H1 Math Syllabus

8/8/2019 H1 Math Syllabus

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MATHEMATICS

HIGHER 1

(Syllabus 8864)

CONTENTS

Page

AIMS 1

ASSESSMENT OBJECTIVES (AO) 2

USE OF GRAPHIC CALCULATOR (GC) 2

LIST OF FORMULAE 2

INTEGRATIONS AND APPLICATIONS 2

SCHEME OF EXAMINATION PAPER 2

CONTENT OUTLINE 3

MATHEMATICAL NOTATION 7


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8864 H1 MATHEMATICS (2010)

1

AIMS

The syllabus provides a foundation in mathematics for students who intend to enrol in universitycourses such as business, economics and social sciences. It covers Functions and Graphs, Calculus,and Statistics. The main focus of the syllabus will be the understanding and application of basic

concepts and techniques of statistics. This will equip students with the skills to analyse and interpretdata, and to make informed decisions.

The general aims of the syllabus are to enable students to:

1 acquire the necessary mathematical concepts and skills for everyday life, and for continuouslearning in mathematics and related disciplines;

2 develop the necessary process skills for the acquisition and application of mathematicalconcepts and skills;

3 develop the mathematical thinking and problem solving skills and apply these skills toformulate and solve problems;

4 recognise and use connections among mathematical ideas, and between mathematics andother disciplines;

5 develop positive attitudes towards mathematics;

6 make effective use of a variety of mathematical tools (including information andcommunication technology tools) in the learning and application of mathematics;

7 produce imaginative and creative work arising from mathematical ideas;

8 develop the abilities to reason logically, to communicate mathematically, and to learncooperatively and independently.


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2

ASSESSMENT OBJECTIVES (AO)

There are three levels of assessment objectives for the examination.

The assessment will test candidates' abilities to:

AO1 understand and apply mathematical concepts and skills in a variety of contexts, including

the manipulation of mathematical expressions and use of graphic calculators;

AO2 reason and communicate mathematically through writing mathematical explanation,

arguments and proofs, and inferences;

AO3 solve unfamiliar problems; translate common realistic contexts into mathematics; interpret

and evaluate mathematical results, and use the results to make predictions or comment on

the context.

USE OF GRAPHIC CALCULATORS (GC)

The use of GC, without computer algebra system, will be expected. The examination paper will be setwith the assumption that candidates will have access to a GC. As a general rule, unsupportedanswers obtained from a GC are allowed unless the question specifically states otherwise. Whereunsupported answers from GC are not allowed, candidates are required to present the mathematicalsteps using mathematical notations and not calculator commands. For questions where graphs areused to find a solution, candidates should sketch these graphs as part of their answers. Incorrectanswers without working will receive no marks. However, if there is written evidence of using GCcorrectly, method marks may be awarded.

Students should be aware that there are limitations inherent in GC. For example, answers obtainedby tracing along a graph to find roots of an equation may not produce the required accuracy.

LIST OF FORMULAE

Candidates will be provided in the examination with a list of formulae.

INTEGRATIONS AND APPLICATIONS

Notwithstanding the presentation of the topics in the syllabus, it is envisaged that some examinationquestions may integrate ideas from more than one topic, and that topics may be tested in the contextsof solving problems and in applications of mathematics.

SCHEME OF EXAMINATION PAPER

For the examination in H1 Mathematics, there will be one 3-hour paper marked out of 95 as follows:

Section A (Pure Mathematics 35 marks) will consist of about 5 questions of different lengths andmarks based on the Pure Mathematics section of the syllabus.

Section B (Statistics 60 marks) will consist of 6 8 questions of different lengths and marks basedon the Statistics section of the syllabus.

Candidates will be expected to answerall questions.


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CONTENT OUTLINE

Knowledge of the content of the O Level Mathematics syllabus is assumed in the syllabus below andwill not be tested directly, but it may be required indirectly in response to questions on other topics.

Topic / Sub-topics Content

PURE MATHEMATICS

1 Functions and graphs

1.1 Exponential andlogarithmic functionsand Graphingtechniques

Include: concept of function

use of notation such as 5)f( 2 += xx

functions xe and lnxand their graphs

laws of logarithms

equivalence of = xy e and = lnx y

use of a graphic calculator to graph a given function characteristics of graphs such as symmetry, intersections

with the axes, turning points and asymptotes

Exclude: concepts of domain and range

the use of notation 5:f 2 +xx a

1.2 Equations andinequalities

Include: solving simultaneous equations, one linear and one

quadratic, by substitution conditions for a quadratic equation to have real or equal

roots solving quadratic inequalities

conditions for cbxax ++2 to be always positive (or always

negative) solving inequalities by graphical methods formulating an equation from a problem situation finding the numerical solution of an equation using a

graphic calculator


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2 Calculus

2.1 Differentiation Include:

derivative of )f(x as the gradient of the tangent to the

graph of )f(xy = at a point

use of standard notations )(f x andx

y

d

d

derivatives of nx for any rational n, xe , lnx, together with

constant multiples, sums and differences use of chain rule

graphical interpretation of 0)(f > x , 0)(f = x and 0)(f


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STATISTICS

3 Probability

3.1 Probability Include: addition and multiplication of probabilities mutually exclusive events and independent events use of tables of outcomes, Venn diagrams, and tree

diagrams to calculate probabilities calculation of conditional probabilities in simple cases use of

= ( ) 1 ( )P A P A

= + P( ) P( ) P( ) P( ) A B A B A B

=

P( )P( )

P( )

A BA B

B

4 Binomial and normal distributions4.1 Binomial distribution Include:

knowledge of the binomial expansion of +( )na b for positive

integern

use of the notations !n and

n

r

concept of binomial distribution B ( , )n p and use of

B ( , )n p as a probability model

use of mean and variance of a binomial distribution (withoutproof)

solving problems involving binomial variables

Exclude calculation of mean and variance for other probabilitydistributions

4.2 Normal distribution Include: concept of a normal distribution and its mean and variance;

use of ),(N 2 as a probability model

standard normal distribution

finding the value of 0}

+

0

the set of positive real numbers and zero, {x : x0}

n

the real n tuples

`= the set of complex numbers

is a subset of

is a proper subset of

is not a subset of

is not a proper subset of

union

intersection

[a, b] the closed interval {x : axb}

[a, b) the interval {x : ax< b}

(a, b] the interval {x : a < xb}

(a, b) the open interval {x : a < x < b}


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2. Miscellaneous Symbols

= is equal to

is not equal to

is identical to or is congruent to

is approximately equal to

is proportional to

< is less than

Y; is less than or equal to; is not greater than

> is greater than

[; is greater than or equal to; is not less than

infinity

3. Operations

a + b aplusba b aminusba b, ab , a.b amultiplied byb

a b,b

a, a/b adivided byb

a : b the ratio ofato b=

n

i

ia

1

a1 + a2 + ... + an

a the positive square root of the real numbera

a the modulus of the real numbera

n! nfactorial forn +

U {0}, (0! = 1)

r

n

the binomial coefficient)!(!

!

rnr

n

, forn, r + U {0}, 0 YrYn

!

1)(1)(

r

rn...nn +

, forn , r +

U {0}


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4. Functions

f function f

f(x) the value of the function fat x

f: A B f is a function under which each element of set A has an image in set Bf: xy the function fmaps the element xto the element yf1 the inverse of the function f

g o f, gf the composite function offand g which is defined by(g o f)(x)orgf(x) = g(f(x))

limax

f(x) the limit off(x) as xtends to ax ; x an increment ofx

x

y

d

d the derivative ofywith respect to x

n

n

x

y

d

d the nth derivative ofywith respect to x

f'(x), f'(x), ,f(n)(x) the first, second, nth derivatives off(x) with respect to x

xyd indefinite integral ofywith respect to x

b

a

xyd the definite integral ofy with respect to xfor values ofxbetween a and bx& , x&& , the first, second, derivatives ofxwith respect to time

5. Exponential and Logarithmic Functions

e base of natural logarithms

ex, exp x exponential function ofx

logax logarithm to the base aofx

ln x natural logarithm ofx

lg x logarithm ofxto base 10

6. Circular Functions and Relations

sin, cos, tan,cosec, sec, cot

the circular functions

sin1 , cos1 , tan1 cosec1 , sec1 , cot1

the inverse circular functions


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7. Complex Numbers

i square root of1

z a complex number, z = x +iy

= r(cos + i sin ), r +0 = rei, r +

0

Re z the real part ofz, Re (x + iy) = x

Im z the imaginary part ofz, Im (x + iy) = yz the modulus ofz, yx i+ = (x2 + y2), rr =)sini+(cos

arg z the argument ofz, arg(r(cos + i sin )) = , <

z* the complex conjugate ofz, (x + iy)* = x iy

8. Matrices

M a matrix M

M1 the inverse of the square matrix M

MT the transpose of the matrix M

detM the determinant of the square matrix M

9. Vectors

a the vectora

AB the vector represented in magnitude and direction by the directed line segment AB

a unit vector in the direction of the vectora

i, j, k unit vectors in the directions of the cartesian coordinate axes

a the magnitude ofa

AB the magnitude of AB

a.b the scalar product ofaand b

aPb the vector product ofa and b

10. Probability and Statistics

A, B, C,etc. events

A B union of events Aand B

A B intersection of the events Aand B

P(A) probability of the event A

A' complement of the event A, the event notAP(A | B) probability of the event Agiven the event B

X, Y, R,etc. random variables


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x, y, r,etc. value of the random variables X, Y, R, etc.

1x ,

2x , observations

1f ,

2f , frequencies with which the observations, x1,x2 occur

p(x) the value of the probability function P(X= x) of the discrete random variable X

1p ,

2p probabi l i t ies of the va lues

1x ,

2x , of the discrete random variableX

f(x), g(x) the value of the probability density function of the continuous random variable X

F(x), G(x) the value of the (cumulative) distribution function P(X x) of the randomvariableX

E(X) expectation of the random variable X

E[g(X)] expectation ofg(X)

Var(X) variance of the random variable X

B(n,p) binominal distribution, parameters nand p

Po() Poisson distribution, mean N(, 2) normal distribution, mean and variance 2

population mean

2 population variance

population standard deviation

x sample mean

s2 unbiased estimate of population variance from a sample,

( )221

1xx

n

s

=

probability density function of the standardised normal variable with distributionN (0, 1)

corresponding cumulative distribution function linear product-moment correlation coefficient for a populationr linear product-moment correlation coefficient for a sample

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H1 Math Syllabus