Download PreCal Summer Math 2015.tst

Name____________________________________
PRECALCULUS SUMMER MATH 2015
Summer Math covers skills or concepts from your math courses previously taught. Do all
problems because there will be a Summer Math Test given by the end of the first week of
school next year. This will be the first test grade of the first nine weeks. Please make sure
that your start on this early to allow enough time to review concepts.
SHOW WORK
Simplify the algebraic expression.
1) -3(2x - 10) - 4x + 9
A) 2x + 39
1)
B) -10x - 21
C) 10x + 39
1
B)
12
1
C)
81
D) -10x + 39
Evaluate the exponential expression.
2) 3-4
A) -81
2)
D) 81
3) 5-3 ∙ 5
A) 25
3)
B)
1
25
C) 125
D)
1
125
Approximate the number using a calculator. Round your answer to three decimal places.
4) 6-2.1
A) 0.023
5) 2.2 !
A) 11.906
B) 0.323
C) -12.600
D) 85.766
B) 12.409
C) 36.462
D) 6.911
4)
5)
Solve the problem.
6) The function f(x) = 300(0.5)x/50 models the amount in pounds of a particular radioactive
material stored in a concrete vault, where x is the number of years since the material was put
into the vault. Find the amount of radioactive material in the vault after 130 years. Round to the
nearest whole number.
A) 230 pounds
B) 49 pounds
C) 390 pounds
D) 58 pounds
6)
7) The rabbit population in a forest area grows at the rate of 9% monthly. If there are 290 rabbits in
September, find how many rabbits (rounded to the nearest whole number) should be expected
by next September. Use y = 290(2.7)0.09t.
7)
A) 861
B) 835
C) 846
D) 848
B) 7
C) b
D) 1
Simplify the exponential expression.
8) (7b)0
A) 0
8)
1
SHOW WORK
3
-12x10y 7
6x14y -2
9)
9)
-5
4x3
y2
10)
A)
10)
1024x15
y 10
B)
y 10
C)
1024x15
1024y 10
x15
D)
y2
1024x15
11) (-4x4y -5)(2x-1y)
11)
-8x3
B)
y4
A) -8x3y 6
-2x3
C)
y4
-8x5
D)
y6
12) (x4)-6
12)
Evaluate the radical expressions or indicate that the root is not a real number.
5
13) 243
Use the product rule to simplify the expression.
14) 45
A) 5 3
B) 6
Add or subtract terms whenever possible.
15) 3 2x - 8 2x
A) 11 2
B) -5
16) 6
17)
3 +2
13)
14)
C) 3
5
D) 15
15)
2x
C) -24
4x
D) -5x
75
4
16)
100 + 50 + 144 + 242
A) 50 + 242 + 22
C) 146 2 + 22
17)
B) 16
D) 16
2 + 22
2 + 100 +
144
18) The time, in seconds, that it takes an object to fall a distance d, in feet, is given by the algebraic
expression
d
. Find how long it will take a ball dropped from the top of a building 38 feet tall
16
to hit the ground. Write the answer in simplified radical form.
A)
C)
38 seconds
16
6+
4
38 seconds
4
B)
2 seconds
D)
2
6
4
2 seconds
18)
SHOW WORK
Find the degree of the polynomial.
19) -9x + 5x3 + 4x2 - 2
A) degree 3
19)
B) degree 5
C) degree 2
D) degree 4
Perform the indicated operations. Write the resulting polynomial in standard form.
20) (8x6 - 8x5 - 4x4 - 1) - (5x6 - 3x5 + 6x4 + 3)
A) 13x6 - 11x5 + 2x4 - 4
C) 3x6 - 11x5 + 2x4 + 2
21) (5x2 + 4x + 7) + (2x2 + 3x - 7) - (5x + 2)
A) 5x2 + 2x + 2
B) 7x2 + 2x + 2
Find the product.
22) (2x + 1)(7x + 8)
A) 14x2 + 23x + 8
20)
B) 3x6 - 5x5 - 10x4 - 4
D) 13x6 - 11x5 + 2x4 + 2
21)
C) 7x2 + 2x - 2
D) 5x2 + 2x - 2
C) 9x2 + 23x + 8
D) 9x2 + 23x + 23
22)
B) 14x2 + 23x + 23
23) (x - 15)2
23)
B) 225x2 - 30x + 225
D) x2 + 225
A) x + 225
C) x2 - 30x + 225
24) (x + y)(x2 - xy + y 2)
A) x3 + y 3
24)
B) x3 + 2x2y + 2xy 2 + y 3
D) x3 - y 3
C) x3 - 2x2y - 2xy 2 + y 3
25) (1 + x5)(1 - x5)
A) 1 - x10
25)
B) 2 - x25
C) 2 - x10
D) 1 - x25
Factor out the greatest common factor.
26) 5x2 - 15x
A) x(5x - 15)
26)
C) 5(x2 - 3x)
B) 5x(x - 3)
D) 5x(x - 3x)
Factor by grouping. Assume any variable exponents represent whole numbers.
27) 5x3 - 10x2 + 8x - 16
A) (x - 2)(5x + 8)
B) (x - 2)(5x2 + 8)
C) (x + 2)(5x2 + 8)
27)
D) (x - 2)(5x2 - 8)
Factor the trinomial, or state that the trinomial is prime.
28) x2 - 5x + 6
28)
29) x2 - x - 90
30) x2 - x - 40
A) (x - 5)(x + 8)
29)
30)
B) (x + 5)(x - 8)
C) (x - 40)(x + 1)
3
D) prime
SHOW WORK
31) x2 - 13x + 42
A) (x + 7)(x - 6)
B) (x + 7)(x + 1)
C) (x - 7)(x - 6)
D) prime
32) 7x2 - 23x - 20
A) (7x - 5)(x + 4)
B) (7x + 5)(x - 4)
C) (7x - 4)(x + 5)
D) prime
31)
32)
33) 20x2 - 27x + 9
33)
Factor the difference of two squares.
34) x2 - 100
A) (x + 10)2
34)
B) (x - 10)2
C) (x + 10)(x - 10)
B) (x - 8)2
C) (x + 8)2
D) prime
Factor the perfect square trinomial.
35) x2 - 16x + 64
A) (x - 8)(x + 8)
35)
D) prime
Factor using the formula for the sum or difference of two cubes.
36) x3 + 27
A) (x - 3)(x2 + 3x + 9)
C) (x + 3)(x2 + 9)
36)
B) (x + 3)(x2 - 3x + 9)
D) prime
Factor completely, or state that the polynomial is prime.
37) 5x2 - 5x - 30
A) prime
37)
B) 5(x + 2)(x - 3)
C) 5(x - 2)(x + 3)
D) (5x + 10)(x - 3)
38) x3 - 4x2 - 9x + 36
38)
39) 4x3 - 4
39)
A) 4(x3 - 1)
B) 4(x + 1)(x2 - x + 1)
C) 4(x - 1)(x2 + x + 1)
D) prime
Solve the problem.
40) Write an expression for the area of the shaded region and express it in factored form.
8
y
8
y
A) y 2 + 64
B) (y + 8)2
C) (y - 8)2
4
D) (y + 8)(y - 8)
40)
SHOW WORK
Find all numbers that must be excluded from the domain of the rational expression.
6
41)
x+ 3
A) x ≠ -6
42)
x- 9
x2 - 64
43)
x- 7
2
x + 3x - 28
A) x ≠ -7, x ≠ 4
B) x ≠ -3
C) x ≠ 3
41)
D) x ≠ 0
42)
43)
B) x ≠ 0
C) x ≠ 7
D) x ≠ -4, x ≠ 7
Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational
expression.
x2 + 9x + 20
44)
44)
x2 + 13x + 40
45)
A)
x + 4 , x ≠ -8, -5
x+8
2
B) - x + 9x + 20 , x ≠ -8, -5
x2 + 13x + 40
C)
9x + 20 , x ≠ - 40
13
13x + 40
D)
9x + 1 , x ≠ - 2
13
13x + 2
3x2 - 33x + 54
x-9
45)
A)
1 , x≠9
x-9
B) 3x2 - 39, no restrictions on x
C)
3x2 - 33x + 54 , x ≠ 9
x-9
D) 3x - 6, x ≠ 9
Multiply or divide as indicated.
x2 + 13x + 42 ∙ x2 - 49
46)
x2 + x - 42
x2 - x - 42
A)
47)
x- 7
x- 6
46)
B)
x+7
x- 6
C)
x- 7
x+6
D)
x+6
x- 7
21x - 21 ÷ 7x - 7
10
90
A)
1
27
47)
B)
9(21x - 21)
7x - 7
C) 27
5
D)
147(x - 1)2
900
SHOW WORK
Simplify the complex rational expression.
1
x+6
48)
3
2
x - 36
A)
x+6
3
B)
48)
3
x- 6
C)
x- 6
3
D) x - 6
Solve the problem.
49) Express the perimeter of the square as a single rational expression.
49)
7
x +6
A)
7
x + 24
B)
28
x +6
C)
28
x + 24
D)
28
x + 12
Add or subtract as indicated.
x - 4 - 2x - 5
50)
x-8
x-8
A) - x + 1
x-8
51)
B) - x - 1
x-8
C)
x+1
x-8
D)
x- 1
x-8
x+5
+ 5x - 2
2
x + 6x + 8 x2 + 7x + 12
Solve the linear equation.
52) -7x - 7 - 2(x + 1) = 2x - 7
2
A) 11
53)
50)
51)
52)
B)
12
11
2
C) 7
D)
12
7
5x - x = x - 7
3
18 6
53)
First, write the value or values of the variable that make a denominator zero. Then solve the equation.
7 + 3 =
6
54)
x + 1 x - 1 (x + 1)(x - 1)
A) none; {1}
B) -1; {1}
C) -1, 1; ∅
6
D) -1, 1; {2}
54)
SHOW WORK
Solve the rational equation.
5 - 2 =
5
55)
y + 4 y - 4 y 2 - 16
55)
Solve the formula for the specified variable.
56) S = 2!rh + 2!r2
for h
A) h = S - 1
2!r
57) F =
9
5
C + 32
56)
B) h = 2!(S - r)
C) h = S - r
D) h =
S - 2!r2
2!r
for C
A) C = 5 (F - 32)
9
57)
B) C = F - 32
9
C) C =
5
F - 32
Solve the absolute value equation or indicate that the equation has no solution.
58) 6x + 4 + 2 = 7
9 1
1 3
A) - ,
B) ∅
C) - ,
4 4
6 2
D) C = 9 (F - 32)
5
58)
3 1
D) - ,
2 6
Solve the equation by factoring.
59) x2 = x + 30
60) 4x2 - 2x = 0
1
A) - , 0
2
59)
60)
B)
1
1
,2
2
C) {0}
D) 0,
1
2
61) 6x2 + 19x + 15 = 0
61)
Solve the quadratic equation by the square root property.
62) 3x2 = 33
63) (2x - 1)2 = 81
A) {-10, 8}
62)
63)
B) {-8, 10}
C) {-5, 4}
Solve the quadratic equation by completing the square.
64) x2 + 12x + 26 = 0
D) {-4, 5}
64)
A) {6 - 26, 6 + 26}
C) {-6 - 10, -6 + 10}
B) {-12 + 26}
D) {6 + 10}
7
SHOW WORK
Compute the discriminant. Then determine the number and type of solutions for the given equation.
65) x2 - 5x + 4 = 0
A) -41; no real solution
C) 9; two unequal real solutions
65)
B) 0; one real solution
D) 9; one real solution
66) 8x2 = -5x - 2
A) -89; no real solution
C) 0; one real solution
66)
B) -39; no real solution
D) 89; two unequal real solutions
Solve the quadratic equation using the quadratic formula.
67) 2x2 + 10x + 3 = 0
67)
A)
-5 - 19 , -5 + 19
4
4
B)
-10 2
19 , -10 +
2
19
C)
-5 - 31 , -5 + 31
2
2
D)
-5 - 19 , -5 + 19
2
2
68) x2 + 4x = 7
68)
Solve the radical equation, and check all proposed solutions.
69) x - 9 = 4x + 9
A) {18}
B) {4, 18}
C) ∅
69)
D) {9}
Solve the problem.
70) For a culture of 90,000 bacteria of a certain strain, the number of bacteria N that will survive x
hours is modeled by the formula N = 9000 100 - x. After how many hours will 54,000 bacteria
survive?
A) 46 hr
B) 36 hr
C) 64 hr
D) 94 hr
71) Sybil is having her yard landscaped. She obtained an estimate from two landscaping companies.
Company A gave an estimate of $240 for materials and equipment rental plus $45 per hour for
labor. Company B gave an estimate of $320 for materials and equipment rental plus $35 per hour
for labor. Determine how many hours of labor will be required for the two companies to cost the
same.
A) 11 hr
B) 7 hr
C) 8 hr
D) 12 hr
Give the domain and range of the relation.
72) {(19, -2), (3, -1), (3, 0), (4, 1), (12, 3)}
A) domain: {19, 4, 3, 12}; range: {-2, -1, 1, 3}
B) domain: {-2, -1, 1, 3}; range: {19, 4, 3, 12}
C) domain: {19, 4, 3, 12}; range: {-2, -1, 0, 1, 3}
D) domain: {-2, -1, 0, 1, 3}; range: {19, 4, 3, 12}
70)
71)
72)
Determine whether the relation is a function.
73) {(-6, 6), (-1, 2), (2, -3), (5, 5)}
A) Not a function
73)
B) Function
8
SHOW WORK
Determine whether the equation defines y as a function of x.
74) y 2 = 6x
A) y is a function of x
74)
B) y is not a function of x
Evaluate the function at the given value of the independent variable and simplify.
3
75) f(x) = x + 1 ;
f(-5)
x2 + 3
76) f(x) = 4x2 + 4x - 5;
A) 4x2 - 4x - 5
75)
f(x - 1)
76)
B) 4x2 - 4x + 3
C) -4x2 + 4x - 5
D) 4x2 - 16x + 3
Solve the problem.
77) The function P(x) = 0.95x - 60 models the relationship between the number of pretzels x
that a certain vendor sells and the profit the vendor makes. Find P(600), the profit the
vendor makes from selling 600 pretzels.
Find the inverse of the one-to-one function.
78) f(x) = 7x - 1
6
77)
78)
Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to the
graph of f.
79) f(x) = x2, g(x) = x2 + 1
79)
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6 x
-2
-3
-4
-5
-6
Suppose the point (2, 4) is on the graph of y = f(x). Find a point on the graph of the given function.
80) The reflection of the graph of y = f(x) across the x-axis
A) (-2, -4)
B) (-2, 4)
C) (2, 4)
D) (2, -4)
81) y = 4f(x)
A) (3, 8)
80)
81)
B) (8, 4)
C) (2, 16)
9
D) (5, 3)
SHOW WORK
Write the equation that results in the desired transformation.
82) The graph of y = x3, vertically compressed by a factor of 0.7
A) y = (x + 0.7)3
B) y = 0.7x3
C) y = (x - 0.7)3
Write the equation of a sine function that has the given characteristics.
83) The graph of y = x, shifted 7 units to the right
A) y = x - 7
B) y = x - 7
C) y = x + 7
82)
D) y = 0.7
3
x
83)
D) y =
x +7
Identify the intercepts.
y
10
5
-10
-5
10 x
5
-5
-10
84)
84)
A) (2, 0), (0, -8)
B) (2, 2), (-8, -8)
C) (-2, 0), (0, -8)
D) (2, 0), (0, 8)
Use the graph to determine the function's domain and range.
y
4
2
-4
-2
2
4
x
-2
-4
85)
85)
y
4
2
-4
-2
2
4
x
-2
-4
86)
86)
10
SHOW WORK
Find the domain of the function.
87) f(x) = 1
x+1
87)
88) f(x) = x2 + 3
88)
For the given functions f and g , find the indicated composition.
89) f(x) = 17x2 - 10x,
g(x) = 9x - 9
(f∘g)(8)
A) 63,504
B) 57,780
C) 9063
90) f(x) = -5x + 7,
g(x) = 6x + 4
(g∘f)(x)
A) -30x - 38
B) -30x + 27
91) f(x) = 4x2 + 6x + 7,
(g∘f)(x)
D) 66,843
90)
C) -30x + 46
g(x) = 6x - 8
D) 30x + 46
91)
Graph the line whose equation is given.
92) y = - 2 x - 1
5
6
89)
92)
y
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
Use the given conditions to write an equation for the line in slope-intercept form.
93) Slope = -3, passing through (7, 4)
A) y - 4 = -3x - 7
B) y = -3x - 25
C) y = -3x + 25
Use the given conditions to write an equation for the line in the indicated form.
94) Passing through (2, -4) and parallel to the line whose equation is y = -8x + 3;
slope-intercept form
1
3
A) y = - x B) y = 8x - 12
C) y = - 8x - 12
8
2
11
93)
D) y - 4 = x - 7
94)
D) y = - 8x + 12
SHOW WORK
Find the slope then describe what it means in terms of the rate of change of the dependent variable per unit change in
the independent variable.
95) The linear function f(x) = -6.6x + 40 models the percentage of people, f(x), who eat at fast food
95)
restaurants each week x years after 1998.
A) m = 6.6; the percentage of people eating at fast food restaurants each week has increased
at a rate of 6.6% per year after 1998.
B) m = -6.6; the percentage of people eating at fast food restaurants each week has decreased
at a rate of -6.6% per year after 1998.
C) m = 6.6; the percentage of people eating at fast food restaurants each week has increased
at a rate of -6.6% per year after 1998.
D) m = 40; the percentage of people eating at fast food restaurants each week has increased at
a rate of -6.6% per year after 1998.
Add or subtract as indicated and write the result in standard form.
96) 4i - (-9 - i)
A) -9 + 3i
B) 9 + 5i
C) 9 - 3i
97) (-6 + 6i) + (3 + 4i) + (-6 - 6i)
A) 3 + 16i
96)
D) -9 - 5i
97)
B) -15 - 4i
Find the product and write the result in standard form.
98) (8 - 3i)(-2 - 3i)
A) -7 - 30i
B) -7 - 18i
C) -9 + 4i
D) -3 + 10i
C) -25 - 18i
D) -25 - 30i
98)
Divide and express the result in standard form.
9 + 4i
99)
4 - 8i
A)
1 + 11
i
20 10
B) -
99)
1 - 11
i
48 24
C) -
17 - 11
i
12 24
D)
17 + 14
i
5
5
Solve the quadratic equation using the quadratic formula. Express the solution in standard form.
100) x2 + 6x + 25 = 0
A) {-3 ± 16i}
B) {-3 + 4i}
C) {-7, 1}
101) 7x2 = 5x - 3
101)
Find the distance between the pair of points.
102) (6, 5) and (-4, -6)
A) -1
100)
D) {-3 ± 4i}
102)
B) 110
C)
21
Find the midpoint of the line segment whose end points are given.
103) (9, 1) and (5, 9)
A) (4, -8)
B) (14, 10)
C) (7, 5)
12
D)
221
103)
D) (2, - 4)
SHOW WORK
Write the standard form of the equation of the circle with the given center and radius.
104) (0, 0); 6
A) x2 + y 2 = 6
B) x2 + y 2 = 3
C) x2 + y 2 = 36
D) x2 + y 2 = 6
104)
105) (-9, 8); 3
105)
A) (x - 8)2 + (y + 9)2 = 3
C) (x - 9)2 + (y + 8)2 = 9
B) (x + 9)2 + (y - 8)2 = 9
D) (x + 8)2 + (y - 9)2 = 3
Solve the problem.
106) Find the complement of an angle whose measure is 48°.
A) 138°
B) 48°
C) 132°
106)
D) 42°
107) Find the supplement of an angle whose measure is 67°.
107)
108) A 14-foot ladder is leaning against a house with the base of the ladder 3 feet from the house.
How high up the house does the ladder reach? If necessary, round to the nearest tenth foot.
108)
14 ft
3 ft
A) 11 ft
B) 14 ft
C) 13.7 ft
D) 14.3 ft
Find the exact value of the indicated trigonometric function of the angle θ in the figure. Rationalize the denominator
where necessary.
8
109) Find cos θ.
A)
7
7
8
109)
B)
8
15
15
C)
7
15
15
D)
15
8
5
110) Find sin θ.
2
110)
13
SHOW WORK
Find the exact values of the indicated trigonometric functions. Write fractions in lowest terms.
72
21
75
111)
111)
Find sin B and tan B.
A) sin B = 7 ; tan B = 7
25
24
B) sin B = 7 ; tan B = 7
24
25
C) sin B = 24 ; tan B = 24
25
7
D) sin B = 25 ; tan B = 24
7
7
Solve the problem.
112) A square playground has a white line drawn through its diagonal. This creates two triangles
whose angle measures are 45°, 45°, and 90°. If the sides of the playground are 67 feet, use special
right triangles to find the length, x, of the diagonal.
A) 94.8 ft
B) 116 ft
C) 38.7 ft
112)
D) 47.4 ft
113) The path between three landmarks in a national park form a right triangle. The oldest
living evergreen tree is at point A, the ranger station is at point B, and the natural
hotspring is at point C. The right angle is at point C, and there is a 60° angle at point B. If
the distance between the hotspring and the evergreen tree is 609 yards, find the distance,
x, from the hotspring to the ranger station. Round to the nearest hundredth.
113)
609 yd
Use a calculator to find the function value to four decimal places.
114) cos 36.8°
A) 0.5990
B) 0.6223
C) 0.7481
115) tan 24.36°
114)
D) 0.8007
115)
14
SHOW WORK
Find the acute angle θ, to the nearest hundredth of a degree, for the given function value.
116) sin θ = 0.1172
117) cos θ = 0.8769
118) tan θ = 0.1732
A) 9.97°
116)
117)
118)
B) -9.97°
C) 99.97°
D) 9.83°
119) Your math class is going to test new digital clinometers by measuring the angle of
elevation of a kite you will fly. The kite flies to an angle of 63.6° on 540 feet of string.
Assuming the the string is taut, how high is the kite to the nearest foot?
119)
540 ft
63.6°
Solve the right triangle for all missing sides and angles to the nearest tenth.
120)
c = 6, A = 61°
A) B = 29°, a = 2.9, b = 5.2
C) B = 29°, a = 10.8, b = 5.2
120)
B) B = 29°, a = 5.2, b = 2.9
D) B = 29°, a = 2.9, b = 10.8
121) b = 180, c = 380
121)
15
SHOW WORK
122) What is the length of the diagonal of a square with side lengths of 7
2?
123) What is the length of an altitude of an equilateral triangle with side lengths 8
122)
3?
123)
124) What is the angle of elevation of the sun when a 75-foot flag pole casts a 16-foot shadow? Round
to the nearest tenth of a degree.
GR:PreCal Summer Ma:CT148035816
A) 77.7°
B) 12°
C) 12.3°
D) 78°
125) Find the length of the leg. If your answer is not an integer, leave it in simplest radical
form.
125)
126) Find the length of the leg. If your answer is not an integer, leave it in simplest radical
form.
126)
127) Find the value of x and y. No decimals
127)
16
124)