tips for mathematics s paper 1

7/30/2019 Tips for Mathematics s Paper 1

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TIPS FOR MATHEMATICS S PAPER 1

Chapter 1 NUMBER AND SETS

Absolute value ||

|| {

|| {

||

|| ||||

| | || ||| | | | || || ||||where ||

Inequalities involving

absolute values.

| | | |

Exponents

if

Surds*answers with

denominators in surds

are considered as not

simplified.

an expression containing a root with an irrational

solution, Rationalising the denominator means getting rid

of the surds

is the conjugate surdof and vice versa.

*When solving equation by squaring

both sides of the equation, the

solutions needed to be checked to get

the correct answers.

example:

Logarithms Laws of logarithms remember that

if 0< a < 1, , so inequality signmust be reversed when both sides of an

inequality is divided by it.

Changing base of logarithms

Complex number

z = a + bi

a = real part

b= imaginary part

i = i2

= -1

(a + bi)( a - bi)=a2+b

2since i =

and

i2

= -1

(a + bi) is the conjugate of

( a - bi)and vice versa. **Equation involving complex numbers can be

solved by equalizing real and imaginary parts

from both sides of the equation.

Argand diagram to represent complex

numberpoint or vector.


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modulus-argumentform of the complex

number

Modulus ofz , || argz = When , determine whether liesin the second or fourth quadrant.

is in radians

- *measure clockwise or anticlockwise

from the positive x-axis

negative indicates that the angle is

measured in the clockwise direction.Sets Algebraic laws of sets.

De Morgans laws

Using definitions,

represents orrepresents and

Chapter 2 Polynomialsp(x)

Factors ofp(x) Given p(a)=0 hence (xa) is the factor ofp(x).

Zeros ofp(x) and a is the zero ofp(x)

Roots of an equation. For the equationp(x) = 0, ifp(a) = 0,

Hence a is the root of the equation.

Factorisation The process of expressingp(x) in terms of its

factors.

p(x)= (x-a)(x-b)(x-c)

if p(x)= (x-a)(x-b)(x-c)= 0

x = a, x = b, x = c, are roots of

p(x)= 0.

Remainder Theorem When a polynomial,p(x) is divided byxa, the

remainder isp(a).

When a polynomial,p(x) is divided by ax + b,

the remainder isp( ).* must know how to do long division

to find q(x) & remainder

p(x) =(xa)q(x) + remainder

q(x) = quotient.

Factor Theorem Ifp(a)= 0, then (x- a) is a factor ofp(x) To factorisep(x), use trial and error to

find the first factor, then use long

division // expanding and comparing

coefficients.


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Completing the square

of a quadratic

equation.

can be used to show

whether a quadratic

polynomial is always

positive or negative.

*to complete the square, the coefficient of x2

must be 1.

if a>0 and q>0, then , f(x)always positive for all values of x.If a


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To solve inequalities

in the form of

f(x) > g(x)

Sketch the graph off(x) and g(x), find

intersection points to solve.

Inequalities involving

modulus sign

Squaring method is used when both sides arepositive for both equations and inequalities.

When quadratic inequality cannot be factorised,then use method of completing the square.

|| ||

[] || [] || Inequality involving

get rid of denominator by multiplying both sides

with [].Partial fractions The process of expressing rational function as sum of two or more simpler fractions.

Rational function Proper fraction Degree off(x) is less than g(x)

Improper fraction Degree off(x) is equal or greater than g(x)

Rules 1. Check that is a proper fraction, if not do long division.2. Check that the denominator g(x) is factorised completely.3. For each linear factor, ax+b in the denominator, there exists a partial fraction in

the form

4. For each quadratic factor ax2+bx + c in the denominator, there exists a partialfraction in the form

5. For each linear factor ax+b repeated n times in the denominator, there exists n

partial fractions of the form

Chapter 3 Sequences and Series

Sequence A list of numbers, stated in a particular order, each number can be derived from theprevious number according to a certain rule. eg: 1,2,3,4,5,

Series The sum of the terms of a sequence. eg: 1+2+3+4+5+ Arithmetic

Progression A.P

A sequence where each term differ from the previous term by a certain number (common

difference), d.

First term, a

The nt

term, Un or Tn Tn = a + (n1)d

Sum ofn terms, Sn

where l is the nth

term.

To prove a sequence is an AP show TnTn-1 = constant = d

* if given three terms a,b, c b - a = cb or 2b = a + c

Given a sequence is an AP start from TnTn-1=Tn-1Tn-2

Geometric

Progression G.P

A sequence where each term can be obtained from the previous term by multiplying by a

certain number (common ratio), r.


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The nt

term, Un or Tn Sum ofn terms, Sn

*must know how to derive Sn

can only be used when || *question involving expressing a recurring

decimal as a rational number eg 3.5252

To prove a sequence is a GP show

Given a sequence is a GP start from=

* if given three terms a, b, c ac = b2

For AP and GP:

SnS

n-1= T

n, can be obtained if S

nis given.

and d = TnTn-1 or r = Summation of a finite series(formula will be provided in examination)

Method of differences

if the general term Ur =f(r+1)f(r) , wheref(r) is a function ofr.

[ ]

[ ]

[ ]

Whenever we see that as express as partial fraction

= 1

as

or divide the numerator and denominatorby the highest power ofn.

Binomial expansion

write expansion

= with + without +

ifn is a positive integer,


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ifn is not a positive integer, * this series is valid only for || .** change

before expanding or

in

terms of , valid for || Chapter 4 Matrices

Matrix a set of number arranged in rows and

columns in a rectangular array and

enclosed by a pair of brackets

Elements numbers in a matrix m x n matrix

also known as order

of matrix

a matrix with m rows and n columns Square matrix equal number of rows and columns m = n

Null or zero matrix All elements are zeros Diagonal matrix All elements except those of the leading

diagonal are zeros. Identity matrix, I

*must be square

matrix

A diagonal matrix with elements in the

leading diagonal are 1sAI = IA=A

Symmetric matrix

*must be square

matrix

All elements are symmetrical about the

leading diagonal.

Equal matrix Two matrices are equal if they have the

same order and if corresponding elements

are equal.

hence:

a = 3, b = 4, c= 6

Multiplication of

matrices

Order ofA Order ofB

m x p p x n

Order ofAB

m x nIfA is a matrix of order m x p and B is a

matrix of order p x n, then AB is a matrix

of order m x n.

Multiply each row of the first matrix A with

each column of the second matrix B.

The of the product matrix AB is theproduct of the i

throw of the first matrix A and

thejth

row of the second matrix, B

*the number of columns inA must be the

same as the number of rows inB.

Properties of Matrices A(BC) = (AB)C Multiplication of matrices is associative.

A(B+C) = AB + AC Multiplication of matrices is distributive over

addition.

**AB BA Multiplication of matrices is notcommutative.


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Transpose of a matrix,

A AT

(AT)

T=A

A matrix whose rows are the columns

and whose columns are rows ofA

AT= A, then A is a symmetrical matrix

Determinant of

matrices

||

|| is not modulus ofA.It is a real number

which can be positive

or negative.

The determinant of a 2 x 2 matrix

||

The determinant of a 3 x 3 matrix

|| || ||

|| ||||

Minor , Mi j The minor of , denoted Mij is thedeterminant of the 2 x 2 matrix obtain by

deleting the ith

row andjth

column.

the minor of 6 is(2 x 0)(8 x3) = - 24

Cofactor, Ci j The cofactor of 6 is= (-1)

1+2(-24) = 24

Inverse matrix of 2 x2 IfA and B are square matrices such that

AB = BA = I, B is known as inverse of

A (B = A-1

) and vice versa.

AA-1

=A-1

A = I

* if||= 0 then A-1do not exists. and A isknown as a singular matrix. || given

Matrix of cofactors Matrix that is formed with the cofactorsas its elements.

Adjoint matrix ofA,

adj A

Transpose of the matrix of cofactors Inverse matrix 3 x 3 if|| thenexists || System of linear

equations

If given M and N and ask to find MN

and MN = n I then M-1

= , then dont

have to use long formula to find M-1

.

If A is a square matrix An= AAAA

n times

Am

An

= Am+n

(Am)

n= A

mn


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Chapter 5 Coordinate Geometry

Distance between 2

points, d Gradient of a line

segment, m

Ratio formulae

Given internal division

external division

* if ask to find ratio then assume the ratio is

m:1 not m:n

midpoint of AB

Straight lines y = mx +c, represents a straight line with

gradient m, cutting they-axis at the point

(0, c).

* if parallel to y-axis, m =(undefined)

if parallel to x-axis, m = 0

y - y1 = m( xx1 ) passing through

(x1, y1) a fixed point.

General equation is ax+by+c =0

Parallel lines Same gradients then m1= m2

Perpendicular lines If two straight lines with gradients m1

and m2are perpendicular, then m1m2= -1

Distance from a point

to a line ax+by+c =0

It is advisable to draw a rough sketch when

answering questions.

Curves

Circle Equation of a circle with centre (a, b) and

radius runits. when O (0,0) is the centre.

*coefficient of x

2and y

2are the same and

no term in xy.

Centre = (-g, -f) and r =

If a circle with centre ( h,k) touches a line

ax +by + c = 0 then Parabolaequal distance from a

fixed point ( the

focus) and fixed line

(the directrix)

*distance from focus

and directrix = a

A (x1, y1)

B (x2,y2)

P (x, y)

m

n

A (x1, y1)

B (x2,y2)

P (x, y)n

m

P(x,y)

.C(a,b)

x2= 4ay

F(0,a) Vertex (0,0)

Directrix, y = -a

(x- h)2= 4a(y-k)

F(h,k+a)

Vertex (h,k)

Directrix, y = k -a


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* if a


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b = c - a

centre (0,0)

foci (c,0), (-c,0)

vertices (a,0), (-a,0)

*the centre is O(0,0) and if the centre

moves to (h,k) then

foci (h c, k)

vertices (h a, k)

b = c - a

centre (0,0)

foci (0,c), (0,-c)

vertices (0,a), (0,-a)

*the centre is O(0,0) and if the centre moves

to (h,k) then

foci (h, k c )

vertices (h, k a )

Parametric equations Cartesian equation can be obtained byeliminating the parameter tfrom the

parametric equations

Chapter 6 Functions

Function a one to one or many to one relationship

each element in set A is mapped to its imagein set B.

*the range of the function can be obtained

easily from the graph of the function within

the domain given.

Domain set ACodomain set B

Range set of images which is a subset of B

*can be determined by sketching the

graph of the function with its given

domain the values of y.

Equality of function Two functions are equal if and only if

they have the same rule and the same

domain.

To test isfis a

function

the vertical line test If any vertical line in the specified domain

cuts the graph at exactly one point thenfis a

function.

To test iffis a one to

one function

the horizontal line test fis a one to one function if and only if any

horizontal line cuts the graph at most one

point.

Operations on

functions

*(fg) is not composite function


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Onto function a function is an onto function if the range

off= codomain off

Algebraic functions Linear functions f(x) = mx + c

m > 0 m < 0

Quadratic functions f(x) = ax + bx + c

a > 0 a < 0

cubic functions

f(x) = ax3+ bx

2+ cx + d

a > 0 a < 0

f(x) = kxn

n =positive odd integer

k > 0 k < 0

Root function wherex

f(x) = kxn

n = (eg : f(x) = 3x1/2)

p = positive even integer >1

k > 0 k < 0

f(x) = kxn

n = (eg : f(x) = 3x1/3 )

p = positive odd integer >1

k< 0 k < 0

Reciprocal function f(x) = kx

n

n =negative odd integer(eg : f(x) = 3x-1

)

k > 0 k < 0

f(x) = kx

n

n =negative even integer(eg : f(x) = 3x- 4

)

k > 0 k < 0

Exponential functions

logarithmic functions

Rational functions

*there are asymptotes where

bassume a=0

-b/a


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Even function symmetrical about the y-axis

Odd function symmetrical about the origin*the graph is unchanged under a 180

o

rotation about the orgin.

Graphs ofy=f(x) and || * the negative part of the graph is reflectedupon the x-axis. No line or curve below thex-axis.

Graphs of Iff(x) = 0 whenx =a, then is not

defined forx=a andx = a is the vertical

asymptote of.

iff(x) cuts they-axis at the point ( 0,a)then

will cut they-axis at the point(0,.

Asf(x)

, Iff(x) has a maximum / minimum atx=a than

has a minimum/ maximumatx=a.

y = f(x) y = - f(x)reflection of f(x) on

the x-axis.

y = f(-x)reflection of f(x) on

the y-axis.

Composite functions

Function gf(x) exists if and only if

Df Rf Dg

Dgf = Df gf

Functionfg(x) exists if and only if

Rg

Rgf

(0, -a) (0, -1/a)


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Inverse functionf-1

*For f-1

to exists, f

must be a one to one f.

Identify functionff

-1and f

-1f

To every one to one function there exists an inverse function such that

ff-1

(x) = x , xdomain of f-1

and f-1f(x) = x, xdomain of f

since the domain is different hence

ff-1

f-1f

y=f(x) y=x

y=f-1

(x)

(the graph of f-1

is the reflection of the graph

f in the line y =x.

Domain of f-1

= Rf Range of f-1

=Df

y = x

if and only if

A function f is

continuous atx = a

if and only if

Limits

[ ]

[]

*it is wrong to substitute the symbol into

the function.

Chapter 7 Differentiation derivative offunctionfwith respect

tox.

also the tangent of the

curve.

* is not dy is the gradient of the tangent of thecurve.


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Differentiation of

standard functions

**

[]

**

** ()

**

** ** []} [ ]

Differentiation of

products of functions

[] the product ruleu = f(x) and v = g(x)Differentiation of

quotients of functions

*

+

the quotient rule

Differentiation of

composite functions the chain rule **Differentiation of

implicit functions

eg

xy + y3

= x2y

To differentiate implicit functions,

1. differentiate both sides of theequation with respect tox.

2. place all the terms withf(x) on oneside,

3. solve forf(x).


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Differentiation of

parametric functions

x = f(t)

y = g(t)

Applications of

differentiations

1. Tangents and Normals

is the gradient of the tangent at a point, P

on the curve.When the coordinates of P is given,

equation of the tangent and normal at P can

be found.

2. Increasing and decreasing functionsif y = f(x) is differentiable in the

interval (a, b) and , then f(x) is

an increasing function.

if y = f(x) is differentiable in the interval

(a, b) and , then f(x) is an decreasing

function.

3. Stationary points

Minimum point - 0 + ( signs of)

Maximum point + 0 - ( signs of

)

4. point of inflexion, x = xo

**

do not change signs for all values

ofx nearxo

but changes signs passing throughxo.

point of inflexion

*there exists a point of inflexion between a

maximum point and a minimum point for a

continuous curve.

Do not waste time determining whether the

stationary point is maximum or minimum if

the question does not ask for it.

To state the stationary points or inflexion

point give the answer in the form (x, y).

5. Curve sketchingFind

axis of symmetry asymptotes* intersections with the axes stationary points 15ehavior of curve asx

*For rational function, ,

a. horizontal asymptotes equatingthe highest power ofx to 0.

b. vertical asymptotes equating thehighest power ofy to 0.

** y-intercept ( whenx = 0) andx-intercept

(wheny = 0)

6. Newton-Raphson method * there exists a real root between

x = a and x = b if the signs of f(a)

and f(b) is different.

( one + the other -)

to find an approximate value for a root of a

non linear equation.

= n+ 1 approximation

X


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7. Rate of change > 0 y increases when t increases. < 0 y decreases when t increases.

Chapter 8 Integration

Integration is the reverse operation of differentiation

Standard integrals || | |

[ ]

[] []

|| | |


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Integration techniques

Always check to see if the

function is in the form of

||

[]to use

[] []

Integration by Substitution *common substitutiona. ( ax + b )n let u = ax + bb. letx = a sin c. letx = a tan d.

let u = cos x or u = sin x

Integration by Partial

Fractions

* for rational functionwhich is not in the form of

Express a rational function as partial

fractions, then use the formula:

||

Integration by Parts *u should be simpler after differentiation

and it must be possible to integrate the

function to get v.

for =x v = x2and u = ln x

* cant integrate ln x

Definite integrals [] Trapezium rule Divide the interval from a to b into n

trapezium, Area under the curve = sum of all trapeziums

= [ ] a b


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Applications of integration

Area of regions Diagram 1

a b

Area is positive above the x-axis

if below the axis change the value to

positive.

Bounded by a curve, thex-axis,x=a

andx= b

b

a

Diagram 2

Area is positive to the right of the y-axis if

to the left of the axis change the value to

positive.

Bounded by a curve, they-axis,y=a

andy= b

[ ]

y=f(x)

y= g(x)

Diagram 3

a b

Bounded by the two curves betweenx =a

andx = b.

* find the intersection points first.

in this interval f(x) is above g(x) or

f(x) > g(x)

Volume of Revolution rotated 360o about the x-axis

refer to Diagram 1

rotated 360o about the y-axis

refer to Diagram 2

[] [] rotated 360

oabout the x-axis

refer to Diagram 3

in this interval f(x) is above g(x) or

f(x) > g(x)

Set by Cg Shi Pei Pei

with XTLCSMKDBDS

Sept 2010

tips for mathematics s paper 1

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