑥 2 − 5𝑥 + 3 = 5 determine the discriminant
A. 17
B. 33
C.
13
D. –3
56. For the equation 3𝑥 2 − 4𝑥 + 6 = −5 determine the discriminant.
57. (5B-3) What are the 2 points of intersection for this system of equations?
𝑥2 + 𝑦2 = 5
{
𝑥 + 𝑦 = −1
A. (–1 ,2), (–2 ,1)
B. (1,2), (2,1)
C. (1, –2), (–2, 1)
D. (–1 ,–2), (–2, –1)
58. (5B-3) What are the 2 points of intersection for this system of equations?
𝑦 = 𝑥 2 − 4𝑥 + 5
{
𝑦−𝑥 =1
59. (5B-3) Which graph represents the solution set of this system of inequalities?
𝑦 ≥ −(𝑥 − 3)2 − 2
{
𝑦 ≤ (𝑥 + 1)2 − 1
60. (5B-3) Which system of inequalities describes the shaded region in this graph?
A. 0.5 ≤ 𝑥 ≤ 1.5 𝑎𝑛𝑑 0 ≤ 𝑦 ≤ 1
B. 𝑦 ≥ 𝑥 2 𝑎𝑛𝑑 𝑦 ≤ −𝑥 2
C. 𝑥 2 ≤ 𝑦 ≤ −𝑥 2 + 1
D. 𝑦 ≥ (𝑥 − 1)2 𝑎𝑛𝑑 𝑦 ≤ −(𝑥 − 1)2 + 1
61. (5B-3) Which is the graphical representation of the solution set for this system of inequalities?
𝑦 ≥ 𝑥 2 + 6𝑥 − 5
{
𝑦 ≤ −𝑥 2 − 2𝑥 + 3
62. (5B-3) This system of equations:
{
𝑦 − 2 = −(𝑥 + 3)(1 − 𝑥)
2𝑥 2 − 𝑦 + 5 = 0
has no real-valued solutions; if the equations were graphed in the x-y plane, they would be
represented by two parabolas that do not intersect. But in the complex numbers, there are two
solutions. What are the values of x for the solutions set to this system?
A. {1, −3}
B. {±𝑖}
C. {1 ± 𝑖√5}
D. {−3 ± 4𝑖√5}
63. This system of equations:
{
𝑦 + 4 = (𝑥 + 1)(2 − 𝑥)
−2𝑥 2 + 2𝑦 − 9 = 0
has no real-valued solutions; if the equations were graphed in the x-y plane, they would be
represented by two parabolas that do not intersect. But in the complex numbers, there are two
solutions. What are the values of x for the solutions set to this system?
64. (5B-4) This is the equation of a parabola: 𝑦 = −3𝑥 2 + 2𝑥 − 10
Determine the x-value for the vertex and whether this value is a maximum or a minimum.
A. Maximum at x = 0.33
B. Maximum at x = - 9.67
C. Minimum at x = 0.33
D. Minimum at x = - 9.67
65. This is the equation of a parabola: 𝑦 = −𝑥 2 + 5𝑥 + 2
Determine the y-value for the vertex and whether this value is a maximum or a minimum.
66. (5B-5) Sally observes that the data derived from an experiment seems to be parabolic when plotted
on ordinary graph paper. Three of the observed points are (1,-1), (2, 5), and (3, 13). Use the
equation of the parabola that contains these 3 points to determine the y-value at x = 5.
A. y = –7.25
B. y = –1.5
C. y = 5
D. y = 35
67. Sally observes that the data derived from an experiment seems to be parabolic when plotted on
ordinary graph paper. Three of the observed points are (0,4), (2, 1), and (-2,3). Use the equation
of the parabola that contains these 3 points to determine the y-value at x = 3.
68. (5B-6) The height above ground of an object thrown upward from an initial height of s ft with an
initial velocity of v ft/sec is modeled by ℎ(𝑡) = −16𝑡 2 + 𝑣𝑡 + 𝑠. Joe throws a baseball upward
at 60 ft/sec from a platform 25 ft above the ground. To the nearest tenth of a second, when will
the baseball hit the ground?
A. - 0.4
B. 1.9
C. 4.1
D. 81.3
69. The height above ground of an object thrown upward from an initial height of s ft with an initial
velocity of v ft/sec is modeled by ℎ(𝑡) = −16𝑡 2 + 𝑣𝑡 + 𝑠. Joe throws a baseball upward at 75
ft/sec from a platform 3 ft above the ground. To the nearest tenth of a second, when will the
baseball hit the ground?
1
1
70. (5B-6) John throws an inflated ball up in the air. The function ℎ(𝑡) = − 24 𝑡 2 + 12 𝑡 + 2 models
the ball’s height in terms of time t, in seconds. After how many seconds will the ball hit the
ground?
A.
-6
B.
1
C.
2
D.
8
71. (5B-6) Mary is standing on top of a building. She throws a ball up from the top of the building.
The height of the ball is modeled by ℎ(𝑥) = −16𝑡 2 + 11𝑡 + 30 . How many seconds after
release will the ball hit the ground?
A. –1.1
B.
0.3
C.
1.8
D.
31.9
72. The formula 𝐿 = 0.1𝑠 2 − 3𝑠 + 22 gives the approximate runway length required to land a small
plane. L is the length of the runway, in feet and s is the landing speed of the airplane in feet per
second. The pilot knows that the runway is 3,200 ft long. To the nearest foot per second, what is
the maximum safe landing speed?
A. -164
B.
50
C. 160
D. 194
73. (5B-6) Carol has a small rectangular herb garden that has a length 5 yd longer than twice its width.
If the area of the garden is 45 yd2, what is the length of the garden, to the nearest tenth of a yard?
A. 3.7
B. 6.1
C. 12.4
D. 17.2