AST20112 Electronic Devices and Circuits
Midterm
Time allowed: 1 hour and 15 min
Total Marks: 50
Answer all 4 questions
Show all calculation steps and drawings
Q1
Using the constant voltage drop model, find Vo, ID1 and ID2 in Figure Q1. You are
required t

AST21111 Discrete Mathematics
Assignment 2 (Due: 5 December 5:00PM)
Solution
1. Prove that for all sets A, B, and C, (A B) (C B) = (A C) B.
Sol:
2. Prove that for all sets A, B, and C, if A B then A C B C.
Sol:
3. For any sets A, B and C, prove that C (A

AST21114 Digital Electronics
Additional Exercise Solution
1) Perform the arithmetic operations (-128) + (-1) in binary using
a) 8-bit signed 2's complement representation for negative numbers
b) 9-bit signed 2s complement representation for negative numbe

AST21114 Digital Electronics
Additional Exercise
1) Perform the arithmetic operations (-128) + (-1) in binary using
a) 8-bit signed 2's complement representation for negative numbers
b) 9-bit signed 2s complement representation for negative numbers
2) Use

AST21111 Discrete Mathematics
Midterm
Time allowed: 1 hour
Total Marks: 60
Answer all questions.
1. Let p and q be propositions. By standard logic identities, show that ~(p V (~p q)
and (~p ~q) are logically equivalent. (Clearly show all steps but no need

AST21111 Discrete Mathematics
Assignment 2 (Due: 5 December 2014 5:00PM)
1. Prove that for all sets A, B, and C, (A B) (C B) = (A C) B.
2. Prove that for all sets A, B, and C, if A B then A C B C.
3. For any sets A, B and C, prove that C (A B) (C A) (C B)

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 8
Multiple Integrals
Part 3
Triple Integrals
In single integral, we calculates the area between
f(x) and x=0. In double integral, we calculates the
area for f(x,y) in axb and y. In triple
integral, we wou

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 7
Multiple Integrals
Part 2
Area by Double Integration
If we take f(x,y)=1 in the definition of double
integral over a region R in the preceding section,
the Riemann sums reduce to
n
n
n
n
Sn = f (xij , y

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 9
Multiple Integral
Part 4
Spherical Coordinate
System
In three dimensions, we can describe the
coordinate by considering a sphere.
The spherical coordinates of a point in space are
shown.
p is the dist

Tutorial 7
Double Integral
Mr. S.T.Wan
Example 1
Evaluate the following integral
2 1
0
1
x ydydx
Example 2
Evaluate the following integral
1 2
0 1
x
xye dydx
Example 3
Evaluate the following integral
y
dydx
0 0 1 xy
1 1
Example 4
Evaluate the double in

Tutorial 6
Partial Derivatives
Mr. S.T.Wan
Tutorial 1
Find dw/dt at t=0
where
y
w = sin(xy + e )
t
x = e , y = ln(t +1)
Tutorial 2
Find dy/dx at (3,4)
1 x
y1
x
2
e y sin xy = 0
Tutorial 3
Find the direc<ons in which

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 7
Multiple Integrals
Part 1
Review of Integral
If f(x) is defined for a x b, we start by dividing the
interval [a,b] into b subintervals [xi-1,xi] of equal
width x=(b-a)/n
Then, we form the Riemann sum f

Tutorial 5
Partial Derivatives
Mr. S.T.Wan
Tutorial 1
Find out the following limit
2
x 2xy + y
lim
( x,y)(1,1)
xy
2
Tutorial 2
Find out the following limit
2
2
sin(x + y )
lim
( x,y)(0,0)
x 2 + y2
Tutorial 3
Find the poi

Tutorial 3
Application of First
and Second order
ODE
Mr. S.T.Wan
Example of Bernoulli
Differential Equation
Solve the equa2on:
dy
+ xy = xy 3
dx
Where
y(0) =
1
2
Vibrating Springs
We consider the mo2on of an object

Tutorial 4
Second Order ODE
Mr. S.T.Wan
Example 1
Solve
d2y
4y = xe x + cos2x
dx 2
Example 2
Solve
d2y
dy
4 + 4y = 2e 2 x +12 cos3x 5sin 3x
dx 2
dx
where
y(0) = 2, y'(0) = 4
Example 3
Solve
d2y
4y = xe 2 x + cos2x
dx 2
W

Tutorial 2
Application of First
order ODE
Mr. S.T.Wan
Exponential Growth and Decay
There are many situa2ons where the rate of change of some
quan2ty x is pro-por2onal to the amount of that quan2ty, for
example,

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 5
Partial Derivatives
Part 2
The Increment Theorem
For y=f(x), the differential of y was defined as
dy = f '(x)dx
Similar terminology is used for a function of two
variables, z=f(x,y). That is, x and y a

Tutorial 1
Integration by Parts
Mr. S.T.Wan
Prove of the Equation
Consider
d(uv) du
dv
= v+ u
dx
dx
dx
d(uv) = vdu + udv
d(uv) = vdu + udv
uv = vdu + udv
udv = uv vdu
Example 1
ln xdx
Example 2
3 x
x e dx
Example 3
x n ln(x

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 9
Multiple Integral
Part 5
Multiple Integral Application
The mass of an object is the quantity of matter in
an object regardless of its volume or other forces
acting on it.
If you leave the planet Earth,

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 2
Ordinary Differential
Equation
Part 2
Homogeneous Functions
The function given by f(x,y) is homogeneous of
degree n, if
n
f (tx, ty) = t f (x, y)
Where n is an integer.
Example 1
Find our the degree o

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 1
Advanced Algebra
Mr S.T.Wan
Linear Combination
Let v 1, v 2, v r be vectors in Rn . A linear
combination of these vectors is any expression of
the form
v=k1v1+k2v2+knvn
w=2u+3v
Example
Write the vect

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 4
Partial Derivatives
Part 1
Functions of Several
Variable
A real-valued function of n real variables is a
function that takes as input n real number,
represented by the variable x1,x2,xn for producing
an

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 13
Fourier Series
Taylor Series
A polynomial is a function that can be written in the
form
p(x) = a0 + a1 x +. + an x
n
For some coefficients ak is not equal to 0.
Polynomials are just about the simples

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 2
Ordinary Differential
Equation
Part 3
Second order linear non-homogeneous
differential equations with constant
coefficients
A second order linear equation is an equation of
the form:
2
d y
dy
a 2 + b +

AST 21112
Mathematics Analysis
Mr S.T.Wan
Lecture 2
Ordinary Differential
Equation
Part 1
Definition of ODE
A differential equation is an equation which contains
derivatives of the unknown
An ODE is an equation containing a function with just one
indepe

Laplace Transform
Definition
Transforms - a mathematical conversion from
one way of thinking to another to make a
problem easier to solve
problem
in original
way of
thinking
transform
solution
in transform
way of
thinking
inverse
transform
solution
in ori