CHAPTER 1 REVIEW
For the exam you should know:
· Matrix Dimensions (rows and columns)
· Identity Matrix (I)
· Matrix Addition, Subtraction and when you can do this
· Matrix Multiplication and when you can do this (including squaring a matrix)
· Finding the determinant of a matrix (2x2’s)
· Finding the inverse of a matrix (2x2’s)
· Setting up matrix equations and solving by inverse matrix method (including algebra steps)
· Solving systems of equations by substitution, elimination, and graphing for 2 x 2 systems
· Solving systems of equations by elimination and substitution for 3 x 3 systems
· Graphing 3-D planes using Intercept method and trace method
· Graphing points in 3D
· Word problems – stating variables, setting up equations, etc
· Using the graphing calculator to solve systems of equations (inverse matrix method)
1) Given the following matrices, find the following without the calculator:
A = B = C = D =
a) The dimensions of matrix C ______b)
c) A C d) A
e) f) B C
g) IA h)
i) 2A + 3B j) D
2) Use the inverse matrix method to solve each system.
a. y – 2 = 3x b. y + 2x = 8
x + y = 6 3y + 2x = 12
3) Lauren and Hilary were in charge of ticket sales for the school concert. Lauren sold 3 adult tickets and 20 student tickets. She collected $59.75. Hilary sold 4 adult tickets and 27 student tickets. Hilary collected $80.50. Determine the cost of each adult ticket and each student ticket. Use any method.
4) If you multiply a 3x5 matrix by a 5x2 matrix what are the dimensions of the solution matrix?
5) a) Solve the following system of
equations by substitution
6) Justin and Lucy are a young couple working on the assembly line of a major automotive company. Two weeks ago Justin worked 37 hours, Lucy worked 41 hours, and their combined salary prior to deductions was $2307. Last week Justin worked 42 hours, Lucy worked 39 hours, and their combined salary prior to any deductions was $2385. How much is each making per hour?
7) Solve the following system of equations using the graphing calculator.
Calls to the US cost 4 times as much as calls within Canada. Bob talked for 47 minutes to friends in Canada and 19 minutes to relatives in the US. Find the per minute cost for each type of call if Bob's last bill was $9.84.
8) Find the x-intercept of the following:
a. 3x + y = 9 b. x – 2y +6 = 4
9) Find the y-intercept of the same equations from 11).
10) Solve the following system of equations by graphing
11) Graph the plane which contains 9) Determine the intercepts of the following
the three points (4, 0, 2), (-2, 0, 0), plane and draw a sketch.
and (0, 3, -4) 2x + y +3z = 6
CHAPTER 3 REVIEW
For the exam you should know:
· Parts of a sinusoidal curve (period, amplitude, sinusoidal axis, y-intercept)
· Transformations of a sinusoidal curve (horizontal stretch and translation, vertical stretch and translation, reflection) and what part of the curve they affect. Also know where they affect the original y=sin(x) curve and y=cos(x) curve. Remember that in transformational form, we use reciprocals, etc.
· Mapping rule to show transformations
· Table of values to show transformations
· Interpreting sinusoidal curves and sketching sinusoidal functions
· Determining equations from sinusoidal graphs
· Application problems (Ferris wheel, etc)
1) Determine the period, Amplitude and Sinusoidal axis, without graphing.
2) Given the mapping rule: x,y ®(2x + 3, -y – 4). Write the equation for y = sinx.
3) Mathman got his cape caught on the blade of a windmill and was hoisted up into the air, around and around he went. The graph shows his height above the ground in meters versus time in seconds.
a) What is the amplitude of the function?
b) What does the amplitude represent?
c) What is the period?
d) What does the period represent?
e) What is the height intercept?
f) What does the height intercept represent?
g) What is the equation of the sinusoidal axis?
h) What does the sinusoidal axis represent?
4) a) Graph: b) Graph:
5) For each graph below, list the transformations that are occurring, and then determine the equation of the line.
a) b) b)
Transformations: Rx = ______Transformations: Rx = ______
VS = ______HS = ______VS = ______HS = ______
VT = ______HT = ______VT = ______HT = ______
Cosine Equation: ______Sine Equation: ______
6) Find the sine equation for the curve in #5a. Find the cosine equation for the curve in 5b.
7) A spring is oscillating sinusoidally back and forth from a motion sensor. At t=3 sec, the spring is at its maximum distance from the sensor, 12 cm. At t=7 sec, the spring is at its minimum distance from the sensor, 3 cm.
a) Find an equation to represent the relationship.
b) How far away is the end of the spring at t=10 sec?
8) A Ferris wheel makes a complete revolution every 15 seconds. At the top, a seat is 22m above the ground. At the bottom it is 1m above the ground.
a) Sketch a graph of the seats movement.
b) What is the radius of the Ferris wheel? ______
c) How high is the axle above the ground? ______
d) What is the equation of the sinusoidal axis? ______
CHAPTER 4 REVIEW
For the exam you should know:
· Operations on radicals
· Trig Expressions using unit circle (finding exact values not using decimals)
· Coterminal angles
· Radian measure (converting degrees to radians, radians to degrees)
· Trigonometric equations – finding all possible answers
· Applications
· Trig Identities
· Rational expressions (+, -, x, ÷) and restrictions on these (what can’t x be?)
1) Give two co- terminal angles for: a) 75º b) -130º
2) Given the following angle measurements, express in radians: a) -25° b) 625°
3) Given the following angle measurements, express in degrees: a) b)
4) Simplify the following radicals:
a) b) c) d)
5) Determine the exact value for each of the following expressions:
a) b) sin2120° + cos(-225°) c)
d) e) f)
g) h) 1- sin²(-45º)
6) Solve for all possible values of x.
a) cos4x = b) 2sinx - = 0 c) 2 cos²x - 1 = 0
d) 4sin6x + 2 = 3 e) 2sinx + 3 = 4 f) 3cosx = -2
7) Solve for the values of x, where
a) 2cosx - = 0 b) sinx + 3 = 4
8) Prove each of the following trig identities:
a) 2tan²qcosqcotq = 2sinq b)
c) secq (cosq + sinq) = 1 + tanq d)
9) Tarzan is swinging on a vine in the jungle. His height above the ground with respect to time is modeled
by a sinusoidal function. He is 6 m above the ground at 2 sec and 10 m above the ground at 6 sec.
a) Find an equation to represent the situation.
b) At what times is his distance 9 m? (all possible values)
CHAPTER 5 REVIEW
For the exam you should know:
· Mean, median, mode, range, standard deviation (of population and sample)
· Normal distribution curves and the 68-95-99 rule
· Sampling methods (bias and unbias)
· Using the graphing calculator to find mean, standard deviation, etc
· Confidence intervals
· Central limit theorem
· BE SURE TO KNOW THE CORRECT SYMBOLS FOR THIS UNIT!!!
1) Given the following data:
45 / 68 / 97 / 68 / 73 / 80 / 39 / 51 / 98 / 72 / 79 / 64 / 68 / 89 / 8133 / 48 / 51 / 66 / 73 / 78 / 85 / 83 / 77 / 68 / 65 / 72 / 77 / 90 / 60
a) Find the mean, median, mode, and range
b) Create a histogram
c) Does your data appear to be normal?
2) Find the standard deviation for the following: 15, 14, 18, 12, 14, 18, 20, 17
3) The January exams for math 11 were all piled together and found to have a mean of 68% with a standard deviation of 6. Assuming normal distribution, draw the normal curve and answer the following questions.
a) What percent of the students had marks between 68% and 74%?
b) What percent had marks between 56% and 74%?
c) If the results are for 240 students, how many had a mark over 74%?
4) Ryan collects a random sample of size 67 and calculates the mean to be 23.7 with a standard deviation of 3.3. Based on this sample, determine:
a. The 99% confidence interval.
b. The 95% confidence interval
c. The 90% confidence interval
5) A random sample of size 80 was selected from a large known population with a mean of 185 and a standard deviation of 27. Samples of the same size are repeatedly collected so that a sampling distribution of sample means could be drawn. Calculate the mean of the sample means and the standard deviation of the sample means.
6) A sample with a sample size of 30 is taken from a population where μ = 47 and σ = 4. The data collected are shown in the following chart.
44.7 / 47.3 / 47.2 / 50.2 / 52.2 / 48.4 / 45.6 / 51.4 / 45.3 / 56.046.0 / 49.4 / 56.1 / 41.3 / 43.1 / 49.6 / 44.6 / 49.7 / 52.1 / 44.5
50.1 / 46.0 / 43.3 / 41.0 / 47.1 / 52.2 / 48.0 / 45.5 / 48.5 / 43.4
a. What is the population mean?
b. Determine the sample mean.
a. What is the population standard deviation?
b. Determine the sample standard deviation.
c. Describe the shape of the resulting distribution if you repeatedly collected samples of the same size.
d. If you repeatedly collected samples of the same size, what would be the value of the mean of the sample means?
e. If you repeatedly collected samples of the same size, what would be the value of the standard deviation of the sample mean?.3 in the following chart.ken yrnuary. Problemo y
CHAPTER 6 REVIEW
For the exam you should know:
· Solving right triangles
· Area of a triangle
· Solving non-right triangles (Law of Sines/Cosines)
· The ambiguous case
· Application problems
1. In DABC, a = 8 m, b = 10 m, and c = 12 m. Determine, to the nearest degree, the measure of the largest angle.
2. In DABC, a = 8 m, b = 10 m, and c = 12 m. Determine, to the nearest degree, the measure of the largest angle.
3. A soccer net is 4.6 m wide. When Beckham gets possession of the ball he immediately notices that he is 12 m from one goal post and 16 m from the other goal post. Within what angle must he shoot in order to score a goal?
4. Find the area of the following triangle a = 27, b = 25 and c = 18
5. Firefighters are called to the scene of a fire that has broken out on the fourth floor of an apartment
building. Their only access to one of the apartments is through a window that is located at a height of 12m.
a) If the base of the ladder must be 5 m from the wall, how long must the ladder be in order to reach the required height?
b) What is the angle of elevation formed by the ladder with the ground?
6. Find the value of the unknown in each triangle:
a) b)
x
15
35°
c) d)
75° 48°
1.6 2.8 x 53
52°
x
7. Find the perimeter of DLMN, where M = 64°, N = 47° and l = 50km
8. Find C in DABC, where = 10cm, = 8cm, and B = 53°.