Mathematical Methods
Exercises and Solutions for Lecture 2
Exercise 2.7.1
(i) Use a truth table to prove that the proposition (p q) is logically equivalent
to the proposition (p) (q).
(ii) Let p be the proposition 4 is an even integer and q be the proposi
Calculus Lecture Notes, Week One
1 Vectors
1.1 Visualising the set R2 using position vectors
a
b
)l e 6 Kb 6 a}. We call the
The set R2 is dened as the set of all 2 x 1 column matrices ( ) where e and b are
a
real numbers. In set notation, we write R2 = 6
AS & A2 PHYSICS FORMULAE YOU HAVE TO REMEMBER
AS & A2 LEVEL
A2 LEVEL ONLY
speed = distance
time
wave speed = frequency wavelength
v=f
acceleration = change in velocity
time taken
force = mass x acceleration
centripetal force = mass x speed2
radius
F = m v
Exercise Set 5
1.
(a)
z=
Solutions
x3 + y 3
x+y
z
3x2 (x + y) (x3 + y 3 ) 1
3x3 + 3x2 y x3 y 3
2x3 + 3x2 y y 3
=
=
=
x
(x + y)2
(x + y)2
(x + y)2
2y 3 + 3xy 2 x3
z
=
y
(x + y)2
by symmetry
To check that the function z = f (x, y) is homogeneous:
(tx)3 + (t
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
18 and 19
Exam-style question 18.5 (a) Dene what we mean by a set cfw_v1 , v2 , . . . , vk of
k linearly independent vectors in a vector space V .
(b) Dene what we mean by a set cfw_v1
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
16 and 17
Exam-style question 16.6 (a) Prove that if a linear system Ax = b has two
distinct solutions, then it has innitely many solutions.
(b) Dene what is meant by the rank of a matri
A Calculus Refresher
v1. March 2003
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An Algebra Refresher
v3. February 2003
This work is licensed under the Creative Commons
Attribution-Noncommercial-Share Alike 3.0 Unported License.
To view a copy of this license, visit http:/creativecommons.org/licenses/by-nc-sa/3.0/
or send a letter to
Mathematical Methods
Practice Questions and Solutions for Lecture 7
Practice question 7.5.1 (i) Show carefully that the derivative of g(x) = arcsin(x)
is given by
1
.
g (x) =
1 x2
(ii) Consider the non-bijective function f : R R dened by
f (x) = sin(x).
Calculus Lecture Notes7 Week Three
1 Functions, Derivatives, Taylor Series and Sta
tionary Points
1 .1 Intervals
A subset I of R is called an interval if, Whenever it contains two real numbers, it
contains all the real numbers between them. An interval ca
Exercise Set 1
1. The lines are coincident.
2. (a) The lines are collinear.
(b) The lines are not collinear.
4. (a) Ab is not dened.
(b) CA is (3 3)(3 2) giving a 3 2 matrix,
1 2
CA = 3 0
4 1
1
2
1 1
1
0
1
4 6
1 = 6 0
3
9 8
(c) A + Cb is not dened since
Exercise Set 1
Exercise Set 2
1. The rst line intersects the plane at the point ( 3 , 2, 11 ). The second line does not intersect
2
2
the plane. (Why? What is the relation between the line and the plane?)
1. The lines are coincident.
2. (a) The lines are
ST102 Exercise 11
Given the importance of sampling distributions to the remainder of the course, in this exercise
you will continue to practise deriving and working with sampling distributions of statistics.
Questions 1 and 2 require you to write out in f
ST102: Text for the gaps in the course pack Chapter 2
This document contains the text which has been omitted from the course pack and lled in
during the lectures, referenced by page number.
Page 25:
x A means that object x is an element of set A.
x A mean
ST102: Text for the gaps in the course pack Chapter 3
This document contains the text which has been omitted from the course pack and lled in
during the lectures, referenced by page number.
Page 65:
Here we will represent the observations as a sequence of
ST102: Text for the gaps in the course pack Chapter 6
This document contains the text which has been omitted from the course pack and lled in
during the lectures, referenced by page number.
Page 161:
Suppose we have a sample of n observations of a random
ST102: Text for the gaps in the course pack Chapter 5
This document contains the text which has been omitted from the course pack and lled in
during the lectures, referenced by page number.
Page 134:
For a discrete multivariate random variable, the joint
ST102: Text for the gaps in the course pack Chapter 4
This document contains the text which has been omitted from the course pack and lled in
during the lectures, referenced by page number.
Page 99:
In statistical inference we will treat observations:
X1
Lecture 5 | Determinants I
Determinants using cofactors
Section 3.2.1
Elementary signed products
Section 3.2.2
Corollaries Results on determinants
These are contained in Section 3.3, pages 104106 as Corollary 3.28,
Corollary 3.29 and Corollary 3.30.
Lecture 1
1.1
Vectors and Lines
Vectors in R2
For this we follow Section 1.3 of the textbook, until line 8 of page 19.
1.2
Vectors in R3
For this we follow Section 1.5 of the textbook.
1.3
Lines
For this we follow Section 1.6 of the textbook.
1.4
Optio
Mmo'l was (3'2 (202433 Q
63/. C(1)=800+'70T iii-+13 Ev 0A eqiaeA- Mc\ Em
We need CLjyIozqovgf
P 1kg! Cmider 3L1\-.CLe0=7o-241+3f
. j(6\:70 0 31(1) 1-14%1
33 12* MM gonzo, Le m 4.44,?0 =77,ch
- yup-4,70 aa 3&4) and 3 i=4 u clad MM
and 3143 : (o- mm + 3,? g
Mathematical Methods
Exam-Style Questions and Solutions for Lectures 4 and 5
Exam-Style Question 4.7
(a) Consider the functions
f : R R dened by f (x) = ex
and
g : R R dened by g(x) = x2 .
Find an expression in terms of x for the composition f g and anoth
Mathematical Methods
Exam-Style Questions and Solutions for Lectures 2 and 3
Exam-Style Question 2.8
(i) Dene what the terms proposition, predicate and existential statement mean.
(ii) Write down the truth table of the conditional proposition p q and the
Mathematical Methods
Exam-Style Questions and Solutions for Lectures 6 and 7
Exam-Style Question 6.7
Let f : S R be a function dened on a set S R.
(i) Dene what we mean by a strict local maximum of f and a global maximum of f .
Let g : S R be a function a
Mathematical Methods, Lecture Fifteen
Practice Questions and Solutions for Lecture 15
Practice question 15.5.1 (a) Find a vector parametric equation for the hyperplane in R4 described by the Cartesian equation
x1 + 2x3 5x4 = 7.
(b) Find a vector parametri
Mathematical Methods, Lecture Fourteen
Practice Questions and Solutions for Lecture 14
Practice question 14.4.1 Consider the plane described by the Cartesian equation
2x + y 3z = 4
(a) Find a vector parametric equation of the plane by treating x and z as
Mathematical Methods, Lecture Twelve
Practice Questions and Solutions for Lecture 12
Practice Question 12.6.1 (a) Evaluate the following determinants using a cofactor expansion by an appropriate row or column.
1
3
|B| =
0
0
2 6 1
|A| = 1 0 5
3 1 4
2
2
1
Mathematical Methods, Lecture Eleven
Practice Questions and Solutions for Lecture 11
Practice Question 11.6.1 Use the algorithm based on elementary row operations to investigate if the following matrices are invertible. For each matrix that is
invertible,
Mathematical Methods
Practice Questions and Solutions for Lecture 8
Practice question 8.7.1
integral
Use a change of variable to compute the arctan-type
1
dx
x2 + 6x + 11
and the arcsin-type integral
1
dx.
2 6x 5
x
Express your nal answers in terms of x.