RYERSON UNIVERSITY
DEPARTMENT
OF
MATHEMATICS
MTH 210
Midterm Test
February 27, 2009
Total marks: 65
Time allowed: 110 Minutes
NAME (Print):
STUDENT #:
Circle your Lab Section:
011
VIC 200
021
VIC 205
031
VIC 608
Instructions:
Verify that your paper conta
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 210 FINAL EXAM - WINTER 2005
NAME:
STUDENT ID:
SECTION:
Section Lab
TA
Section Lab
TA
1
Thursdays at 1
Anca
3
Thursdays at 4
Chris
2
Thursdays at 5
Chris
4
Fridays at 9
Anca
INSTRUCTIONS
This exam has 6 pag
MTH 210 Midterm Test I Solutions
1. Prove by induction that
n
i(i 1) =
i=2
n(n 1)(n + 1)
3
Proof: (By induction on n)
Base Case Let n = 2,
n = 2.
2
i=2
i(i 1) = 2 (2 1) = 2, 1 (2(2 1)(2 + 1) = 2, so true for
3
k
i=2
Inductive Step Let n = k 2 Assume true
MTH 210
W2005 MIDTERM SOLUTIONS
1 of 3
Question A Recursion and Induction 20 Marks
Given the sequence an defined with the recurrence relation:
a0 = 1
an = n (an-1)2 for n 1
Terms of a Sequence (5 marks)
a 1 = 1 . 12 = 1
a 2 = 2 . 12 = 2
a3 = 3 . (2 . 12)2
RYERSON UNIVERSITY
DEPARTMENT
OF
MATHEMATICS
MTH 210
Midterm Test I
Total marks: 50
March 3, 2006
Time allowed: 110 Minutes
NAME (Print):
STUDENT #:
Circle your Lab Section:
Monday 11
BUS 300
021
Wednesday 10
EPH 112
011
Wednesday 10
BUS 210
031
Instructi
6.1 - 6.3
Counting
P. Danziger
Given a set S we will use |S| for the number of elements of S.
1
Simple Probability
A random process is a repeatable process (series of events) whose outcome follows a (known) probability distribution.
The sample space, S, i
MTH 210
1.
Midterm Test Solutions
b
a t
c
t
d
d
t
t
d
d
t
d
t
x
dt
t
z
w
y
G
For the graph G given above give the following, or explain why the graph does not have
the given property. (Quote any theorems you use.)
(a) Find the order, size, total degree an
6.8, 6.9
Probability
P. Danziger
1
Probability
Up to now we have been using the assumption of equal likelyhood, we want a more general denition
of probability.
Denition 1 A probability function, P , on a sample space S is a function which maps events
(sub
6.5 - 6.7
Permutations & Combinations
P. Danziger
1
Permutations
Given a set of objects we may consider how many ways there are of writing that set in order.
Alternately, we may ask how many ways we may rearrange the elements of a set.
If the set has n el
MTH 210 Midterm Test I Solutions
a
t
1.
e1
tb
'
'$
$
t
e2
c
&%
&
t
d
e3
%
G
For the graph G given above give the following, or explain why the graph does not have the given property.
(Quote any theorems you use.)
(a) On the graph circle the connected comp
RYERSON UNIVERSITY
DEPARTMENT
OF
MATHEMATICS
MTH 210
Midterm Test
Total marks: 50
March 1, 2007
Time allowed: 110 Minutes
NAME (Print):
STUDENT #:
Circle your Lab Section:
011
ENG 102
021
ENG 105
031
LG 06
Instructions:
Verify that your paper contains 6
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 210 MIDTERM - WINTER 2005
NAME:
STUDENT ID:
SECTION:
Section Lab
TA
Section Lab
TA
1
Thursdays at 1
Anca
2
Thursdays at 4
Chris
3
Thursdays at 5
Chris
4
Fridays at 9
Anca
INSTRUCTIONS
This exam has 5 pages