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MA  262  Linear Algebra and Differential Equatio...
MA  262  Linear Algebra and Differential Equations  exam MA262 — FINAL EXAM — SPRING 2016 — MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet ﬁll in the information below and use a # 2 pencil to ﬁll in the required information on the scantron. 3. MARK YOUR TEST NUMBER ON YOUR SCANTRON 4. Once you are allowed to open the exam, make sure you h
MA  262  Linear Algebra and Diffe...
MA  262  Linear Algebra and Differential Equations  exam MA262 — FINAL EXAM — SPRING 2016 — MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet ﬁll in the information below and use a # 2 pencil to ﬁll in the required information on the scantron. 3. MARK YOUR TEST NUMBER ON YOUR SCANTRON 4. Once you are allowed to open the exam, make sure you h
Page 3
3.
Let
y
(
x
) be the solution to the initial value problem
dy
dx
=
y
x
(ln
x
)
2
and
y
(
e
)=
e.
Then ln
y
(
e
2
) is equal to
A. 2
B. 3
C.
3
4
D.
3
2
E. 1
4.
Let
y
(
x
) satisfy the following initial value problem
(3
x
2
e
xy
+
x
3
ye
xy
)
dx
+(
x
4
e
xy
)
dy
=0
,
y
(1) = 0
.
Then
y
(2) is equal to
A. ln 2
B.

ln 2
C.
3
4
ln 2
D.

2
3
ln 2
E.

3
2
ln 2
Page 4
5.
Let
y
(
x
) satisfy the following initial value problem
y
′′
+
y

1
(
y
′
)
2
=
y

1
e

y
y
′
,
y
(0) = 1
,y
′
(0) = 1
.
Then
V
(
x
)=
y
′
(
x
) satisﬁes
A.
V
(
x
)=
y

1
(1 +
1
e

e

y
)
B.
V
(
x
)=
1
2
y

1
(2 +
1
e

e

y
)
C.
V
(
x
)=

y

1
e

y
+
1
e
+1
D.
V
(
x
)=
y

1
(1

1
e
+
e

y
)
E.
V
(
x
)=
3
2
y

1
(
2
3

1
e
+
e

y
)
6.
The rank of
1
2
1
4
0
1
3
1
3
7
6
13
2
5
5
9
is equal to
A. 3
B. 4
C. 1
D. 0
E. 2
Page 5
7.
Let
T
:
R
3
→
R
5
be the Linear transformation given by
T
(
x
)=
Ax,
where
A
=
1
2
3
2
1
4
1

1
1
0

3

2
1
1
1
.
Then the dimension of the range of
T
is equal to
A. 1
B. 2
C. 3
D. 4
E. 5
8.
Let
A
=

1
0
0
1
5

1
1
6

2
. Let
λ
1
,λ
2
and
λ
3
denote the eigenvalues of
A
and let
E
1
,E
2
and
E
3
denote the corresponding eigenspaces. Which of the following is correct?
A.
λ
1
=2
,λ
2
= 3 and
λ
3
=

1
,
dim
E
1
= dim
E
2
= dim
E
3
=1
B.
λ
1
=
λ
2
=

1
,λ
3
=4
,
dim
E
1
= dim
E
2
=2
,
dim
E
3
=1
C.
λ
1
=
λ
2
=

1
,λ
3
=4
,
dim
E
1
= dim
E
2
=1
,
dim
E
3
=1
D.
λ
1
=1
,λ
2
=
λ
3
=4
,
dim
E
1
=1
,
dim
E
2
= dim
E
3
=2
E.
λ
1
=
λ
2
=

1
,λ
3
=3
,
dim
E
1
= dim
E
2
=2
,
dim
E
3
=1
Page 6
9.
Find all values of
α
such that the vectors (1
,
0
, α/
4
,
1)
,
(2
,
0
,
1
,α
) and (1
,
0
,
1
,
2) are linearly
dependent
A.
α
= 1 and
α
=2
B.
α
= 2 and
α
=6
C.
α
= 2 and
α
=4
D.
α
=

2 and
α
=4
E.
α
= 3 and
α
=6
10.
Let
A
and
B
be square matrices such that det(
A
) = 5 and det(
A
+
B
) = 20
.
We conclude that
det(
I
+
A

1
B
) is equal to
A. 4
B. 3
C. 5
D. 10
E. 8
Page 7
11.
Let
M
3
(
R
) be the space of 3
×
3 matrices with real entries and let
S
be the subspace of
M
3
(
R
)
such that the sums of the elements of each row is equal to zero. The dimension of
S
is equal to
A. 3
B. 4
C. 1
D. 2
E. 6
12.
A particular solution to the equation
(
D
2

2
D
+ 10)
2
y
=
e
x
sin 3
x
+ cos 2
x,
(which does not contain terms that solve the homogeneous equation) has the form
A.
y
p
(
x
)=
e
x
(
A
1
cos 3
x
+
B
1
sin 3
x
+
A
2
x
cos 3
x
+
B
2
x
sin 3
x
+
A
3
x
2
cos 3
x
+
B
3
x
2
sin 3
x
)+
A
5
cos 2
x
+
B
5
sin 2
x
B.
y
p
(
x
)=
e
x
(
A
1
x
cos 3
x
+
B
1
x
sin 3
x
+
A
2
x
2
cos 3
x
+
B
2
x
2
sin 3
x
)+
A
3
cos 2
x
+
B
3
sin 2
x
C.
y
p
(
x
)=
e
x
(
A
1
cos 3
x
+
B
1
sin 3
x
)+
A
2
cos 2
x
+
B
2
sin 2
x
D.
y
p
(
x
)=
e
x
(
A
1
x
2
cos 3
x
+
B
1
x
2
sin 3
x
)+
A
2
cos 2
x
+
B
2
sin 2
x
E.
y
p
(
x
)=
e
x
(
A
1
x
cos 3
x
+
B
1
x
sin 3
x
)+
A
2
cos 2
x
+
B
2
sin 2
x
Page 8
13.
Find the general solution to the diﬀerential equation
y
(4)

8
y
′′
+ 16
y
=0
.
A.
y
=
c
1
e
2
x
+
c
2
e

2
x
B.
y
=
c
1
xe
2
x
+
c
2
xe

2
x
C.
y
=
c
1
e
2
x
+
c
2
e

2
x
+
c
3
xe
2
x
+
c
4
xe

2
x
D.
y
=
c
1
xe
2
x
+
c
2
xe

2
x
+
c
3
x
2
e
2
x
+
c
4
x
2
e

2
x
E.
y
=
c
1
cos 2
x
+
c
2
sin 2
x
+
c
3
x
cos 2
x
+
c
4
x
sin 2
x
14.
Which of the following is a Green’s function of the diﬀerential equation
y
′′

3
y
′
+2
y
=
F
(
x
)?
(
K
(
x, t
)=
1
W
[
y
1
,y
2
](
t
)
(
y
1
(
t
)
y
2
(
x
)

y
2
(
t
)
y
1
(
x
)))
A.
K
(
x, t
)=(
e
2
x
+4
t

e
x
+5
t
)
B.
K
(
x, t
)=(
e
2(
x

t
)

e
(
x

t
)
)
C.
K
(
x, t
)=(
e
3(
x
+
t
)

e
x
+
t
)
D.
K
(
x, t
)=(
e
4(
x

t
)

e
5(
x

t
)
)
E.
K
(
x, t
)=(
e
3(
x

t
)
+
e
2(
x

t
)
)
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