MA1301
Chapter 2
Calculus and Basic Linear Algebra II
Complex Numbers
As early as 250 A.D., Greek algebraist, Diophantus, attempts to solve quadratic equations of the form
ax 2 + bx + c = 0 . He accepts only positive rational roots and ignores all others.
MA1301
1.
2.
Exercises on Matrices, Determinants and Systems of Linear Equations
Evaluate
7
2
3
4 3
(a) det 2
1 3
7
1 1 1
(b) det a b c
2
a b2 c 2
b
1 + a
a 1+b
(c) det
a
b 1+
Consider the following sets of linear equations. In each case use Gaussi
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Contents
Preface
1. Integrals
1.1. Areas and distances
1.2. The definite integral
1.3. The fundamental theorem of calculus
1.4. Indefinite integrals and the net change theorem
1.5. The substitution rule
1
2
2
7
13
17
20
Preface
Textbooks: Single Variable
Contents
Preface
4. Integrals
4.1. Areas and distances
4.2. The denite integral
4.3. The fundamental theorem of calculus
4.4. Indenite integrals and the net change theorem
4.5. The substitution rule
1
2
2
7
13
17
20
Preface
Textbooks: Single Variable Cal
Contents
5. Applications of Integration
5.1. Areas between curves
5.2. Volumes
5.3. Volumes by cylindrical shells
5.4. Average value of a function
1
2
5
8
10
5. Applications of Integration
This section is to apply integrals in the following applications:
Contents
8. Further Applications of Integration
8.1. Arc length
8.2. Area of a surface of revolution
8.3. Applications to physics and engineering
1
2
7
10
8. Further Applications of Integration
We have studied techniques of integration and some applicatio
MA1301
Solutions
Matrices, Determinants and Systems of Linear Equations
1.
Evaluate
7
2
b
1 1 1
3
1 + a
4 3
a 1+b
(a) det 2
(b) det a b c (c) det
1 3
2
7
a
b 1+
a b2 c 2
Solution:
(a)
3 7 2
0 2 19
2 19
det 2 4 3 = det 0 10 17 = det
= 34 + 19
MA1301
Chapter 1
Calculus and Basic Linear Algebra II
Vector Algebra
1 Review of Basic Ideas (p.1 p.8)
In engineering and science, physical quantities which are completely specified by their magnitude (size)
are known as scalars. Examples are: mass, tempe
MA1301
Solutions
Exercises on Vector Algebra
r r rrrr r rr
rrr
Let a = 2i 2 j + k , b = i + 8 j 4k , c = 12i 4 j 3k . Find
r
ra rr r rrr r r
(a) a , r , a + b , a + b , a b , a b ;
a
rr
r
r
(b) a b and the angle between a and b ;
r
r
(c) the coefficient o
MA1301 Calculus and Basic Linear Algebra II
Matrix Algebra and System of Linear Algebraic Equations
1. Introduction (p.61 p.77)
A matrix of order m n or an m n matrix is a rectangular array of numbers having m rows and n
a11 a12 . a1n
a
21 a22 . a2 n
MA1301 Complex Numbers
Solutions
1. Find the square roots of 5 12i .
Solution:
2
If 5 12i = a + bi , then 5 12i = ( a + bi ) = a 2 b2 + 2abi .
Equating the real and imaginary parts
a 2 b2 = 5
( a2 9) ( a 2 + 4) = 0
36
a 4 5a 2 36
a 4 5a 2 36
2
a 2 =5
=0
=
MA1301
1.
Exercises on Vector Algebra
r r rrrr r rr
rrr
Let a = 2i 2 j + k , b = i + 8 j 4k , c = 12i 4 j 3k . Find
r
ra rr r rrr rr
(a) a , r , a + b , a + b , a b , a b ;
a
rr
r
r
(b) a b and the angle between a and b ;
r
r
(c) the coefficient of the pr
Contents
2. Applications of Integration
2.1. Areas between curves
2.2. Volumes
2.3. Volumes by cylindrical shells
2.4. Average value of a function
1
2
5
8
10
2. Applications of Integration
This section is to apply integrals in the following applications: