Part -1:
1. Solve the right triangle shown in the figure
A = 54o, c = 59
a. What are the lengths of the sides?
b. What are the angles?
2. Convert the angle in degree to radians. Express your answer as a multiple of Π.
115o
3. Find a positive angle less than 360o that is conterminal with the given angle.
-85.
4. Use the right triangle and the given information to solve the triangle
a = 1, c = 8; find b, α and β
5. Find the length x to the nearest whole number.
6. Find the radian measure of the central angle of a circle of radius r = 4 meters that intercepts an arc of length s = 1500 centimeters.
The radian measure of the central angle is __.
7. Find the exact value of the given expression
cos-1 √3/2
8. Find the exact value of each of the remaining trigonometric function of θ.
tanθ = -2/5, cos θ>0
9. The point(-9, -7) lies on the terminal side of an angle θ. Find the exact value of the six trigonometric faction of θ.
10. use reference angles to find the exact value of the following expression. Do not use a calculator.
Sec 495o
11. Find the length x to the nearest whole number
12. Use the right triangle and the given information to solve the triangle.
a = 9; β=62o; find b, c and α
13. Let θ be an angle in standard position. Name the quadrant in which θ lies.
Sin θ < 0, tan θ < 0
The angle θ lies in which quadrant?
1. I
2. IV
3. II
4. III
14. if θ = 27o, find the distance between two cities, A and B, to the nearest mile. The radius of the Earth is approximately 4000 miles.
The distance between the two cities, And B, is approximately____ miles.
15. Find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. Express arc length in terms of Π. then round your answer to two decimal places.
Radius, r= 18 feet; Central angle θ = 205o
16. Find the reference angle for the angle 325o
17. Find the length x to the nearest whole number.
18. Find the length x to the nearest whole number.
19. Draw the angle in standard position. State the quadrant in which the angle lies. Work the exercise without converting to degrees.
11Π/6
Choose the correct graph below.
20. Convert the angle in radians to degrees.
Π/4 radians.
Part -2:
1. Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.
4x2 - 3x + 6/(x-1)(x2 +1)
What is the form of the practical decomposition of the given expression?
A. (A/x-1) + (Bx +c/x2+1) + (Dx + E) /(x2 +1)2
B. A/(x-1) + B/(x2+1)
C. A/(x-1) + Bx +C/(x2 +1)
D. Ax + B/(x-1) + (Cx + D)/(x2 +1)
E. A/(x-1) + Bx/(x-1) + C/(x2 +1)
F. A/x-1 + Bx/(x-1) + Cx + D/(x +1)
2. Solve the system. If there is no solution or if there are infinitely many solutions and the system's equations are dependent, so state.
5x + y = -10
x + y - z = -8
3x + 2y +z = 0
Select the correct choice below and fill in any answer boxes within your choice.
A. There is one solution. The solution set is {-1/2, -15/2, 33/2}
B. There are infinitely many solutions.
C. There is no solution.
3. Solve the sysytem by the addition method.
4x - 3y = -6
3x - 4y = 6
Select the correct choice below and fill in any answer boxes present in your choice.
A. The solution set is {-6, -6}.
B. There are infinitely many solution.
C. There is no soution.
4. The sum of three numbers is 11. The sum of twice the first number, 3 times the second number, and 4 times the third number and the second number is 14. Find the three numbers.
First number
second number
third number
5. A rectangular lot whose perimeter is 240 ft is fenced along three sides. An expensive fencing along the lot's length cost $18 per foot. An inexpensive fencing along the two side widths costs only $3 per foot. The total cost of the fencing along the three sides comes to $1680. What are the lot's dimensions?
The longer dimension is __ feet.
The shorter dimension is __ feet.
6. A company that manufactures small canoes has a fixed cost of $20,000. It costs $80 to produce each canoe. The selling price is $160 per canoe. (In solving this exercise, let x represent the number of canoes produced and sold.)
a. Write the cost function.
C(x) =
b. Write the cost function.
R(x) =
c. Write the cost function.
This means that when the company produces and sells the break - even number of canoes.
A. there is less money coming in than going out.
B. there is more money coming in than going out.
C. the money coming in equals the money going out.
D. there is not enough information.
7. Solve the system by the substitution method.
xy = 10
x2+ y2 = 29
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is {( -2, -5), (2, 5), (-5, -2), (5,2)}.
B. there is no solution.
8. Write the partial fraction decomposition of the given rational expression.
3/(3x2 -5x -2)
3/(3x2 -5x -2 = 3/7(x-2) - 9/7(3x +1)
9. Solve the system by the method of your choice.
9x2 + y2 = 90
xy = 9
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solutions to the system is/are{(-1, -9),(1,9),(-3, -3), (3,3)}
B. There is no solution.
10. Twice a first number decreased by a second number is 9. The first number increased by three times the second number is 15. Find the numbers.
11. A person invested $8100 for 1 year, part at 6%, part at 10 %, and the remainder at 14 %. The total annual income from this investment was $914. The amount invested at each rate.
The person invested $ at 6%, $ at 10 % and $ at 14%.
12. Solve the given system of equations.
x + y + 6z = -19
x + y + 4z = -15
x - 5y - 5z = 27
Select the correct choice below and fill in any answer boxes within your choice.
A. There is one solution. The solution set is {()}
B. there are infinitely many solutions.
C. there is no solution.
13. Solve the system by the method of your choice.
x = 4y + 3
3x - 12 y = 9
Select the correct choice below and fill in any answer boxes present in your choice.
A. The solution set is {}.
B. There are infinitely many solutions.
C. There is no solution.
14. Solve the system of equations by the substitution method.
X = 3y - 8
X = 4y - 5
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
B. There are infinitely.
C. There is no solution.
15. Solve the system of equations by the substitution method.
Y = x2 -7x + 5
Y =-x2 -13x + 13
Select the correct choice below and, if necessary, fill the answer box to complete your choice.
A. The solution set is{}
B. There is no solution.
16. Find the partial fraction decomposition.
(3x + 177)/((x-4)2(x +5))
Choose the correct pattern for the decomposition below. And fill in any answer boxes to complete your choice.
Part -3:
1 .Solve the system using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
x + y - z = -3
3x - y + z = 11
-x + 4y -3z = 1
The solution set is {}.
2. Find BC + CB given that B= 
. If an operation is not defined, state the reason.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. BC + CB =
B. This matrix operation is not possible because product BC is not defined.
C. This matrix operation is not possible because product CB is not defined.
D. This matrix operation is not possible because the orders of the two matrices being added are different.
3. Find A(BC) given that A =
. If an operation is not defined, state the reason.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. A(BC) =
B. This matrix operation is not possible because product BC is not defined.
C. This matrix operation is not possible because product A and BC is not defined.
4. Find 8B - 3C given that B = 
. If an operation is not defined, state the reason.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. 8B-3C =
B. This matrix operation is not possible because the order of two matrices being subtracted is different.
C. This matrix operation is not possible because the scalar by which matrix B is mulipiled is different that h scalar by which matrix C is multiplied.
5. Find A - C given that A = 
.If an operation is not defined, state the reason.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. A - C =
B. The matrix operation is not defined. The difference of two matrices of different orders is undefined.
C. The matrix operation is not defined. The number of columns in the first matrix is not th same as the number of rows in the second matrix.
Part -4:
1. Find the standard form of the equation of the ellipse satisfying the following conditions.
Major axis vertical with length 6: length of minor axis = 4: center ( - 10. - 3)
The standard form of the equation is ___.
2. An explosion is recorded by two microphones that are 3 miles apart. Microphone M1 receives the sound 6 seconds before microphone M2. Assuming sound travels at 1100 feet per second. determine the possible locations of the explosion relative to the locations of the microphones.
Which equation below describes the possible locations of the explosion relative to the locations of the microphones?
A. x2/10,890,000 - y2/62,726,400 =1
B. x2/62,726,400 - y2/10,890,000 =1
C. x2/51,836,400 - y2/10,890,000 =1
D. x2/10,890,000 - y2/51,836,400 =1
E. x2/3300 - y2/7920
F. x2/7920 - y2/3300 =1
3. Find the standard form of the equation of the hyperbola satisfying the given conditions.
Foci at ( - 5, 0) and (5,0): vertices at (4, 0) and ( -4, 0)
The equation is ___.
4. Use vertices and asymptotes to graph the hyperbola. Locate the foci and find the equations of the asymptotes.
16y2 -x2 = 1
Choose the correct graph.
The foci are ____.
(Type an ordered pair. Simplify your answer. Type an exact answer. using radicals as needed. Rationalize all denominators. Use a comma to separate answers as needed.)
The equations of the asymptotes are ___ = ±___.
(Simplify your answer. Type your answer in point-slope form. Use integers or fractions for any numbers in the equation.)
5. Will a truck that is 16 feet wide carrying a load that reaches 11 feet above the ground clear the semielliptical arch on the one-way road that passes under the bridge shown in the figure on the right?
It ___ clear the arch because the height of the archway of the bridge 8 feet from the center is approximately __ feet (Round to two decimal places as needed.)
6. Graph the ellipse and locate the foci.
4x2 +81y2 = 324
Choose the correct graph below.
Locate the foci.
The foci are located at ___.
(Type ordered pairs. Use a comma to separate answers. Type exact answers. using radicals as needed. Simplify your answers.)
7. Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0.- 26): Directrix: y = 26
The equation of the parabola is ___. (Simplify your answer.)
8. Find the vertex. focus. and directrix of the parabola. Then graph the parabola. (y +2)2 = - 8x
The vertex of the parabola is __. (Type an ordered pair.)
The focus of the parabola is ___. (Type an ordered pair.)
The directrix of the parabola is ___.
(Type an equation. Simplify your answer.)
Use the graphing tool to graph the parabola.
9. Use the center, vertices, and asymptotes to graph the hyperbola. Locate the foci and find the equations of the asymptotes.
(x+5)2/49 (y+4)2/64 = 1
Determine the location of the foci.
(Type an ordered pair. Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed )
The equations of the asymptotes are ___ = ±__.
(Simplify your answer. Type your answer in point-slope form. Use integers or fractions for any numbers in the equation.)
10. Convert the equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. Answer parts A-D below.
x2 + y2 - 2x - 4y - 4 = 0
A. y-1= ± (x-2) B. y+ 2= ±(x +1)
C. y -1= ±(x+ 2) D. y+ 2= ±(x - 1)
C. Choose the correct graph.
D. The foci are located at ___.
11. Graph the ellipse and give the location of its foci.
(x - 1)2/9 + (y - 1)2/100 = 1
Choose the correct graph below.
Locate the foci.
(Type ordered pairs. Use a comma to separate answers. Type exact answers, using radicals as needed Simplify your answers.)
12. The cables of a suspension bridge are in the shape of a parabola. as shown in the figure. The towers supporting the cable are 300 feet
apart and 20 feet high. If the cables touch the road surface midway between the towers. what is the height of the cable at a point 75 feet from the center of the bridge?
The cable is _____feet from the road surface at a point 75 feet from the center of the bridge.
13. Find the focus and directrix of the parabola with the given equation. Then graph the parabola.
x2 = -24y
The focus is D.
(Type an ordered pair.)
The directrix is ___
(Type an equation.)
Use the graphing tool to graph the parabola.
14. Convert the equation to standard form by completing the square on x and y Then graph the ellipse and give the location of its foci.
25x2 + 9y2 + 450x + 144y + 2376 =0
The standard form of the equation is __. (Type an equation. Simplify your answer.)
Choose the correct graph below.
The foci are located at ___.
(Type ordered pairs. Use a comma to separate answers. Type exact answers. using radicals as needed. Simplify your answers.)