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Second Midterm Exam
C Ofli Cs
For #1-12, match the nmnbered quadratic equations in two variables with
their lettcrc(l sets of solutions. Worth point each.
1.) =
+1
2
x
y
E
2.) y
2
—
2
4.)x
=
1
y
C
5.) xy=—1
7)x
—
2
0
10.)
2
x
—
2
y
)(
=
8.)x
=
2
1
0
0
1
=
—
(
2
z
6.) =
+
2
x
—1
y
.J3
::*z
11.)
+y
2
=
I—)
3)x
—
2
—1
9.)xy=1A
0
I
12.) y=x
2
F
(
OUr_
I
K.
.
1
1-?
** ************** ** * ** ** *** ****** ** * * ** ** **** ** **
Trigonometry
13.) What is the distance between the
-
1
)
OiIltS
(—2,3) and (1, —2)?
(2)2
14.) Find the length of the unlabled side of the triangle below.
Lf
C
2
Ct5
S
c4
15.) Find sin(O), cos(6), and tan(9) for the angle
3
5
cos(e)
ti
(o)
2
5
given below. (3 points.)
16.) Find siii(9) for the angle 9 given below.
5
vq)
7
7
5
.7
(e)
5
17.) Find cos(9) for the angle 9
given
below.
3
a
2
3
2-
z(z)(3) cos (e)
13
(e)
05
I—I2
1+
34
)Zco(6)
côs(e
-
18.) Find the length c shown below,
3
c3
C
5
2
kS —2(’3)(5)
92S
2
c
-3o-
- 3tt_51
2
c
3
1D.) Write the
VeCtor
(3, —1) in l)olar coordinates.
11(3,-i))!
-
20.) Rotate the
7
poiilt
(co
7( cos(3), siri(3)) clockwise by an angle of 6.
(S-
-,
7(cos
sI
21.) Write the matrix that rotates the plane counterclockwise by an angle
of
.
Simplify your answer so that it does not contain the letters sin or cos.
/cos(Z) .-s,’7)
)
22 The matrix that rotates the plane counterclockwise by an angle of
is
I1
1
I.
Rotate the vector (7, 3) counterclockwise by an angle of
Write your answer as a row vector.
J3/
4
************************************************
Transformations of Solutions of Equations in Two Variables
The
rIl
/2
equation
1nr11i1au(’11 iibi”
=
iS
the set, of
solutions,
S, of the polyiioiiiial
3 + 3x
x
.
2
S
23.) Give an equation for 2
.
3
A(
)
(S),
_ the Tschirnhauseii cubic shifted right
3 and clown 2.
A(
(
2
_
31
s)
A (3.-2
Xx-3
U
24.) Let D
=
scaled by
D(S)
( ).
Give an equation for D(S), the Tschirnhausen cubic
in the x-coordinate and 2 in the y-coordinate.
(3o
1
/
3(3)2
3k3I2
(+
O
5
2
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * ** * * * * * * * * * * * *
Planar Transformations
Shown beh)w is a set S iii the plane. (The x- and y-axes are not part of S.
rIhey are just drawn for perspective.)
25.) Draw 3
,
2
A(_
)
(S)
_
7
7
5
5
‘1
_7_5.af --j
26.) Draw
S
3
a
I
i
-7-’
S67
I
23’5 7
-
-5
-7
()
27.) Draw Ri(S)
7
5
3
2.
.1
I
-7-&-S-’f
a
—3
-‘I.
-s
-6
-7
6
I
I
1£
(1-
3a5 7
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * *
C onics
For #2—30, draw the set, of solutions of the given equation in two variables.
(Label at least one 1
)Oillt, precisely iii #28.)
28.) xy
=
3
2+y
29.) x
2
it
=
4
30.) f+=1
It
.3
.3
2.
.44..
-2
if
.3
14
-
-2.
—3
—3
-It
************************************************
Trigonometric Functions
For #31-36, draw the graphs of the given functions.
31.) cos(6)
32.) sin(s)
33.) tan(O)
I
J/fl
—I
—1
34.) sec(s)
35.) csc(O)
\\J
—
—I
.1
I
?r
/
36.) cot(s)
‘
I
I
2r
2
a
—1
‘N
7
N
*** * *** * ***** ****** ** *** * ** ** *** * ***** ****** *** *
Equations in One Variable
The rcmaimng questious are each
sohit,ions of the given equations, aiid
the
work. If au equation has no
worth 2 points.
show your
For #37—42, find
solution, explaiui why.
37.)
9->x
log
(
9
4—x)=3
‘-f--2 :8
EET
38.) (x
—
3)2
= —1
No so
39.) 2
(x—3)
=
4
a
S
Co
Powaf
x-3=-2
[x5
8
€
q)
>
-i
>b
-(
+
‘—‘c’4
II
,%Oq)
II
o
ii’
0
1-
0
A
+
biD
+
biD
‘%
0
V
0
‘I
+
I,
1;
c
0
4-
0
+1
-
1’
+
cj
1’
“I
NJ
—2
1-
—
Ii
1-
—
II
ç%4
QjVT
1-
j
—
—
>1
(N
(%
0
a
U
II
N
I)
(s,I+
“4.
II
1-
>
..
Is
:1
c
I’
0
c%4
‘I
4-
c’4
I,
0
‘I
3
0