Math 1104-C Fall 2011
Course Schedule
Course Information
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Syllabus: Math 1104C
Instructor: Jeremy Macdonald
Office
Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Senior Math Circles
An Introduction to Inequalities
February 20, 2013
Inequalities in Geometry:
Here are some rudimentary facts:
1) A line is the shortest distance between two points.
2) Thus, we have that
Canadian
Mathematics
Competition
An activity of the Centre for Education
in Mathematics and Computing,
University of Waterloo, Waterloo, Ontario
Euclid eWorkshop # 3
Solutions
c 2006 C ANADIAN M ATHEMATICS C OMPETITION
Euclid eWorkshop #3
S OLUTIONS
S OLU
Canadian
Mathematics
Competition
An activity of the Centre for Education
in Mathematics and Computing,
University of Waterloo, Waterloo, Ontario
Euclid eWorkshop # 4
Trigonometry
c 2006 C ANADIAN M ATHEMATICS C OMPETITION
Euclid eWorkshop #4
T RIGONOMETRY
An Introduction to Inequalities:
Solutions
1. For any real numbers x,y>1:
x2
y2
+
8
y1 x1
Solution: multiply both sides by (y 1)+(x1) so by Cauchy Schwarz (y 1)+(x
y2
x2
+
) ( x2 + y 2 )2 = (x+y)2 . Now we just want (x+y)2 8(x+y2)
1)(
y1 x1
and its easy t
Canadian
Mathematics
Competition
An activity of the Centre for Education
in Mathematics and Computing,
University of Waterloo, Waterloo, Ontario
Euclid eWorkshop # 2
Solutions
c 2006 C ANADIAN M ATHEMATICS C OMPETITION
Euclid eWorkshop #2
S OLUTIONS
S OLU
Canadian
Mathematics
Competition
An activity of the Centre for Education
in Mathematics and Computing,
University of Waterloo, Waterloo, Ontario
Euclid eWorkshop # 1
Logarithms and Exponents
c 2006 C ANADIAN M ATHEMATICS C OMPETITION
Euclid eWorkshop #1
L
Math 1104C Term Test 4
November 21, 2011
Name:
Tutorial (circle): C1
Student ID:
C2
C3
C4
Time: 50 minutes
Note: To receive full marks, you must show your work.
1. (15 marks) For each of the following sets of vectors, determine whether the vectors are lin
Additional practice problems Chapter 3
110
1. Consider the matrix A = 0 1 1 . Find A1 and write A and A1 as products of elementary
232
matrices.
2. Find the matrix A that satises
3(A1 )T 2
3. Let A be a 3 3 matrix such that A1
1
3
2
= 0
1
2
4
1
=
1T
A.
2
Additional practice problems Chapter 8
1. Let z = 4 3i and w = 2 2i. Write each of the following in standard form a + bi.
(a) z 4w
(b) 2iz
(c) zw
(d) z 2
(e) z z
(f)
z
w
Answers
1.
z 4w
=
(4 3i) 4(2 2i) = 4 3i + 8 + 8i = 12 + 5i
2iz
=
2i(4 3i) = 8i 6i2 =
Additional practice problems Chapter 4
1. Let A and B be 2 2 matrices with det(A) = 1 and det(B ) = 2. Compute the determinants of the
following matrices:
(a) A1 BA
(b) A2 B 2
(c) AT (3B ).
ab
2. Let A = p q
xy
c
r and suppose that det(A) = 3. Compute the
Math 1104 Practice Final Exam
Disclaimer: This practice exam does NOT intend to cover all topics that may appear on the nal exam.
Any topic covered during the course may be tested on the nal. Solving these problems alone does not
constitute sucient prepar
Math 1104 Practice Final Exam
Disclaimer: This practice exam does NOT intend to cover all topics that may appear on the nal exam.
Any topic covered during the course may be tested on the nal. Solving these problems alone does not
constitute sucient prepar
Theorem (Invertible Matrix Theorem, Parts 1-4). Let A be an n n matrix.
Then the following statements are equivalent.
(a) A is invertible.
(b) AT is invertible.
(c) For every b Rn , the system Ax = b has a unique solution.
(d) The homogeneous system Ax =