Practice Final Exam
Linear Algebra
12
1.) Give the vector written as a row
—3
1N14
o)
—2
(,-i2)
34o
2.) Find the product
(I+
( (
3)
4)
4C)
-(;
Q
0
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Conics and other solutions of equations in two variables
3.) Give the equation for a line of slope 2 that passes through the origin.
I
/
4.) Give the equation for a line of slope 2 that passes through
the point (3,4).
5.) Give the equation for the unit circle.
2.
2.
(N
6.) Give the equation for the circle of radius 4 centered at the origin.
xl
C
7.) Give the equation for the circle of radius 4 centered at the point (5. 7).
8.) Give the equation for the ellipse obtained by starting with the unit
circle, and then scaling the x-axis by 3, and the y-axis by 2.
()i
()2•
9.) Give the equation for the ellipse from #8, shifted right by 4 and
upby5.
7
:H
0
5
3
‘1-
7
10.) Draw the set of solutions of the equation xy
2
11.) Draw the set of solutions of the equation x
2
12.) Draw the set of solutions of the equation y
=
—
1.
2
y
1.
—
13.) Draw the set of solutions of the equation y
=
.
2
x
14.) Draw the set of solutions of the equation x
=
.
2
y
,1
1
(//
4
15.) R_-/
is the rotation of the plane by angle
=
—.
‘Lii
\//
If
H C
2 is the set of solutions of xy
R
=
1, then draw R_(H).
2 from #13. Give the equation
16.) Let P the set of solutions of y = x
for P shifted right by 2 and up by 3.
3
U
2
jz
()
.3
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Trigonometry
17.) What is the distance between the points (2, 5) and (3, 8)?
18.) Find the length of the unlabeled side of the triangle below.
19.) Find sin(s), cos(O), and tan(E) for the angle
given below. (3 points.)
3
Lf
20.) What’s the Pythagorean Identity?
For #21-29, graph cos(x), sin(x), tan(x), sec(x), csc(x), cot(x), arccos(x),
arcsin(x), and arctan(x).
30.) What’s arccos
()?
31.) What’s arcsin(—1)?
32.) What’s arctan(—1)?
-
Match the functions with their graphs.
33.) sin(x) /\
J
36.) sin
39.) cos(x)
42.) srn(x) H
()
,-
—
35.)
38.)
41.)
44.)
—
A.)
B)
a
it
1
IT
2
—I
—I
-2
-z
D.)
—
4\2
}
V
sin(2x) T
sin(x) 11)
2sin(x)C
sin(—x)
C.)
a
a
Dl
—2
E.)
2.
a
—7T
)
.)
34.) sin (x +
37.) sin(x) + iE
fE
40.) sin(x
43.)
sin(x) p
2.
7Th
1T
2
-z
G.)
H.)
I.)
2
2
Fir
LI
-2
2
-7rf\-
—I
-2
-2
J.)
2
—I
—2
2
7r
-2-
Match the functions with their graphs.
45.)
(
=
f(x)
cos(x)
if
xE [0.oo).
A.)
46.) g(x)
=
cos(x)
sin(x)
B.)
Complex numbers
47.) Find (2+i3)+(4+i5).
48.)Find(2+i3)(1+i4).
2 -i2.
49.) Whats the norm of 7+ 13.
50.) Find 2(cos(3) + I sin(3))4(cos(8) + isin(8)).
if x E (—Dc, 0);
if xE [0,Dc).
5L) Draw an X on your answer sheet on the number 3(cos ()+i sin ())z.
(The number z is drawn on the answer sheet.)
52.) Find
4
if
2.
2
3
53.) Find
.
7
(+)
: (I)
-
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Equations in one variable
Write the solutions of the following trigonometric equations.
54.) tan(x)
=
55.) cos(x)
=
56.) cos(x)
=
57.) sin(x)
=
58.) sin(x)
=
+
—4
1
,fJA(
7
)4
-
—
Z’c TT
-
C
()
ccon ()
e1i.- ( ) 2ev cJ
—2
(- L)-’
See the answer sheet for #59-64.
¶
-
9
()
LL)f
First Name:
1.)
Last Name:
( -4z
11.)
-6
2.)
3.)
t-zo
2x
(x-3)
4.)
12.)
5.)
6.)
7.)
Xyz
(X
-5)
I
2.
2.
4
(y—’?)
L
8.)
() (j)
x-•’1
(——)
13.)
//
-
9.)
2
10.)
14.)
/Z
/
/
—
Iz
.1—
Is)
Is)
—I
—
I
I
%NN
-I---’
iN
_)
I
—
-
—
1-
-
(D
*
*
*
*
*
*
*
*
*
*
*
*
*
*
I
4.
N
Is)
7
D
cD
0
‘.—,
N
N
27.) arccos(x)
29.) arctan(x)
28.) arcsin(x)
7r
3
—‘
3.
3.
-air
30.)
41.)
I
7T
31.)
72.
42.)
T’
43.)
32.)
b
33.)
A
44.)
34.)
6-
45.)
A
35.)
J
46.)
B
36.)
47.)
37.)
48.)
38.)
r:
49.)
39.)
(:
50.)
40.)
—
\S 8
(oA (A)
-1)
44
51.)
((o4
..
4:
;r
4
52.)7
56.) i.un (y..2ft-E
J
ccr;() 4
—
53.)
-g
54.)
);icv) 4ry-?rr
57) :j4j
iT-
-
Zrii
-
‘vJ
LrV
().4. 2frt
(XJC&’-
58.) ;fljJj
55.)
ftA.J
I
L) -f 2iT
yJ;A-(-.
)
vwJ
.
—
(i’r’
-.
138)
For the remaining questions, #59-64, give the solutions of the equations.
If there is no solution, explain why there is no solution. For at least one of
the remaining problems, the domain of the equation will play an important
role in the solution.
C’
V
-v-j
0
1
-.
+
C
cD
¶;
+
‘I
1’
0
0
ii
0
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—
I’
0
U
.1
:-
,..\
0
C’
;:‘
c
c
F-..’
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-
)(
;=:
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I,
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rfl
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)<
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C
1
0
a
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0
-p