IX. LAPLACE TRANSFORMS
The method of solution to a problem may be more tedious in one domain than another. Integral transforms enable
the solution from one domain to another. Integral transforms are characterized by a transformation from the time
(t) to t

University of the Philippines
College of Engineering
DEPARTMENT OF CHEMICAL ENGINEERING
Chemical Engineering 106
Long Exam 1
Instructions:
o Show complete solution, state all assumptions (if any) and simplifications.
o Start each problem in a new page.
o

THEOREMS ON
DETERMINANTS
ChE 106 Mathematical Methods in Chemical Engineering
1st Semester AY 2014-2015
ChE
106
Mathematical
1st Semester
AY 2014-2015
Engr.
Kristian
July R. YapMethods in Chemical Engineering
University of the
Philippines
Diliman
THEOREM

University of the Philippines
Department of Chemical Engineering
ChE 106
Linear Systems: Application Examples
1. Aris(1965) developed a technique of writing simultaneous chemical reactions in the form
of linear algebraic equations. For example, the follow

Compiled Problems in Matrices
1. Red, blue, and yellow marbles are placed in 3 separate bags, all of which weigh at 780 kg per
bag. Thrice the number of red marbles plus half of the blue marbles are less than twice the
number of yellow marbles by 22. 75%

ChE 106 LE 1 Answer Key
1. Conceptual problems
I.
Inverse of the transpose
Properties to use:
1. A1 A = I
2. () =
3. I = I
(A )1 = (1 )
(A )1 A = (1 ) A
I = (1 )
I = I
I=I
II.
Determinants
i.
a) 2. The minor with the highest degree of determinant is [ .]

I. MATRICES
A matrix is a rectangular array of numbers, equations or functions arranged in rows and columns. By
convention it is denoted by capital letters and the elements are enclosed in square brackets.
[
]
The size of a matrix is given by the number o

First Exam Coverage
ES 21 Notes
MATRICES
c1
c
.2
.
C .
ci
.
.
c
m
Definition
A matrix is a rectangular array of numbers or functions
arranged in rows and columns usually designated by a
capital letter and enclosed by brackets, parentheses or

First Exam Coverage
ES 21 Notes
MATRICES
D.
Definition
A matrix is a rectangular array of numbers or functions
arranged in rows and columns usually designated by a
capital letter and enclosed by brackets, parentheses or
double bars. A matrix may be denote

cerebral cortex
where the localization of cognitive functions occur
Differentiation
various types of cells that make up the 6 layers of the cortex are
produced
Hebbian Theory of Learning
"Neurons that fire together, wire together"
False
Pruning is the onl

ChE 106 TBC/HQR & TDE/HUV
Problem Set
Deadline: 01 September 2015 (in class)
Show your complete solution to the following items. Use the specified method in your evaluations.
EVALUATION OF DETERMINANTS
Showing the details, evaluate:
4 1 8
1.
0
0
2
0
3
5
2

ChE 106 Mathematical Methods in Chemical Engineering
PRACTICE PROBLEMS 01
1.
Using Theorems on Determinants, prove the following:
2a
2b
2b
2a
a+b a+b
2.
bc
2
a + c = 2 ( a b ) ( a + b )
b
Figure 1 (below) depicts a chemical exchange process consisting of

VIII. LINEAR DIFFERENTIAL EQUATIONS
Linear differential equations with constant coefficients are equations that exhibit linearity in terms of the
independent variable and take the following form:
1
2
+ 1
+ + 2 2 + 1
+ = ()
1
where are constants.
This m

APPLICATION TO PROBLEMS OF GROWTH AND DECAY
Mathematical Formulation:
For problems in growth and decay, the time rate of
change of a given quantity A is proportional to A. When
transformed to a mathematical statement, this gives us the
differential equati

APPLICATION TO PROBLEMS OF GROWTH AND
DECAY
Mathematical Formulation:
For problems in growth and decay, the
time rate of change of a given quantity A is
proportional to A. When transformed to a
mathematical statement, this gives us the
differential equati

F.
First Order Bernoulli Differential Equation
A first order Bernoulli differential equation follows the form
dy
P(x) y Q(x) yn
dx
The solution to this equation may be obtain by transforming the original
equation into a linear differential equation by th

1
F.
First Order Bernoulli Differential Equation
1
x 2 y 3 2x 2 c
A first order Bernoulli differential equation follows the form
dy
P(x) y Q(x) yn
dx
The solution to this equation may be obtain by transforming the original
equation into a linear differen

ES 21
2nd Exam Notes
LINEAR DIFFERENTIAL EQUATION
-an equation in which the dependent variable and its derivatives appear to the first
degree only and the coefficients are either constants or functions only of the
independent variable
Examples:
INTRODUCTI

ChE 106 Mathematical Methods in Chemical Engineering
HOMEWORK 04 (with answers)
1.
Find the solution to the given system of linear equations using
a) Gaussian Elimination
b) Gauss-Jordan Reduction
+ c
a
=4
3a + 2b + 3c + 2d = 24
3b + 2c + 3d = 24
2a
a 1

ChE 106 Mathematical Methods in Chemical Engineering
SEATWORK 02 (with answers)
1.
Given:
x+ y
3x
2y
A = xy
1 xy
x + y + 1 3x 2 y + 1
2
0 0
B= x
1 0
x y y 1
2 x + 1 3 y + 3 x y
C= 1
3 y
x
4x
6
2 y
Solve for AB + C using only the Theorems on De

ChE 106 Mathematical Methods in Chemical Engineering
HOMEWORK 04
1.
Find the solution to the given system of linear equations using
a) Gaussian Elimination
b) Gauss-Jordan Reduction
a
+ c
=4
3a + 2b + 3c + 2d = 24
3b + 2c + 3d = 24
2a
2.
+ 5c + 2d = 21
Us

ChE 106 Mathematical Methods in Chemical Engineering
HOMEWORK 02
1.
Given the following matrices:
3 5 2
A = 0 1 6
7 0 4
a.
b.
2.
2 1
B = 8 0
4 5
7 1 0
C=
4 3 2
Find A A 1 BC + A 2 I n A . Specify the size of In. Label the resulting matrix as Matrix

ChE 106 Mathematical Methods in Chemical Engineering
HOMEWORK 03
1.
Three sons were asked by their mommy as to what they want to have this Christmas. The youngest thought
of a BMW but the eldest son said, Mom wont like that! (meaning, their mom could not

NAME:
SIGNATURE:
SECTION:
LW 14
FILL THE TABLE ABOVE. RETURN THIS QUESTIONNAIRE AFTER THE CLASS
PROBLEM
The truss shown is supported by a pin at H and a roller at K. It
is used to support the roof of a school. The load from the roof
is modeled by the vert