MATHEMATICS 23a, FALL 2007
Practice Quiz # 1
October 2007 Solutions by Julian Bouma
For purposes of this quiz, treat the following approximations as exact:
cos 37
o
= sin 53
o
=
4
5
,
cos 53
o
= sin 37
o
=
3
5
1. The answer is (c). The angle
α
between vectors
~
v
and
~
w
is defined as
cos
α
=
~
v
·
~
w

~
v

~
w

4
2
·
2

1
is equal to the dot product 4
·
2 + 2
·
(

1) = 6, and the lengths
of the vectors are
√
4
2
+ 2
2
and
p
2
2
+ (

1)
2
, making the expression for
the angle
arccos
6
√
20
·
√
5
= arccos
3
5
= 53
o
2. The answer is (d). From the equations
T
[1]
3
0
=
[2]
3
[1]
3
T
0
[1]
3
=
[1]
3
[1]
3
you can deduce that [T] is equal to
[2]
3
[1]
3
[1]
3
[1]
3
.
The value of
~
v
that
T~
v
=
0
[1]
3
is
[2]
3
[2]
3
3. The answer is (d). All numbers in the set satisfy (2
n
2
> m
2
) = (2
>
m
2
/n
2
) However, since there are an infinite amount of rational numbers
that satisfy this property and thus there is no greatest element, the least
upper bound must be a number outside the set but is still greater than
any element in the set, namely
√
2.
4.
(a) Original:
∃
T
∈
S
such that
∀
A
∈
S, TA
=
A
Negation:
∀
T
∈
S,
∃
A
∈
S
such that
TA
6
=
A
(b) The original given statement
P
is true. What it is saying is that
there exists a 2 x 2 matrix such that for any 2 x 2 matrix, the
first matrix times the second matrix will equal the second matrix.
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 Spring '10
 Bamberg
 Real Numbers, Approximation, Vectors, Rational number, Countable set

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