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7/31/2019 Form 4 Notes
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MathematicForm 4
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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
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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
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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.
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Operations on Set
(A B) C (P Q) R
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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
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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.
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(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.
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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
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(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.
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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, .
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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
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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 =
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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
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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.
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(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
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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
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From the ogive :
Median = 29.5 minFirst quartile = 24. 5
min
Third quartile = 34 min
Interquartile range =34 24. 5 = 9.5 min.
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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
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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
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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
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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
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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
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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
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CHAPTER 9: TRIGONOMETRY
sin =Opposite
hypotenuse
AB
AC
=
cos =adjacent BC
hypotenuse AC=
tan = oppositeadjacent
ABBC
=
Add Sugar To Coffee
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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
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TRIGONOMETRIC GRAPHS
y = cos x
y = sin x
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y = tan x
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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
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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.
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CHAPTER 11: LINES AND PLANES
Angle Between a Line and a Plane
In the diagram,
BCis the normal line to the
plane PQRS.
AB is the orthogonal projectionof the lineACto the plane
PQRS.
The angle between the lineAC
and the plane PQRS is BAC
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Angle Between Two Planes
In the diagram,The plane PQRS and the plane
TURS intersects at the line RS.
MNand KNare any two lines
drawn on each plane which are
perpendicular to RS and intersectat the point N.
The angle between the plane
PQRS and the plane TURS is
MNK.
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The angle between the
plane MRS and the
base RSTU is MNK
Note that MNK MRU MST
What is the angle between the plane MRS and the
base RSTU?
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The diagram shows a cuboid with a horizontal base ABCD.
The angle between the planes BQD and ABQP is
A. PQD
B. ABD
C. PBD
D. QBD
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Clone June 2005 P2
The diagram shows a right prism with a rectangle PQRS as
its horizontal base. The right angled triangle UQR is the
uniform cross section of the prism. The rectangle PQUT is
inclined.
Calculate the angle between the
plane TRQ and the base PQRS.
Solution:
The angle between the plane TRQ
and PQRS is TRS
15
8tan TRS =
TRS = tan-1 = 28o 4'
15
8
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