Seèond Midterm Exam
Instructions: You have 50 minutes to complete the test, every question
is worth one point, unless otherwise stated.
Conies
For #1-12, match the numbered quadratic equatioiis in two variables with
their lettered sets of solutioiis. Worth point each.
1.) 2
x+y
2
=
1
E
2.) x
2
4) y
2
=
1
3
5.) xy
7.)
—
10.) x
2
E
I,
1
2
X
8.) y
1
—
C
=
0
K
3.) 2
x+y
2
—1
G-
6.)
=
F
=
11.) x
2
=
—
0
D
12.) x
2
12
F.
H
1
2
+y
9.) xy
&
i<..
2
=
=
—1
=
0
A
1
1
6
T
Trigonometry
13.) What is the distance between the points (4, —1) and (—2, 3)?
-.
(
L
4.
(-
3
-.
14.) Find the length of the unlabeled side of the triangle below.
.
2
gz5
6253
5
15.) Find sin(O), cos(O), and tan(O) for the angle 0 given below. (3 points.)
Oc’v”A
2
-
16.) Find sin(O) for the angle 0 given below.
A9
1
17.) Find cos(0) for the angle 0
given
‘1
below.
/\
/
\
I- 6 i
-
60
c
S
60
18.) Find the length c shown below.
3
LeAAJ
ci
\
‘3
/1
1
V
3
19.) Write the vector (3, —2) in polar coordinates. That is, write (3, —2)
as a vector of the form r( cos(O), sin(O) ) where r > 0.
II(3,-z)l\(s l3T
(3,-2)
(3/
)
20.) Write the matrix that rotates the plane clockwise by an angle of
D1>()
-
)\
(-)
)
4
—.
************************************************
Equations in Two Variables and Their Solutions
:
The “Droplet c1uartic’ is the set of solutions, S, of the polynomial equation
= x
(1
3
—
//
_ the Droplet quartic shifted right 2
4
,
2
A(
(S),
21.) Give an equation for )
and down 4.
-
A
9o[S
(L4)(S)
22.) Let D
=
scaled by
-
(a,)
A(.2c
L
( ).
I
z (4
-
(X I 2)
Give an equation for D(S), the Droplet quartic
in the -coordinate and 3 iii the y-coordinate.
D(S)
-1)
13
C)
3
Xv— 3x
3
5
(
)2 =
(3 x
)3(
-
3 ,)
************************************************
Planar Transformations
Shown below is a set S in the plane. (The x- and y-axes are not part of S.
They are just drawn for perspective.)
_
,(S)
2
A(
)
23.) Draw 1
/
I
3
1
1
-L
..3
:
24.) Draw
()
25.) Draw R(S)
Li
2-
II
-Z
.-
-Li
—
6
Conies
-
-
Fth #26-28, Draw the set of solutions of the given equation in two variables.
(Label at least one point precisely in #26.)
26.) xy=2
28.) 4y21
27.) =
+9
2
x
y
I
I
-
1
************************************************
Trigonometric Functions
For
29.)
#29-31, draw the graphs of the
sin(O)
givell functions
31.)
30.) cos(O)
tan(O)
H
-
I
4
/
)2
II
Equations in One Variable
The remaining questions are each worth 2 points. For #35-40, find the
solutions of the given equations, and show your work.
(x
2
32.) log
33.) (3x
—
—
5)
5)2
=
=
3
Orrr\CL&p’\:
X >
9
3.-53
oR
3x53
2
3
34.)
V2
q (x)
7
—
4
—-
rfto
4oorJ
8
Ii
U’-
Um )N
>s\
•cj.)
><
4—
1V%)
--
I-
X
1.
ii
:11
I?
C)
‘1
I
4-.
S
—
I
x
I,
çt
-
I
x
+
A
II
c
S—’
ct’j
cj
0
><
0
I
N
5-
Cl
1J
><
><
x
D
+
1
************************************************
Extra credit
The following problems are worth 2 points each, with no partial credit.
38.) Compute the following product of matrices:
(3 7 (v ON /0 ON /3 7’\
o o) i 2)
I”
/3
)
(0 C)
0
2
C)
A(oo)
‘o
r(fl;X
QV11
A
(oo)
()O
+e
(o
Q
-
39.) Consider a triangle whose edges have all the same length a. Prove
that all of its angles are the same as well.
eLwof
- I-
ft
-
+o 2
2
ft
7
::=)
-
C
C-o
O
1
11
C-oA cx
2-4
ft
2
—
CAY CO4
Cjv.
OAj
1
4
6
1
(o,Tr)
s,Y,u_
¶
J U
10
oiL
tr
(o -n)
oim
CffQ
G4
1
opQ
40.) Consider the trapezoid drawn in the picture. Compute the length of
h, using the inforinations in the picture. (Observe that the triangles ABD
and ABC are equal).
‘q
J
4
Wc
f
4
A24
z
I
Cor
±
1
è.
3
A
I
rc
>
2
a
-
=
—
L
\12..
ca:
-1
/
0
CoAo
2%
11
I ii
U I Iii
.
.
2.)
14.)
.
.....
.-
..
.
3 •1
15.) sin(&)
4.)
cos(O)
5”
tai(8)
6.)
16.)
8.)
18.)
10.)
20.)
12
=
=
-
-