University of London The London School of Economics and Political Science
Actuarial Science
MATH 117

Spring 2011
1
STOCHASTIC PROCESSES: LEARNING THE LANGUAGE
By A. J. G. Cairns, D. C. M. Dickson, A. S. Macdonald,
H. R. Waters and M. Willder
abstract
Stochastic processes are becoming more important to actuaries: they underlie much of modern finance, mortality analys
University of London The London School of Economics and Political Science
Actuarial Science
MATH 117

Spring 2011
CALCULUS
ITO
J. MICHAEL STEELE
Abstract. This entry for the Encyclopedia of Actuarial Sciences
provides an introduction to the It
o Calculus that emphasizes the
definition of the It
o integral and the description of It
os Formula, the
most widely used re
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
MA107: Quantitative Methods (Mathematics) 201617
Exercises 3: Recurrence equations and limits
For all homeworks in this course please note that no marks will be given for answers only. Always
show the method used to obtain your answers.
Please write clear
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
MA107: Quantitative Methods (Mathematics)
Solutions to the examination (2015)
Question 1.
(a) The equilibrium quantity occurs when pS (q) = pD (q) and so we have
2q + 3 = 6 q
=
3q = 3
=
q = 1,
and then, using the inverse demand function (say), this gives
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
Summer 2013 examination
MA107
Quantitative Methods (Mathematics)
(Half Unit)
Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours.
This paper contains 5 questions. You may attempt as many questions as you wish,
but only the best 4
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
MA107: Quantitative Methods (Mathematics) 201617
Exercises 4: The cobweb model and derivatives
For all homeworks in this course please note that no marks will be given for answers only. Always
show the method used to obtain your answers.
Please write clea
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
Summer 2015 examination
MA107
Quantitative Methods (Mathematics)
Half Unit
Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours.
This paper contains 5 questions. You may attempt as many questions as you wish,
but only the best 4 qu
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
MA107: Quantitative Methods (Mathematics) 201617
Exercises 6: Partial derivatives
For all homeworks in this course please note that no marks will be given for answers only. Always
show the method used to obtain your answers.
Please write clearly. Dont jus
University of London The London School of Economics and Political Science
Quantitative Methods (Mathematics)
MATH MA107

Fall 2014
MA107: Quantitative Methods (Mathematics) 201617
Exercises 1: Revision
For all homeworks in this course please note that no marks will be given for answers only. Always
show the method used to obtain your answers.
Please write clearly. Dont just use formu
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
1
Consider the function f : R2 R defined by
f (x, y) = 2x2 + y 2 .
(a) Find a Cartesian equation in R3 for the tangent plane T to the surface
z = 2x2 + y 2 at the point (x, y, z) = (1, 1, 3). Simplify your answer.
(b) Write down a Cartesian equation in R3
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
5 Let
3 2
A_QQ.
(a) Find the solution of the system of differential equations y = Ay: (a
yi(t)=3y1(t)+2y2(t) _ yi
cfw_y(t)=2y1(t)+6y2(t) y_( >
Which satises the initial conditions 311(0) = 5, 312(0) = 5.
(b) (i) Find an orthogonal matrix Q and a diagonal
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
London School of Economics
MA100
Mathematical Methods
Mock Exam 201415
This test does not count towards your final grade.
Name:
Class Teacher:
Class Group:
Instructions to candidates
Time allowed: 1 hour.
Answers should be justified by showing work.
Use
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
8. 1/ L7 (a) If the variable a: E R+ , solve the homogeneous equation
\_h/'
(t; (b) Show that the following differential equation is not exact:
\./
(y ~ 3,3 sin 3:)da: + (5y2 cosa: + 3x)dy = 0 Where x E 1R+
It is known to have an integrating factor of th
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
Mathematical Methods, Lecture TwentySeven
Practice Questions and Solutions for Lecture 27
2 1 2
Practice question 27.3.1 Consider the matrix A = 0 1 2.
0 0 3
(a) Find the eigenvalues and the corresponding eigenvectors of A.
(b) Hence, find an invertible
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
A AgVgwavwg y
6.
(a) Consider the matrix
1 2 1 3
A = 2 3 O 1 .
4 5 2 3
Z (i) Put the matrix A into reduced row echelon form.
("It (ii) Find a basis of the column space, CS(A). State why CS(A) is a plane in R3,
and nd the Cartesian equation of this plane.
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
3 A function f : R3 > R is dened by
fm,y,z = lar:32.'1:ycosz x2y2lz2x+1.
3 2 _
(a) Show that (0,0,0) is not a stationary point of f, whereas (1, 1,0) and (1, 1,0) are (+
stationary points. ( You dont need to look for other stationary points.)
(b) Classi
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
Mathematical Methods
ExamStyle Questions and Solutions for Lectures
22 and 23
Examstyle question 22.8 (a) Given an m n matrix A, state the RankNullity
theorem associated with A.
Consider the linear system Ax = b where
1 3 1 0
A = 2 6 0 2 ,
4 12 1 3
x
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
Let
A=(
2. (a) Show that v1 is an eigenvector of A and nd its corresponding eigenvalue.
(23 3
Find the other eigenvalues and corresponding eigengiehtors of A.
l I Orthogonally diagonalise A. ('5) (322! l
a:
(b) Let f denote the quadratic form f (23,31, 2)
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
4:L(a) Let M be an n x n matrix. State precisely what is meant by the statement
A is an eigenvalue of M with corresponding eigenvector v.
(b)Let
613 8 1
A: 252, V: 0
717 9 1
7 (i) Using the denition of eigenvector, show that v is an eigenvector of A and n
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
Mathematical Methods, Lecture ThirtyThree
Practice Questions and Solutions for Lecture 33
Practice question 33.6.1 Consider the cost function C : R2 R defined by
C(x, y) = 4x2 + 4y 2 2xy 40x 140y + 1800
for a firm producing two goods x and y.
(a) Show th
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
MA100 Mathematical Methods
Mock Exam 201415 with Solutions
1. (a) Give the definition of a basis of a 2dimensional subspace of R3 .
Solution: A basis of a 2dimensional subspace of R3 is a set of 2 linearly independent vectors which span the given subsp
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
6.
A function f : R2 ) R is dened by
ay) = 5y  32
f .
4(a) Find its gradient vector, and evaluate it at the point (2,3,!) = (1, 1). Find the
rate of change of the function in the direction (2, 1)T at the point (1, 1).;
Q (h) In what directions is the ra
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
6 Let
5 9 1 15 6
A = 3 1 7 4 and d = 5 .
1 2 _ 0 3 l
(a) Find the general solution of Ax = d. (Put the augmented matrix into reduced row
echelon form. ) Express your solution in vector form.
(b) Let T denote the linear transformation T(x) 2: Ax.
(i) Writ
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
(C5) Partial derivatives, tangent planes, homogenous
functions
MA100 HIGHLIGHTS: CALCULUS
Rate of change at (a,b) in the direction u/Directional
derivatives in the direction u
(C2) Planes, curves and conics
Tangent lines
Parabola:(! = ! ! + !" + !
Ellipse
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
f"'
2 (a) A company uses two goods in its production. When it uses x of good 1 and y of good 2,
it achieves a prot P given by
P(x, y) = 40.5w x2 + y.
(1) Carefully sketch some contours of the prot function.
Good 1 costs 0.5 per unit; good 2 costs .8 1 per
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
MA100 Solutions and Commentary 2014
General Comments
Questions are marked out of a total of 25 points each, and the best six marks are added,
giving a total out of 150, which is then scaled so that 150 points=100%. At least half of
each question is design
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
MA100
Mathematical Methods
Solutions to summer 2015 Examination
1
1. (a) The derivative of f : [1, ) R is given by
f 0 = (2x 9)ex + (x2 9x + 19)ex = (x2 7x + 10)ex = (x 2)(x 5)ex ,
so the stationary points of f are
x=2
and
x = 5.
The derivative satisfies
University of London The London School of Economics and Political Science
Mathematical Methods
MATH 100

Summer 2016
THE LONDON SCHOOL
OF ECONOMICS AND
POLITICAL SCIENCE I
Summer 2015 examination
, MA’1OO ‘
Mathematical Methods
Suitable for all candidates
Instructions to candidates
This exam contains 8 questions. You may attempt as many questions as you wish, but Only
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE
University of London
Wednesday 7th June 1995 2.30pm — 5.30pm
B.Sc.(Econ) Part I and Part II Examination 1995
and
B.A./B.Sc. Degree in the Faculty of Arts, Economics and Science Examination 1995
Mathematical
University of London The London School of Economics and Political Science
Mathematical Methods
MATH MA100

Summer 2016
LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE
University of London
._.—_._._—————————————————m
B.Sc.(Econ) Part I and Part II Examination 1996
and
B.A./B.Sc. Degree in the Faculty of Arts. Economics and Science Examination 1996
Mathematical Methods