Math 1060: Final Exam
First Name:____________________________ Last Name:______________
Linear Algebra
13
2\(1
—4) —3
1.) Give the vector written as a row j
2.) Find the product
(
)( )
Conics and other solutions of equations in two variables
3.) Give the equation for a line of slope
that passes through the origin.
4.) Give the equation for a line of slope
that passes through
the point (2,3).
3
-
2.
5.) Give the equation of the unit circle.
I
6.) Give the equation of the circle of radius 3 ccntered at the origin.
7.) Give the equation of the circle of radius 3 centered at the point (—2, 3).
8.) Give the equation of the ellipse obtained by starting with the unit circle,
and then scaling the x-axis by 4, and the y-axis by 2.
9.) Give the equation of the ellipse from #8, shifted right by 1 and
up by 5.
zz
I
-.3
S
10.) Draw the set of solutions of the equation y
=
.
2
x
the set of solutions of the equation x
=
.
2
y
1L)
Draw
12.) Draw the set of solutions of the equation xy
=
1.
2
13.) Draw the set of solutions of the equation y
2
14.) Draw the set of solutions of the equation x
15.) R
14
=
(
2
y
=
1.
=
1.
is the rotation of the plane by angle
2 is the set of solutions of xy
If H C R
=
.
1, then draw R(H).
x from #13. What’s the equation
16.) Let P the set of solutions of y = 2
for P shifted right by 1 and down by 3?
I
3
I
************************************************
Trigonometry
17.) What is the distance between the points (3, 5) and (—1, 8)?
18.) Find the length of the unlabeled side of the triangle below.
LNNN
19.) Find sin(0), cos(8), and tan(O) for the angle 6 given below. (3 points.)
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.) \Vhat’s arcéin(1)?
32.) Whats arctan(—1)?
Match the functions with their graphs.
33.) sin(x)
36.) sin(—x)
39.) cos(x)
sin(x)
42.)
34.)
37.)
40.)
43.)
—
A.)
35.) sin()
38.) sin(x)
41.) 2sin(x)
44.) sin(2x)
sin (x+)
sin(x) + 1
sin(x—)
sin(x)
1
—
C)
3)
—,
P.)
E.)
Dj
I
2.
-t
I.)
H.)
z
—I
J.)
a
2
—z
a
2
Match the fu.nctions with their graphs.
45) f(x)
=
sin(x)
cos(x)
< 0;
f
X
if
x > 0.
46.) g(x)
=
cos(x)
sin(x)
B.)
A.)
Complex numbers
47.) Find (1 + i3) + (2
—
i4).
48.) Find (2+i3)(1+i5).
49.) WIiat’s the norm of 5 + i3.
50.) Find 3(cos(—2) + isin(—2))4(cos(5) + isin(5)).
if x <0;
if x > 0.
51.) (2 points) Draw the number z
=
(
2 cos
() + i sin (i)) on your answer
sheet. Draw an X on your answer sheet on the number 3(cos(7r)+isin(ir))z.
52.) Find (+i)
.
4
53.) Find
+
Equations in one variable
7 state how many solutions the equation
For each of tile following equations
can have at most.
2
54.) x
55.)
—
2x + 1
=
0
/1O+eX=7
56.) sin(x)
=
57.) cos(x)
=
58.) loge(x)
=
4
5
See the answer sheet for #59-64.
First Name:__________________________ Last Name:
11.)
1..)
2.)
3.)
12.)
4.)
5.)
6.)
7.)
13.)
8.)
9.)
10.)
14.)
15.)
17)
18.)
19.)
20.)
16.)
************************************************
23.) tan(8)
22.) sin(s)
2L) cos(9)
2
2
2
—r--
‘r
----!‘-—
I
I
1-
1.
—I
—I
-2
-l
2
26.) cot(9)
25.) csc(6)
24.) sec(6)
2
2
2
I
I
I
I
--—-)--q----
r-
I
a.
a.
—I
—2
—I
-1
—2
c.,wi(x)
v(x)
27.)
;::)
28.)
c,kc-(x)
29.)
-
-
—t.-,
30.)
31.)
32.)
33.)
41.)
42.)
43.)
44.)
34)
45.)
35.)
46.)
36.)
47.)
37.)
48.)
38.)
39.)
40.)
49.)
50.)
51.)
/;
/
56.)
52.)
53.)
57.)
54.)
58.)
55.)
#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. Each problem is worth two points.
For
the
remaining questions,
(ex) + 3(ex)
59) 2
—
4
=
0
60.) 1og(x+1)+1og(x—3) =0
X_le
2
e
61.) 7
=
1
1
62.) (x
63.)
—
5)2
2 =
e2T
2
64.) (x
—
=
9
—1
1)(2x —4)
=
2
(x
—
1)(x
—
3)