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Set 1 Exam: 050306RR - Exponents, Logarithms, Sequences, and Series
1. Solve the equation below by expressing each side as a power of the same base and then equating
exponents.
2. Find the sum, if it exists, of the infinite geometric series below:
B. A sum doesn't exist for this geometric series.
3. Find the first five terms of the arithmetic sequence in which a
A. 5, 4, 3, 2, 1
B. 6, 5, 3, 3, 2
C. 7, 6, 5, 4, 3
D. 6, 5, 4, 3, 2
4. Make a table of coordinates for the function below and then graph it.
f (x) = 5
5. Find the accumulated value of an investment of $7,000 at 7% compounded continuously for 6 years.
To solve question 3, use the correct formula for compound interest:
A. $10,653.73
B. $9,940.00
C. $10,753.73
D. $10,505.11
6. Write the first four terms of the sequence whose general term is given by the following formula:
a
n
= 4n 1
A. 5, 9, 13, 17
B. –3, –7, –11, – 15
C. 3, 7, 11, 15
D. 3, 4, 5, 6
7. Use summation notation to express the sum below. Use 1 as the lower limit of summation and i for the index of summation.
8. Write the first five terms of the arithmetic sequence below.
A. 15, 11, 6, 3, –1
B. 15, 11, 7, 3, –1
C. 11, 7, 3, –1, –5
D. 19, 15, 11, 7, 3
9. Solve the logarithmic equation below. Be sure to reject any value that isn't in the domain of the original logarithmic expressions. Give the exact answer.
(x 2) = 3
A. {6}
B. {11}
C. {7}
D. {10}
10. Write the equation below in its equivalent exponential form:
11. Find the sum of the first 60 terms of the arithmetic sequence 14, 19, 24, 29, . . . .
A. 9,690
B. 9,840
C. 9,700
D. 314
12. Use properties of logarithms to expand the logarithmic expression below as much as possible.
A. 5 – ln 9
B. 5 + ln 9
C. ln e
D. ln e
13. Find the sum of the first 70 terms of the arithmetic sequence 17, 23, 29, 35, . . . .
A. 15,685
B. 437
C. 15,890
D. 15,680
14. What type of transformation would occur with the following exponential functions?
f(x) = a
f(x) = a
A. Reflection about the x-axis
B. Reflection about the y-axis
C. Vertical shrinking
D. Vertical stretching
15. Solve the exponential equation below. Express the solution set in terms of natural logarithms.
16. Evaluate the expression below without using a calculator:
log
A. –4
B.
C. 4
D. 40
17. Find the common ratio of the geometric sequence given below.
18. Use properties of logarithms to expand the logarithmic expression below as much as possible. Where
possible, evaluate logarithmic expressions without using a calculator.
A. ln e
3
+ ln 11
B. ln e
3
– ln 11
C. 3 – ln 11
D. 3 + ln 11
19. Write the equation below in its equivalent logarithmic form.
20. Evaluate the expression below without using a calculator.
Set 2 Exam: 050307RR - Trigonometric Functions
1. Convert the angle below to degrees. Round your answer to two decimal places.
A. 307.58°
B. 310.58°
C. 308.58°
D. 309.58°
2. Given sin t = and cos t = , use identities to find sec t.
3. Use a sketch to find the exact value of the expression .
4. Classify the angle below as acute, right, obtuse, or straight.
A. Right
B. Acute
C. Obtuse
D. Straight
5. Solve the right triangle in the figure below if A = 51.4° and c = 57.1. Round lengths to one decimal place and express angles to the nearest tenth of a degree.
A. B = 51.4°, a = 35.6, b = 44.6
B. B = 38.6°, a = 35.6, b = 44.6
C. B = 38.6°, a = 44.6, b = 35.6
D. B = 51.4°, a = 44.6, b = 35.6
6. From a boat on the river below a dam, the angle of elevation to the top of the dam is 25°. If the dam is 2,039 feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)?
A. 4,370 feet
B. 4,738 feet
C. 4,730 feet
D. 4,373 feet
7. Find the radian measure of the central angle of a circle with radius r = 1.8 meters that intercepts an arc s
= 4.14 meters.
A. 2.3 radians
B. 1.6 radians
C. 0.56 radians
D. 0.9 radians
8. In the right triangle below, C is the right angle and the two sides are given. Find cos . Give an exact
answer with a rational denominator.
9. Solve the right triangle shown in the figure below when A = 51.9° and c = 51.2. Round lengths to one decimal place and express angles to the nearest tenth of a degree.
A. B = 38.1°, a = 31.6, b = 40.3
B. B = 51.9°, a = 31.6, b = 40.3
C. B = 38.1°, a = 40.3, b = 31.6
D. B = 51.9°, a = 40.3, b = 31.6
10. Convert the angle of 160° to radians. Express your answer as a multiple of .
11. Graph the function
12. Find the reference angle for 96°.
A. 94°
B. 16°
C. 6°
D. 84°
13. Use the given graph to obtain the graph of
14. Draw the angle below in standard position.
15. Find the exact value of the expression cos
16. Find a positive angle less than 360° that's coterminal with the angle .
17. Convert the angle in radians to degrees.
18. A car wheel has a radius of 16 inches. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward 4 feet?
A. 181.9°
B. 186.9°
C. 176.9°
D. 171.9°
19. Find the radian measure of the central angle of a circle of radius r = 1.3 meters that intercepts an arc of length s = 2.34 meters.
A. 1.7 radians
B. 0.65 radians
C. 0.77 radians
D. 1.8 radians
20. Convert the angle 54° to radians. Express your answer as a multiple of
Set 3 Exam: 050308RR - Analytic Trigonometry
1. Use a half-angle formula to find the exact value of the expression sin 165°.
2. Use the given information to find the exact value of the expression sin 2 .
3. Use trigonometric identities to find the exact value of the following expression.
4. Complete the following identity:
A. sin x tan x
B. sec x csc x
C. –2tan
D. 1 + cot x
5. Identify and in the following expression, which is the right-hand side of the formula for cos ( – ).
cos (170°) cos (50°) + sin (170°) sin (50°)
A. = –170°, = 50°
B. = –50°, = 170°
C. = 50°, = –170°
D. = 170, = 50°
6. Use trigonometric identities to find the exact value of the expression below.
7. Complete the following identity:
8. use a half-angle formula to find the exact value of the expression cos 112.5°.
9. Solve the following equation on the interval
10. Use a calculator to solve the following equation on the interval Round the answer to two
decimal places.
sin x = 0.29
A. 0.29, 5.99
B. 0.29, 3.44
C. 0.29, 1.87
D. 3.44, 5.99
11. Rewrite the following expression as a simplified expression containing one term.
12. Use the figure below to find the exact value of the trigonometric function cos 2 .
13. Find the exact value of the expression cos (245° – 5°).
14. Find the exact value of the expression below.
15. Solve the problem sin 8x – sin 2x.
A. 2 sin 3x cos 5x
B. 2 sin 5x cos 3x
C. 2 cos 2x cos 5x
D. 2 sin 3x
16. Solve the following equation on the interval
17. Use the figure to find the exact value of the trigonometric function cos 2 .
18. Complete the following identity:
19. Find the exact value of the following by using a sum or difference identity.
sin (185° 65°)
20. Find the exact value of the expression .
Set 4 Exam: 050309RR - Additional Topics in Trigonometry
1. Solve the triangle in the figure below.
A. B = 50°, a = 8.25, c = 6.55
B. B = 55°, a = 8.25, c = 6.55
C. B = 55°, a = 6.55, c = 8.25
D. B = 60°, a = 6.55, c = 8.25
2. Find the absolute value of the complex number z = 14 8i .
3. Convert the polar equation r = 4 csc to a rectangular equation.
4. Convert the rectangular equation y = 1 to a polar equation that expresses r in terms of .
A. sin =1
B. r cos = 1
C. r sin = 1
D. r = 1
5. A vector v has initial point P1 = (0, 0) and terminal point P2 = (4, 6). Write v in terms of ai + bj.
A. v = 4i + 6j
B. v = 6i 4j
C. v = 4i 6j
D. v = 6i 6j
6. Solve the triangle in the figure below. Round lengths to the nearest tenth and angle measures to the nearest degree.
A. A = 30.8°, B = 125.1°, C = 24.1°
B. A = 30.8°, B = 24.1°, C = 125.1°
C. A = 125.1°, B = 24.1°, C = 30.8°
D. A = 125.1°, B = 30.8°, C = 24.1°
7. Use Heron's formula to find the area of a triangle in which a = 16 yards, b = 13 yards, and c = 16 yards.
Round to the nearest square unit.
A. 95 square yards
B. 98 square yards
C. 104 square yards
D. 101 square yards
8. Use a polar coordinate system to plot the point with polar coordinates of .
9. Use the vectors below to find the specified scalar. u = 8i + 5j and v = 11i 6j; Find u • v.
A. 88
B. 118
C. 58
D. 30
10. Convert the polar equation r = 9 csc to a rectangular equation.
11. Convert the rectangular equation y = 3 to a polar equation (that is, in terms of r and ).
A. sin = 3
B. r cos = 3
C. r = 3
D. r sin = 3
12. Find the product of the complex numbers below.
13. Find the unit vector having the same direction as v if v = 12i + 5j.
14. A surveyor standing 52 meters from the base of a building measures the angle to the top of the building and finds it to be 35°. The surveyor then measures the angle to the top of the radio tower on the building and finds that it's 50°. How tall is the radio tower?
A. 13.93 meters
B. 10.01 meters
C. 25.56 meters
D. 9.17 meters
15. Solve the triangle below.
A. B = 45°, a = 8.18, c = 12.6°
B. B = 50°, a = 8.18, c = 12.6°
C. B = 40°, a = 12.68, c = 8.1°
D. B = 45°, a = 12.68, c = 8.1°
16. A vector v has initial point P
A. v = 6i + 5j
B. v = 5i + 6j
C. v = 6i + 6j
D. v = 5i 6j
and terminal point P2. Write v in terms of ai + bj.
17. The rectangular coordinates of a point are given below. Find polar coordinates of the point.
18. Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.
v = 3i + 2j, w = 2i 3j
A. Parallel
B. Orthogonal
C. Not enough information
D. Neither orthogonal nor parallel
19. Two sides of an angle (SSA) of a triangle are given below. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle measures to the nearest degree.
A = 30°, a = 22, b = 44
A. B = 60°, C = 60°, c = 38.1
B. No triangle
C. B = 90°, C = 60°, c = 38.1
D. B = 60°, C = 90°, c = 38.1
20. The graph of a polar equation is shown below. Select the polar equation for the graph.
A. r = 6 sin
B. r = 3 + cos
C. r = 3 + sin
D. r = 6 cos
Set 5 Exam: 050310RR - Systems of Equations and Inequalities
1. Graph the inequality x
2
+ y
2
> 49.
2. Write the partial fraction decomposition of the rational expression below.
3. A steel company produces two types of machine dies, part A and part B, and is bound by the following constraints:
• Part A requires 1 hour of casting time and 10 hours of firing time.
• Part B requires 4 hours of casting time and 3 hours of firing time.
• The maximum number of hours per week available for casting and firing are 100 and 70, respectively.
• The cost to the company is $0.75 per part A and $3.00 per part B. Total weekly costs can't exceed
$45.00.
Let x = the number of part A produced in a week and y = the number of part B produced in a week. Write a system of three inequalities that describes these constraints.
4. Write the form of the partial fraction decomposition of the rational expression below. It's not necessary to solve for the constants.
5. Use the addition method to solve the system below.
x 2y = 14
2x 3y = 22
A. {(–3, 7)}
B. {(–2, 6)}
C. {(2, 7)}
D. Ø
6. Solve the system of equations below.
x + y + z = 2
x y + 5z = 6
2x + y + z = 3
A. {(–5, 4, 3)}
B. {(3, –5, 4)}
C. {(4, –5, 3)}
D. {(3, 4, –5)}
7. Graph the inequality x + y < –4.
8. Use the substitution method to solve the system below.
x + y = 3
y = x
2
6x + 9
A. {(2, 1), (3, 0)}
B. {(3, 0)}
C. {(–2, 5), (–3, 6)}
D. {(2, 5), (3, 0)}
9. Write the partial fraction decomposition of the rational expression below.
10. Graph the inequality –2x – 3y
11. Write the partial fraction decomposition of the rational expression below.
12. Which one of the following ordered pairs is a solution of the system below?
x + y = 1
x y = 9
A. (5, –4)
B. (–5, –4)
C. (–5, 4)
D. (5, 4)
13. Use the substitution method to solve the system of equations below.
5x + 2y = 68
x = 3y
A. {(–4, –12)}
B. {(–12, –4)}
C. {(–12, 4)}
D. {(–11, –4)}
14. A steel company produces two types of machine dies, part A and part B. The company makes a $3.00 profit on each part A that it produces and a $6.00 profit on each part B that it produces. Let x = the number of part A produced in a week and y = the number of part B produced in a week. Write the objective function that describes the total weekly profit.
A. z = 9(x + y)
B. z = 3x + 6y
C. z = 6x + 3y
D. z = 3(x – 6) + 6(y – 3)
15. Write the partial fraction decomposition of the rational expression below.
16. Write the partial fraction decomposition of the rational expression below.
17. Use the addition method to solve the system below.
A. {(5, 0), (–5, 0)}
B. {(5, 0)}
C. {(0, 5)}
D. {(0, 5), (0, –5)}
18. Solve the system of equations below.
x + y = 6
3 x 4y 4z = 9
x z = 3
A. {(1, –2, 5)}
B. {(1, 5, –2)}
C. {(–2, 1, 5)}
D. {(–2, 5, 1)}
19. Graph the solution set of the system of inequalities below.
y < x + 8
y > 8x 3
20. Solve the system below by the method of your choice. Identify systems with no solution and systems with infinitely many solutions using set notation to express their solution sets.
y = 6 2x
4x + 2y = 12
A. {(x , y)|2x + y = 6}
B. {(5, –4)}
C. Ø
D. {(0, 6)}
