Math 111 2015-16
Assignment 6 Solutions
Due Thursday Jan 21, 2016
1. Find a formula for the nth term of the sequence defined by the third-order recursion
tn+2 = tn+1 + 10tn + 8tn1
with initial values t0 = 1, t1 = 7, t2 = 11. Check that your answer works f
Math 111 2016-17
Assignment 8 Solutions
Due Thursday Feb 9, 2017
1. Find the solution to the dynamical system:
xn 1 1 2 1 xn
y 2 1 1 y
n 1
n
zn 1 3 0 1 zn
x0 0
y 1
0
z0 0
Note that you already found the eigenvalues and the eigenvectors in A
Math 111 2016-17
Assignment 7 Solutions
Due Thursday Jan 26, 2017
The last two questions were hard! Kudos to those who managed to get them.
1. Find a formula for the nth term of the sequence defined by the third-order recursion
tn+2 = tn+1 + 10tn + 8tn1
w
Math 111 2016-17
Assignment 9 Solutions
Due Thursday March 9, 2017
1. By calculating the first few cases, find a simple expression for the sum:
Sn = 11! + 22! + 33! + nn! .
[Hint: they grow like factorials so calculate the quantities 1!, 2!, 3!, 4!, 5!, 6
_
NAME: LAST
FIRST
Math 111
_
Student #
Test 5 Solutions
March 2 2017
There are 4 questions worth a total of 20 marks.
1. [9 marks] The sequence xn is tabulated at the right. It follows the rule that
each term is the sum of the previous two. We are intere
Page 1 of 4 pages
MATH 111 Test 4 Solutions Feb 2 2017
_
NAME: LAST
Math 111
FIRST
_
Student #
Test 4 Solutions
Feb 2 2017
There are 4 questions on 4 pages each worth 5 marks. Answer in the space provided. If you
need extra space advise the marker where t
Geometric interpretation of the product.
(a) What happens in the diagram when we multiply two complex
numbers z and w? Write them in polar form. Use r and s for their
moduli and (theta) and (phi) for their arguments:
z = r(cos + isin)
w = s(cos + isin)
Th
Math 111 2016-17
Assignment 6 Solutions
Due Thursday Dec 1, 2016
1. (a) Find an affine transformation T which maps the black triangle V
onto the grey triangle W with vertex vi mapping to vertex wi.
4
(b) Now imagine that each point x in the v-triangle mov
MATH 111 Test 2 Preparation 2015-16
Math 111
Test 2 preparation
Nov 3 2015
Test 2 will be written in class Friday Nov 13. It will cover the Chapter 2 material. The
questions will be technical in nature, that is do you know how to solve certain types of
pr
Math 111
Assignment 5 Solutions
Due Tuesday Nov. 17, 2016
1. The linear transformation T: R 2 R 2 is pictured at the right.
The starting box is a square with sides x=1 and y=1. The
image is a parallelogram with corners at (2, 3/2), (0, 3/2),
(2, 3/2), (0,
MATH 111 Complete Notes
Winter 2017
Chapter 5
Games and State Transitions
6
5
4
1
2
3
50 Series
51 Board Games
52 Reproduction
53 Complex Numbers
54 Complex Systems
MATH 111 Chapter 5C 2016-17 (revised March2017)
1
50C Series
Series: general concepts
An i
MATH 111 Midyear exam preparation Dec. 2013
MATH 111
page 1
MIDYEAR EXAM PREPARATION
DEC 2013
1. For this question, only the answer will be marked. 2 marks for each
3 2
(a) Find the inverse of the matrix
.
2 1
3 2 1 a
(b) Calculate the matrix product
page 1
MATH 111 Complete Notes
Winter 2016
Chapter 5
Games and State Transitions
6
5
4
1
2
3
50 Series
51 Board Games
52 Reproduction
53 Complex Numbers
54 Complex Systems
MATH 111 Chapter 5C 2015-16
1
page 2
50C Series
Series: general concepts
An infinit
Math 111
Problems for Chapter 5
2015 - 2016
1. Consider the "win or lose" game with the transition matrix at the
right.
(a) Suppose you have a choice whether to start on node X or node Y.
Which should you choose to maximize your probability of winning?
(b
Math 111
Test 6 Preparation Problems
March 2016
1. Find the 15th power of z = -1 + i .
z = -1 + i
Write z in polar form
1
- 1 + i = 2 cos(135) + i sin(135)
(-1 + i)15 = ( 2 )15cos(15 135) + i sin(15 135)
To simplify this, note that 8135 is a multiple of 3
Math 111
Problems for Chapter 2
2015-16
1.(a) Find a parametric equation for the line passing through the point A = (3, 1, 2) in the direction of the vector
v = [1, 2, 1].
(b) Is the point B = (1, 3, 0) on this line?
2.(a) Find a parametric equation for t
Fall Review Solutions
1. Find all numbers x for which
(a)
x2 + x + 1 < 0
(b)
1
1
+
>0
x 1 x
(c) |x
1| + |x
(d) |x
2| > 1
1| |x + 2| = 3
Solution.
(a) Since x2 + x + 1 = x + 1 )2 + 3 > 0, the function x2 + x + 1 is always positive.
2
4
1
1
1
(b) The expres
MATH 111 April Exam Preparation Solutions
MATH 111 FINAL EXAMINATION PREPARATION SOLUTIONS
page 1 of 34 pages
APR 2015
1. This is a multi-part question with short answer parts.
1 2
(a) For the matrix A =
find the eigenvalues and for each of these find a
MATH 111 Test 2 Preparation 2014-15
Math 111 Test 2 preparation
Oct 23 2014
Test 2 will be written in class Thursday Oct 30 at 12:30. It will cover the Chapter 2
material. The questions will be technical in nature, that is do you know how to solve
certain
Math 111
Problems for Chapter 3
2016 - 2017
1. Find the matrix of the following linear transformations.
x
x
x y 2z
x y
(b) T x y
y y x
(a) T y
x y
z
(c) A reflection in the plane in the line x = y.
(d) A dilation in the plane along the x-axis
Math 111
Problems for Chapter 2
2016-17
1.(a) Find a parametric equation for the line passing through the point A = (3, 1, 2) in the direction of the vector
v = [1, 2, 1].
(b) Is the point B = (1, 3, 0) on this line?
2.(a) Find a parametric equation for t
Math 111 2013-14
Assignment 8 Solutions
Due ThursdayMarch 6, 2014
1. By calculating the first few cases, find a simple expression for the sum:
Sn = 11! + 22! + 33! + nn! .
[Hint: they grow like factorials so calculate the quantities 1!, 2!, 3!, 4!, 5!, 6!
Math 111
Assignment 9 Solutions
1. Consider a population in which individuals progress through three
stages: juvenile X, subadult Y and adult Z. Suppose that each year
juveniles survive with probability p and become subadults, subadults
survive with proba
Math 111 2013-14
Assignment 2 Solutions
Due Thursday Oct. 10.
Note: Many of you made errors in your work so got the wrong answer. Please note that there is a
simple way to check the correctness of the answer for most of these problems. In such cases
there
Math 111
Test 6 Solutions
March 31 2014
1. A complex number z has modulus r = 10 and argument = 100. For both parts below give an exact
answer and a 2-place decimal approximation
(a) Calculate the real part of z4.
(b) Calculate the imaginary part of one o