Date post: | 04-Apr-2018 |
Category: |
Documents |
Upload: | shi-pei-pei |
View: | 216 times |
Download: | 0 times |
of 18
7/30/2019 Tips for Mathematics s Paper 1
1/18
1
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.
7/30/2019 Tips for Mathematics s Paper 1
2/18
2
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.
7/30/2019 Tips for Mathematics s Paper 1
3/18
3
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
7/30/2019 Tips for Mathematics s Paper 1
4/18
4
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.
7/30/2019 Tips for Mathematics s Paper 1
5/18
5
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,
7/30/2019 Tips for Mathematics s Paper 1
6/18
6
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.
7/30/2019 Tips for Mathematics s Paper 1
7/18
7
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
7/30/2019 Tips for Mathematics s Paper 1
8/18
8
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
7/30/2019 Tips for Mathematics s Paper 1
9/18
9
* if a
7/30/2019 Tips for Mathematics s Paper 1
10/18
10
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
7/30/2019 Tips for Mathematics s Paper 1
11/18
11
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
7/30/2019 Tips for Mathematics s Paper 1
12/18
12
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)
7/30/2019 Tips for Mathematics s Paper 1
13/18
13
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.
7/30/2019 Tips for Mathematics s Paper 1
14/18
14
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).
7/30/2019 Tips for Mathematics s Paper 1
15/18
15
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
7/30/2019 Tips for Mathematics s Paper 1
16/18
16
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 || | |
[ ]
[] []
|| | |
7/30/2019 Tips for Mathematics s Paper 1
17/18
17
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
7/30/2019 Tips for Mathematics s Paper 1
18/18
18
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