Form 4 Notes

download Form 4 Notes

of 36

  • date post

    05-Apr-2018
  • Category

    Documents

  • view

    221
  • download

    0

Embed Size (px)

Transcript of Form 4 Notes

  • 7/31/2019 Form 4 Notes

    1/36

    MathematicForm 4

  • 7/31/2019 Form 4 Notes

    2/36

    CHAPTER 1: STANDARD FORM AND SIGNIFICANT

    FIGURES

    Standard Form: a 10n

    1 a < 10, n is an integer

    215000 = 2.15 105

    0.000324 = 3.24 10-4

    Significant Figure: 1268954 = 1270000 to 3

    significant figure

    0.003674 = 0.00367 to 3 sig. fig.

    Big number, n is

    positive

    Small number, n isnegative

  • 7/31/2019 Form 4 Notes

    3/36

    CHAPTER 2: QUADRATIC EXPRESSION AND

    EQUATIONS

    Expansion: (2x 5)(x + 3) = 2x2 + 6x 5x 15

    = 2x2 + x 15

    Factorisation: 3x2 - x 2 = (3x + 2)(x 1)

    Quadratic Equation:

    2x

    2

    5x 3 = 0(2x + 1)(x 3) = 0

    2x + 1 = 0, x =2

    1

    x 3 = 0, x = 3

  • 7/31/2019 Form 4 Notes

    4/36

    CHAPTER 3: SET

    The complement of set A

    = A'

    Symbol

    - intersection

    - union

    - subset

    - universal set

    ,{ }

    - empty set

    - is a member of

    n(A) number of element in setA.

    A complement of setA.

  • 7/31/2019 Form 4 Notes

    5/36

    Operations on Set

    (A B) C (P Q) R

  • 7/31/2019 Form 4 Notes

    6/36

    Clone 2005

    The Venn diagram shows the sets K, L and M with the

    elements. It is given that the universal set = K L M andn(K) = n(L M).

    K

    L

    M

    a - 3

    a - 2

    8

    2

    2

    2

    Find the value of a

    n(K) = 2 + 8 + 2 =

    12

    n(L M) = a 2 + 8

    = a + 6

    a + 6 = 12

    a = 6

  • 7/31/2019 Form 4 Notes

    7/36

    CHAPTER 4: MATHEMATICAL REASONING

    (a) Statement

    A mathematical sentence which is either true or false butnot both.

    3 + 4 = 7 A true statement

    32 = 6 A false statement

    x + 5 = 8 Not a statement because it is not known

    whether it is true or false.

  • 7/31/2019 Form 4 Notes

    8/36

    (b) Implication

    Ifa, then b

    a antecedent

    b consequent

    Ifx is an even number, then x is divisible by two

    Antecedent: x is an even number

    Consequent: x is divisible by two.

  • 7/31/2019 Form 4 Notes

    9/36

    p if and only ifq can be written in two implications:

    1. Ifp, then q

    2. Ifq, thenp

    x + 2 = 5 if and only if x = 3

    1. If x + 2 = 5, then x = 3

    2. If x = 3, then x + 2 = 5

  • 7/31/2019 Form 4 Notes

    10/36

    (c) Argument

    Three types of argument:

    Type IPremise 1: AllA are B

    Premise 2 : CisA

    Conclusion: Cis B

    Type II

    Premise 1: IfA, then B

    Premise 2:A is true

    Conclusion: B is true.

    Type III

    Premise 1: IfA, then BPremise 2: Not B is true.

    Conclusion: NotA is true.

  • 7/31/2019 Form 4 Notes

    11/36

    INDUCTION:

    Make a general conclusion by induction for the numerical

    sequence:

    7 = 6(1)2 + 1

    25 = 6(2)2 + 1

    55 = 6(3)2 + 1

    97 = 6(4)2 + 1

    .

    6(n)2 + 1, n = 1, 2, 3, .

  • 7/31/2019 Form 4 Notes

    12/36

    CHAPTER 5: THE STRAIGHT LINE

    (a) Gradient

    Gradient ofAB =

    m =12

    12

    xx

    yy

    (b) Equation of a straight lineGradient Form:

    y = mx + c

    m = gradient

    c= y-intercept

  • 7/31/2019 Form 4 Notes

    13/36

    Intercept Form:

    -int ercept

    -intercept

    y

    x =

    Intercept Form: 1=+b

    y

    a

    x

    a

    b

    Find the equation of the straight line which passes through

    the point A(1, 2) and has a gradient of 3.

    Solution:

    Equation of straight line: y = mx + c

    Substitute A(1, 2) and m = 3, 2 = 3 + c

    c = -1 Equation of straight line: y = 3x 1.Eqn

    a =xintercept

    b = yintercept

    Gradient =

    http://var/www/apps/conversion/current/tmp/scratch27822/C:/Documents%20and%20Settings/user/Local%20Settings/Temp/Rar$DI00.968/Equation%20of%20straight%20line.gsphttp://var/www/apps/conversion/current/tmp/scratch27822/C:/Documents%20and%20Settings/user/Local%20Settings/Temp/Rar$DI00.968/Equation%20of%20straight%20line.gsp
  • 7/31/2019 Form 4 Notes

    14/36

    Parallel Lines

    The gradient of two parallel lines are equal.

    m1 = m2

    OPQR is a parallelogram. Find

    (a) the coordinates of Q

    (b) the equation of QR5

    Solution:

    (a) Q(4, 7)

    2

    1

    04

    02=

    c = 5, Eqn of QR: y = x + 521

    (b) Gradient of OP = mOP = = mQR

  • 7/31/2019 Form 4 Notes

    15/36

    CHAPTER 6: STATISTICS

    (a) Mean

    for ungrouped data.n

    xx

    = n = number of data

    x = mid point

    f

    fx

    = for grouped data.

    (b) Mode

    Mode is the data with highest frequency.

  • 7/31/2019 Form 4 Notes

    16/36

    (d) Class, Modal Class, Class Interval Size, Midpoint,

    Cumulative frequency, Ogive

    Example :

    The table below shows the time taken by 80 studentsto type a document.

    Time (min) Frequency

    10-1415-1920-24

    25-2930-3435-3940-4445-49

    1712

    21191262

    Midpoint of modal class

    = = 272

    2925+

    For the class 10 14 :Lower limit = 10 min

    Upper limit = 14 min

    Lower boundary = 9.5 minUpper boundary = 14.5 min

    Class interval size = Upper

    boundary lower boundary

    = 14.5 9.5 = 5 minModal class = 25 29 min

  • 7/31/2019 Form 4 Notes

    17/36

    Ogive

    To draw an ogive, a table of upper boundary and cumulative

    frequency has to be constructed.Time(min)

    Frequency Upper boundary

    Cumulativefrequency

    5-9

    10-1415-1920-2425-29

    30-3435-3940-4445-49

    0

    17

    1221

    191262

    9.5

    14.519.524.529.5

    34.539.544.549.5

    0

    182041

    60727880

  • 7/31/2019 Form 4 Notes

    18/36

    From the ogive :

    Median = 29.5 minFirst quartile = 24. 5

    min

    Third quartile = 34 min

    Interquartile range =34 24. 5 = 9.5 min.

  • 7/31/2019 Form 4 Notes

    19/36

    Histogram, Frequency Polygon

    (e) Histogram, Frequency Polygon

    Example:The table shows the marks obtained by a group of

    students in a test.

    Marks Frequency

    1 1011 2021 30

    31 4041 50

    28

    16

    204

  • 7/31/2019 Form 4 Notes

    20/36

    CHAPTER 7: PROBABILITY

    (b) Complementary Event

    P(A ) = 1 P(A)

    ( )

    ( ) ( )

    nAPA

    nS=

    (c) Probability of Combined Events

    (i) For mutually exclusive events,AB = P(A orB) = P(AB) = P(A) + P(B)

    (ii) For Independent Events.

    P(A and B) = P(A B) = P(A) P(B)

    Definition of Probability

    (a) Probability that event A happen,

    S = sample space

  • 7/31/2019 Form 4 Notes

    21/36

    CHAPTER 8: CIRCLES III

    Circle Theorems

    O

    A B

    C

    x

    y O

    O

    Angle at the

    centre = 2 angle

    at the

    circumferencex = 2y

    AB

    C

    x

    yO

    O

    D

    Angles in the

    same segment

    are equal

    x = y

    OA B

    C

    9 0O

    Angle in a semicircle

    ACB = 90o

  • 7/31/2019 Form 4 Notes

    22/36

    a

    b

    O

    O

    Sum of opposite

    angles of a cyclicquadrilateral = 180o

    a + b = 180o

    a

    b

    O

    O

    The exterior angle of

    a cyclic quadrilateralis equal to the

    interior opposite

    angle.

    a = b

    O

    P Q

    Angle between a

    tangent and a radius =

    90o

    OPQ = 90o

  • 7/31/2019 Form 4 Notes

    23/36

    x

    y o

    o

    The angle between atangent and a chord is

    equal to the angle in the

    alternate segment.

    x = y

    P

    T

    S

    O

    IfPTand PS are tangents to acircle,

    PT = PS

    TPO = SPO TOP= SOP

  • 7/31/2019 Form 4 Notes

    24/36

    CLONE 2004:

    CDE is a tangent to the

    circle. Find the value of x.

    A. 16o B. 18o

    C. 20o D. 22o

    Solution:

    G = 48oo

    oo

    GFD 66

    2

    48180=

    =

    x = 66o 48o = 18o Answer: B

  • 7/31/2019 Form 4 Notes

    25/36

    CHAPTER 9: TRIGONOMETRY

    sin =Opposite

    hypotenuse

    AB

    AC

    =

    cos =adjacent BC

    hypotenuse AC=

    tan = oppositeadjacent

    ABBC

    =

    Add Sugar To Coffee

  • 7/31/2019 Form 4 Notes

    26/36

    CLONE 2005:

    Find the value of cos .

    A.12

    5 B.

    5

    13

    C.5

    12 D.5

    13

    Solution:

    cos = - cos EGF =5

    13 Ans: B

  • 7/31/2019 Form 4 Notes

    27/36

    TRIGONOMETRIC GRAPHS

    y = cos x

    y = sin x

  • 7/31/2019 Form 4 Notes

    28/36

    y = tan x

  • 7/31/2019 Form 4 Notes

    29/36

    CHAPTER 10: ANGLE OF ELEVATION AND

    DEPRESSION

    Angle of Elevation

    The angle of elevation is the

    angle betweeen the

    horizontal line drawn from

    the eye of an observer andthe line joining the eye of the

    observer to an object which

    is higher than the observer.

    The angle of elevation ofB

    fromA is BAC

  • 7/31/2019 Form 4 Notes

    30/36

    Angle of Depression

    The angle of depression is theangle between the horizontal line

    from the eye of the observer an

    the line joining the eye of the

    observer to an object which islower than the observer.

    The angle of depression ofB

    fromA is BAC.

  • 7/31/2019 Form 4 Notes

    31/36

    CHAPTER 11: LINES AND PLA