Download Final Exam Study Guide

Final Exam Study Guide
33 of the 99 questions of this review will be on your final exam!!!
What I need to know from Unit 1 – Matrices
0 𝑥𝑦 −3𝑥
𝑦 + 𝑥𝑦 2𝑦
𝑦𝑥
−𝑦 −𝑥
1. Simplify [
]+[
]∙[
]
−𝑥
−3 𝑥 + 3𝑦
3𝑥 −𝑥 −2 5𝑦 −3𝑥
2
−3𝑦
3𝑦
4
−4𝑥𝑦
𝑦𝑥 ])
[−5𝑥
−3𝑦𝑥
1]
−5𝑥
−1
2. Simplify
∙ ([
]+[
2
𝑥𝑦
𝑥
−5𝑦
1
4𝑏 𝑏
0
−3 −3𝑎
3. Simplify −4 [
] ∙ [ 3 −1]
𝑏𝑎 −3𝑏 𝑎2
−𝑏 2
9
6
4. Simplify: [ −1 ] = [ 4 ] − 𝐶
−12
−10
−9 8
5. Find the inverse: [
]
−3 2
5
5 −2
6. Find the determinant: |−1 2 −7|
−2 −1 −6
−2𝑎 − 3𝑏 + 4𝑐 = 25
7. Solve the system: −𝑎 + 𝑏 + 5𝑐 = 27
4𝑎 + 3𝑏 + 5𝑐 = −1
−11𝑥 − 7𝑦 = 22
8. Solve the system:
12𝑥 = −4𝑦 + 16
9. The school that Totsakan goes to is selling tickets to a spring musical. On the first day of ticket sales
the school sold 7 senior citizen tickets and 9 child tickets for a total of $83. The school took in $109
on the second day by selling 8 senior citizen tickets and 15 child tickets. Find the price of a senior
citizen ticket and the price of a child ticket.
What you need to know from Unit 2 – Conics
10. Write the standard from equation of the ellipse: Center: (9, −10), Vertex: (9,1), Co-Vertex:
(−1, −10)
11. Write the standard form of the equation: −𝑥 2 + 𝑦 2 − 6𝑥 − 18𝑦 + 47 = 0.
12. Write the standard form of the equation: 𝑥 2 + 𝑦 2 − 14𝑥 − 26𝑦 + 209 = 0.
2𝑥 2 + 𝑦 2 + 11𝑥 − 76 = 0
13. Solve the system: 2
2𝑥 + 𝑦 2 + 18𝑥 − 104 = 0
𝑥 2 − 6𝑥 + 𝑦 = 0
14. Solve the system: 2
𝑥 + 7𝑦 2 − 6𝑥 + 50𝑦 = 0
15. Identify the center, vertices, co-vertices, and foci of the ellipse:
(𝑥+2)2
9
+
16. Identify the vertices and foci of the equation, then graph the equation:
17. Graph the equation: 16𝑥 2 − 25𝑦 2 + 50𝑦 − 425 = 0
18. Graph the equation: 𝑦 2 + 𝑥 − 5 = 0
(𝑦+1)2
36
(𝑥−1)2
4
=1
−
(𝑦+1)2
16
=1
What you need to know from Unit 3 – Introduction to Trigonometry
19. Given a right triangle where C is the right angle, find 𝑚∠𝐴 if 𝑎 = 13 and 𝑏 = 14
20. Convert to decimal degree: 230°1′12"
25𝜋
21. Convert to degrees: − 6
22. Find cos 𝜃 if tan 𝜃 =
21√22
5
44
23. Find tan 𝜃 if sec 𝜃 = 3
24. Find the reference angle: −
7𝜋
9
14𝜋
25. Find the reference angle: 9
26. A suspension bridge has two main towers of equal height. A visitor on a tour ship approaching the
bridge estimates that the angle of elevation to one of the towers is 24°. After sailing 477 ft closer he
estimates the angle of elevation to the same tower to be 55°. Approximate the height of the tower.
27. A 40 foot tree is hit by lightning and the top of the tree falls, making a 35° angle with the ground.
How high above the ground did the tree break?
What you need to know from Unit 4 – Graphing Trigonometric Function
28. Identify the 5 characteristics for: 𝑦 = 1 − cos 𝑥
29. Identify the 5 characteristics for: 𝑦 = 10 sin(2𝑥 + 𝜋) − 5
𝜋
30. Graph the function: 𝑦 = 4 cos (𝑥 − 4 )
𝜋
31. Graph the function: 𝑦 = sin (𝑥 − 3 ) − 2
𝜋
32. Graph the function: 𝑦 = 2 sin (3𝜃 + 2 ) + 2
𝜋
33. Graph the function: 𝑦 = 3 sec (2𝜃 − 6 ) + 2
5
34. Find the exact value of the expression: sin (cos−1 (13))
𝑢
35. Find the function: cos (tan−1 2)
6
36. Find the exact value of the expression: sin (tan−1 7)
What you need to know from Unit 5 – Trig Identities
csc 𝑡−1
cot 𝑡
1. Prove the identity: cot 𝑡 = csc 𝑡+1
1
1
2. Prove the identity: 1−sec 𝑥 + 1+sec 𝑥 = −2 cot 2 𝑥
3. Prove the identity: 8 csc 2 𝑥 − 3 cot 2 𝑥 = 3 + 5 csc 2 𝑥
4. Evaluate the expression: sin 105°
5.
6.
7.
8.
9.
𝜋
Evaluate the expression: cos (− 12)
Solve the equation for 0 ≤ 𝜃 ≤ 2𝜋: −3 + 6 cos 𝜃 = 0
Solve the equation for 0 ≤ 𝜃 ≤ 2𝜋: √2 cos 𝑥 sin 𝑥 − cos 𝑥 = 0.
Solve the equation for 0 ≤ 𝜃 ≤ 2𝜋: 2 cos2 𝑥 − sin 𝑥 − 1 = 0
Solve the equation for 0 ≤ 𝜃 ≤ 2𝜋: 6 sin 𝑥 = sin 𝑥 + 3
What you need to know from Unit 6 – Law of Sines and Cosines
1. Given triangle ABC with a = 7, C = 37°, and B = 18°, find c. Round the answer to two
decimal places.
2. Solve triangle ABC given that A = 45°, B = 54°, and b = 70.
3. Solve ΔABC with A = 69°, b = 34, and c = 46.
4. Find the area of a triangle with sides 4 m, 5 m, and 7 m.
5. In triangle 𝑋𝑌𝑍, 𝑦 = 41.3, 𝑋 = 28°, and 𝑍 = 43°. Determine the length of side x.
6. In ∆𝑍𝑋𝑌, 𝑥 = 10 𝑓𝑡, 𝑚∠𝑍 = 33.9°, 𝑎𝑛𝑑 𝑦 = 24.5 𝑓𝑡. Solve the triangle.
7. In ∆𝐹𝐷𝐸, 𝑚∠𝐹 = 136°, 𝑚∠𝐸 = 39° 𝑎𝑛𝑑 𝑑 = 4 𝑖𝑛. Solve the triangle.
8. Find the area of the triangle: In ∆𝐹𝐷𝐸, 𝑒 = 7 𝑚, 𝑓 = 9 𝑚, 𝑑 = 6𝑚
9. Two ships leave port at 4 pm. One is heading at a bearing of N 380 E and is traveling at 11.5 miles
per hour. The other is traveling 13 miles per hour at a bearing of S 470 E. How far apart are they
when dinner is served at 6 pm?
What you need to know from Unit 7 – Complex Numbers and Polar Equations
1. Convert to a rectangular equation: 𝑟 sin 𝜃 = 12
2. Convert to a polar equations: 3𝑥 + 7𝑦 = 9
3. Convert to a polar equation: 𝑥 2 + 𝑦 2 = 9
4. Raise the number to the given power and write the answer in trigonometric notation.
𝜋
𝜋
4
[3 (cos 2 + 𝑖 sin 2 )]
5. Divide and leave the answer in trigonometric form.
15(cos 72°+𝑖 sin 72°)
5(cos 9°+𝑖 sin 9°)
6. Multiply and leave the answer in trigonometric form. 6.5(cos 37° + 𝑖 sin 37°) ∙ 7.5(cos 29° +
𝑖 sin 29°)
7. Find the cube roots of −2√2 + 2√2𝑖
8. Find the square roots of 7√2 − 7√2𝑖
9. Find the fifth roots of unity.
What you need to know from Unit 8 – Vectors
1. Find the vector that has the given magnitude and direction angle. ‖𝑣‖ = 7, ∝= 2700
2. u = <2, -5>, v = <-4, 1>. Find 3u + 2v.
3. Write the vector with initial point (-2,4) and terminal point (-3,5) in Component form.
4. An airplane has airspeed of 450 kilometers per hour bearing N 45o W. The wind velocity is 45
kilometers per hour in the direction S 30o W. Find the resulting velocity (speed and direction) of the
plane.
5. A Lear Jet has a speed of 400 MPH in still air. Suppose the plane travels east and encounters a 30
MPH wind blowing due North. Find the resulting velocity (speed and direction) of the jet.
6. You are paddling a canoe east on river at a speed of 4 mi/hr. The river flows 10° south of east at 1.5
mi/hr. What is the resulting speed and direction of your canoe?
7. Find the resultant of the vectors <10cos25o, 10sin250> and
<12cos(-230), 12sin(-230)>
8. For the given vectors u and v, find their dot product 𝑢 ∙ 𝑣.
𝑢 =< 3.45,6.72 >, 𝑣 =< −2.15,5.60 >
9. A 200-lb block is suspended by two cables. At point A, there
are three forces acting: W, the block pulling down, and R and
S, the two cables pulling upward and outward. Find the
tension in each cable.
What you need to know from Unit 9 – Sequences and Series
1. Which the explicit equation for the given sequence? 3, 7, 11, 15, 19…
5 10 17
2. Write the summation notation for 2   
2 3 4
193
111
3. Given two terms in an arithmetic sequence find the recursive formula: 𝑎20 = 6 𝑎𝑛𝑑 𝑎34 = 2
4. What are the 3rd and 7th terms of the given sequence? an = 4(1.5)n – 1
5. Given two terms in a geometric sequence find the explicit formula. 𝑎2 = −8 𝑎𝑛𝑑 𝑎5 = −512
6. The seating in an auditorium is set in such a way that the first three rows contain 22,
26, and 30 seats respectively. Each consecutive row increases at the same rate all the
way to the last row (row 36). How many seats does this auditorium contain?
7. Write the equation for the nth term for the given sequence. -7, -2, 9, 26, 49, 78
8. Given two terms in a arithmetic sequence find the explicit formula. 𝑎18 = −1733 𝑎𝑛𝑑 𝑎35 = −3433
22
9. Evaluate:
 2n  1
n 1
What you need to know from Unit 10 - Combinations and Permutations
1. A class has 6 boys and 8 girls. How many ways can Ms. Cleveland choose a group of 2 boys and 2
girls?
2. Ashley is going to dinner at a pasta restaurant. She is allowed to choose one of three pastas, one of
4 sauces, and one of three meats. How many different pasta dishes can she make?
3. How many different ways can you arrange the letters in the word FEBRUARY?
4. To logon to the school computer system each student must have a password with 3 letters and 2
digits in that order. Each letter and number can be used only once. How many different passwords
are possible?
5. Determine whether the scenario involves a permutation or a combination. Then find the number of
possibilities: A team of 17 lacrosse players needs to choose a captain and co-captain.
6. A school council has 16 members, including 4 seniors. There are four members randomly chosen to
represent the student council at a school open house. What is the probability that all four council
members chosen are seniors?
7. Determine whether the scenario involves a permutation or a combination. Then find the number of
possibilities: The student body of 170 students wants to elect three representatives.
8. Find the probability of the event: A technician is launching fireworks near the end of a show. Of the
remaining twelve fireworks, seven are blue and five are red. If she launches five of them in a
random order, what is the probability that exactly three of them are blue?
9. There are 10 players on a softball team. Each game the batting is randomly chosen. Find the
probability that you are chosen to bat first and your best friend is chosen to bat second.
What you need to know from Unit 11 - Limits
1. Evaluate: lim
𝑥→1
2. Evaluate: lim
𝑥→3
3. Evaluate: lim
𝑥 3 +2𝑥−11
𝑥+3
2𝑥 2 −5𝑥−3
𝑥−3
𝑥 4 −16
𝑥→2 𝑥−2
𝑥 2 −2𝑥−15
4. Evaluate: lim
𝑥→2
𝑥−5
1. Evaluate: lim
2𝑥 2 −5𝑥−12
𝑥−4
𝑡−2
𝑥→4
2. Evaluate: lim 𝑡 2 −4
𝑥→2
3. Evaluate: lim
1
−1
1+𝑟
𝑥→1
𝑟
2
4. Evaluate: lim 𝑥 + 4𝑥 + 1
𝑥→1
5. Evaluate: lim (5 − 2𝑥 2 + 7𝑥 3 )
𝑥→∞