HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2012
MATHEMATICS COMPULSORY PART PAPER 1
SAMPLE SOLUTIONS
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Solution
1.
Marks
m12 n8
= m12 n83
n3
n5
= 12
m
1M
1M+1A
Remarks
for
ap
= apq
aq
1M for ap =
1
ap
. . . .

Combinatorics, 2015 Fall, USTC
Homework 8
The due is on Wednesday, Nov. 25, at beginning of the class.
Solve all 5 problems.
1. (a). A fixed point of a permutation is an integer i such that (i) = i. Let X be
the number of fixed points of permutation . W

Combinatorics, 2015 Fall, USTC
Homework 10
The due is on Monday, Dec. 14, at beginning of the class.
Solve all 5 problems.
1. Given a graph G with (G) = n and a proper coloring f : V (G) [n], prove that for any
color i [n], there exists a vertex of colo

Combinatorics, 2015 Fall, USTC
Homework 5
The due is on Monday, October 26, at beginning of the class.
Solve No. 4, 6 and any other 3 problems. If you solve more than 5 problems, do
list the 5 problems which you want to be graded at the very beginning o

Combinatorics, 2015 Fall, USTC
Homework 4
The due is on Monday, October 19, at beginning of the class.
Solve any 5 problems. If you solve more than 5 problems, do list the 5 problems which
you want to be graded at the very beginning of your sheets.
1. L

Combinatorics, 2015 Fall, USTC
Homework 9
The due is on Monday, Dec. 7, at beginning of the class.
Solve all 5 problems.
1. Show that any graph with n vertices and with m edges has at least
1 4m2
mn
3
n
triangles. (No probabilistic method needed.)
2. D

Combinatorics, 2015 Fall, USTC
Homework 2
The due is on Monday, September 21, at beginning of the class.
Solve any 5 problems. If you solve more than 5 problems, do list the 5 problems which
you want to be graded at the very beginning of your sheets.
1.

Combinatorics, 2015 Fall, USTC
Homework 6
The due is on Monday, Nov. 9, at beginning of the class.
Solve any 5 problems. If you solve more than 5 problems, do list the 5 problems which
you want to be graded at the very beginning of your sheets.
1. Prove

Combinatorics, 2015 Fall, USTC
Homework 7
The due is on Monday, Nov. 16, at beginning of the class.
Solve No. 3, 4, 5, 6 and any one of the other problems. If you solve more than 5
problems, do list the 5 problems which you want to be graded at the very

Combinatorics, 2015 Fall, USTC
Homework 1
The due is on Monday, Sep. 14.
Solve any 10 problems. If you solve more than 10 problems, do list the 10 problems
which you want to be graded at the very beginning of your sheets.
1. How many 9-digit numbers can

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HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION
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PRACTICE PAPER
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PAPER 1
Question-Answer Book
(2 ho

MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 2 - Suggested Solution: Dierentiation
Q1. (Tangent line of a curve) Find an equation of the tangent line to the curve y = 5 6x 2x3 at
x = 2.
Solution By dierentiating y , we have dy/dx = 6 6x2 . Substi

MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 2 : Dierentiation
Q1. (Tangent line of a curve) Find an equation of the tangent line to the curve y = 5 6x 2x3 at
x = 2.
Q2. (Normal line of a curve) For what non-negative value(s) of b is the line y =

MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 3 - Suggested Solution: Ch. 4 Applications of Derivatives
Q1. (Increasing and decreasing functions)
increasing and decreasing.
Determine the intervals where the following function is
f (x ) = x 5
Solu

MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 4 : Ch. 5 Integration
Q1. (Fundamental theorem of calculus) Evaluate the following denite integrals by the Fundamental
Theorem of Calculus.
2
Z
(a)
x2 dx
=
1
3
Z
(b)
e
1
dx
x ln x
=
/2
Z
sin 2x dx
(c)

MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 4 - Suggested Solution: Ch. 5 Integration
Q1. (Fundamental theorem of calculus) Evaluate the following denite integrals by the Fundamental
Theorem of Calculus.
2
Z
(a)
x2 dx =
1
3
Z
(b)
e
1
dx =
x ln x

MATH 1013 - Calculus I
Chapter 5 - Integrals
1. Areas Under Curves and Riemann Sum
Consider the case that a car is travelling on a straight road with velocity v given by a function of time t.
Suppose we want to know the distance travelled by the car durin

MATH 1013 - Calculus I
Chapter 3 - Derivatives
1. Rules of Dierentiation
From the denition of derivatives and rules of limit, the following commonly used dierentiation rules
can be derived.
In below, c is a constant, f (x) and g(x) are dierentiable functi

MATH 1013 - Calculus I
Preliminaries
1. Set
A set is a collection of objects (elements). A set may contain innite many elements. A set without any
element is called an empty set, denoted by .
Two sets are equal if and only if they contain exactly the same

MATH 1013 - Calculus I
Chapter 4 - Applications of Dierentiation
1. Newtons Method
The Newtons method is a popular numerical method to nd the roots of the function f (x), i.e., to nd
those x that satisfy the equation f (x) = 0. The popularity of this meth

MATH 1013 - Calculus I
Chapter 2 - Limits and Derivatives
1. Tangent and the Idea of Limit of a Function
#1.1 Tangent Line
Secant line is a line that intersects a curve at more than one points.
Tangent line is a line that touches a curve at only one poi

PAPER 2
HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2012
11.30 am -12.45 pm (1% hours)
INSTRUCTIONS
1.
Read carefully the instructions on the Answer Sheet. After the announcement of the start

FOR TEACHERS USE ONLY
HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION
PRACTICE PAPER
MATHEMATICS
COMPULSORY PART
PAPER 1
MARKING SCHEME
This marking scheme has been prepared by the Hong Kong Examinatio

Combinatorics, 2015 Fall, USTC
Homework 3
The due is on Monday, October 12, at beginning of the class.
Solve all problems.
1. The Fibonacci sequence cfw_F (n) is defined as follows:
F (0) = 0, F (1) = 1 and for n 2, F (n) = F (n 1) + F (n 2).
(1). Prove