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Math 11A Exam Review – January 2011
Chapter 1 – Investigating Equations in 3-Space
1. Mary charges a base rate of $6 and an additional $2/hour to baby sit. Marg charges $7/hour. Kara charges a flat rate of $20.
a) Write equations representing each of the sitter's charges.
b) Marg and Mary earned the same amount and babysat for the same number of hours. For how many hours did they baby
sit? How much did they earn?
c) How much would Kara have earned for the same number of hours?
d) After how many hours would all three sitters earn the same amount? Show this algebraically and graphically.
e) Which sitter should the parents contact if they expect to be out for approximately five hours?
2. Jacob joined Nobody's Health Club this fall which charges a base monthly fee as well as a fee per visit. In September he
went to the club seventeen times and was charged $60.50. In October he went only nine times and was charged $40.50.
a) Write equations representing the amounts paid each month.
b) Find the club's base and per-visit fees.
3. Solve each of the following systems of equations algebraically:
a) x = –4y + 3
b)
c) 4x – 3y + z = –9
6
3
2x + 3y = –4
–2x + y + 3z = –8
x  3y  
5
2
6x – 2y + 2z = –5
3x 
3
117
y 
2
4
4. Solve the following system graphically: x – 2y = 10 , 3x + 2y = 6.
5. Solve the following using matrices (by hand and then by calculator):
4x – 5y = –32, 3x + 2y = 22
6. A system of equations can have no solution, one solution, or infinite solutions.
a) Give examples of systems in two variables which demonstrate each of these situations.
b) Solve your systems algebraically to obtain the resulting answers.
c) Explain what each of the solution types would mean graphically.
7.
Jane found that if she studied for tests, she did better, but if she studied too much, her improvement was minimal. Here are
her latest results: two hours of studying: 85% on test, three hours studying: 95% on test, five hours studying: 96% on test.
a. Assuming her results can be modeled quadratically, set up the equations which could be used to find the function
expressing her results on her tests versus the number of hours spent studying.
b. Using a calculator, solve the matrix to find the quadratic equation.
c. Find what Jane’s grade would be if she studied for only one hour.
Chapter 3 – Sinusoidal Functions
1. Write the term which best describes the graph given ("sinusoidal", "periodic", or "neither").
a)
b)
c)
d)
2. Identify the transformations that have occurred on y = sin x to give the following:
a) –2(y – 5) = sin [ 1/4 (x + 30o)]
b) y = 3 sin (2x – 90o) + 1
3. Given that (x, y)  (3x + 10o, –4y – 5) is the mapping rule for transformations on y = cos x,
a) state the resulting equation
b) name the amplitude, period, and equation of the sinusoidal axis
4. Graph the equation 1/6 (y – 9) = cos [2 (x – 20o)] using the mapping rule.
5. Given the following graph, find four different possible equations.
y
x
6. You are swinging back and forth on a swing. Assume that y (distance in cm from the centre of the swing while in resting
position) varies sinusoidally with t (time in sec the stopwatch reads). When t = 2, you are at one end your swing, where y =
120. It takes you only 1.5 seconds to reach the other end, where y = –120.
a) Find an equation representing this situation.
b) Find your position when you started the stopwatch.
Chapter 4 – Trigonometric Equations
1.
2.
a)
3.
Fully label the unit circle.
Determine the exact values of the following, using the unit circle:
cos 150o
b) tan 210o
c) sin (–60o)
d) cos 600o
Evaluate, giving the exact values:
a)
3 cos120 sin 240
cos180
4.
a)
Find the approximate value(s) for x in the equation –1 = 5 cos [3(x–10o)] – 3 using the graph:
x  [0o, 180o]
b) x  (–, )
y
b) (sin2225o)(sin245o)
c)
5
cos150

3 sin 60 sin 225
x
5.
a)
Solve for x in the domain given:
sin x tan x + sin x = 0 , x  (–, )
c) cos 2 x 
6.
7.
8.
b) 3 – 3 sin x – 2 cos 2 x = 0 , x  [–180o, 180o]
 3
, x  (–, )
2
Name the 8 basic trigonometric identities studied.
Solve this system of equations algebraically y  2  2 sin x , y  2 sin x
solution graphically.
Solve for x (in degree mode, unless otherwise indicated).
using degree measure and then illustrate the


a) 2 cos x   3 , x   ,  
g) sec x  2 cos x  tan x  0 , x  0 ,360
b) cos x  2 sin x  1  0 , x 0 ,360
h) 3 tan x  4 sec x  4 , all solns radian mode
2
2
 
i) cos 3x  
c) sin x  cos x , x  0, 
d)
cos2 x
 1 , x    ,  
1  sin x
2
, x 60 ,60
2
j) sin2x     1 , x , 
 x
 2
2
e) cot x  csc x  1  0 , x 0,2  

k) cos2   

f) sin x  cos x cot x  2 , x  360 ,360
3
, x , 
4
2
l) cos x  0.63 cos x  0 , x 360 ,360
9.
Simplify the following.
4 x 2  4 xy
3xy  3y 2

x 2  2 xy  y 2 x 2  2 xy  y 2
1
1
c) 2
 2
x  2 x  3 x  x  12
x
x3  6x2  5x

x7
x2  49
2x
5
d) 2
 2
x  3x  2 x  1
a)
b)
10. Verify the following identities.
a) tan x  cot x  sec x csc x
g) sec 2 x csc 2 x  sec 2 x  csc 2 x
b) csc2 x  cos2 x csc2 x  1
h)
c) sin x cos x  sec x sin x   sin2 x tan x
i)
cos x
1  sin x

 2 sec x
1  sin x
cos x
d) 5sin2 x  3cos2 x  3  2 sin2 x
j)
tan2 x
 sin2 x
1  tan2 x
e)
cos4 x  sin 4 x
 cos x  sin x
cos x  sin x
k)
f)
sec x sin x

 cot x
sin x cos x
11. a) Convert to radian measure: 25 degrees
cos x sec x
 cot x
tan x
cos x
 cot 2 x1  cos x
sec x  1
b) Express in degrees:

7
12. Evaluate, leaving in simplest radical form:
a)
 10 
 3 
sin
  tan 
 3 
 4
b)
3
 
1  6 cos 
 4
13. Give the arclength, in a circle of radius 5 cm, intercepted by a central angle of 72 .
14. Suppose that the waterwheel shown rotates at 6 revolutions per minute (rpm). Two seconds after staring your stopwatch
point P on the rim of the wheel is at its greatest height. Assume that the distance (d) of point P from the surface of the water
varies sinusoidally with the number of seconds (t) the stopwatch reads.
a) Sketch the graph
b) Write a possible equation for this sinusoid.
c) Determine how far P is from the water when the stopwatch reads 5.5 seconds.
d) Find when point P is 1.5 metres from the surface of the water.
15. A railway company is preparing to build a new line through Rolling Mountains. You set up a Cartesian coordinate system
with its origin at the entrance to the tunnel through Bald Mountain. Your survey crew finds that the mountain rises 250
metres above the level of the track and that the next valley goes down 50 metres below the level of the track. The crosssection of the mountain and valley is roughly sinusoidal with a horizontal distance of 700 metres from the top of the
mountain to the bottom of the valley
e) Write the equation expressing the vertical distance from the track versus the horizontal distance from the tunnel
entrance. (Hint: you will have to use substitution to find the missing variable(s).)
f) How long will the tunnel be?
g) How long will the bridge be?
Chapter 5 – Statistics
1.
Find the mean, median and standard deviation for the following sample. Show all your work do not use graphing calc
technology.
12
14
18
11
25
10
22
21
15
12
26
21
13
14
20
2.
What percentage of data in a normal distribution would be contained within a) 1 standard deviation of the mean b) 2 standard
deviations of the mean c) 3 standard deviations of the mean ?
3.
A population is found to have a mean value of 110 with a standard deviation of 5. What percentage of the data would
measure between
a) 100 and 115 b) 105 and 115 c) less than 110 d) 95 and 120
4.
McDonalds wants to check the weights of its ¼ pounder hambuger for quality control reasons. Using the 6 unbiased
sampling methods describe how you would set up sampling for each.
5.
Explain the three components of the Central Limit Theorem.
6.
A random sample of 74 is taken from a known population where the mean is 64 and the standard deviation is 7. Samples of
the same size are repeatedly collected.
a) Determine the mean of the sample means
b) Determine the standard deviation of the sample means
7.
8.
Determine a 99% confidence interval for the population with a mean of 23.5 and a standard deviation of 3.2 given the
following sample.
20.3
27.2
30.1
27.9
23.6
23.2
27.6
26.6
22.8
23.4
28.8
25.3
25.6
23.4
25.7
Construct a 95% confidence interval for the following data and state the margin of error:
10.2
7.3
7.8
8.6
8.5
8.5
7.8
8.6
8.5
9.6
7.6
8.1
8.1
9.6
9.4
8.9
10.3
10.0
6.7
7.1
8.8
9.4
9.3
7.2
8.5
5.6
7.5
7.8
10.9
8.1
9.1
9.2
7.7
10.3
9.
Calculate the margin of error for question 8.
Chapter 6 – Trigonometry and its Applications
1.
Find the area of ABC if a = 7.5 cm, b = 9 cm, and C = 100o.
2.
State the Law of Sines and the Law of Cosines.
3.
Find the smallest angle of the triangle with sides 7, 9, and 12.
4.
Solve ABC (that is, find the measures of all sides and angles) given: A = 35 o, b = 32, c = 41
5.
Find B in ABC if: A = 40o, a = 15, and b = 30
6.
The longest car ferry is the 173 metre Norland operating on the North Sea. What angle would her length subtend when
viewed from a point at sea which is 300 m from her bow and 220 m from her stern?
7.
An oil well is to be located on a hillside that slopes at 10 o. The desired rock formation has a dip of 27 o to the horizontal in the
same direction as the hill slope. The well is located 3200 feet downhill from the nearest edge of the outcropping rock
formation. How deep will the driller have to go to reach the top of the formation?