- Information
- AI Chat
Notes and formula for easy mathematics
Diploma in Chemical Engineering (EH110)
Universiti Teknologi MARA
Recommended for you
Related documents
- Lab Report 2 (Titration) (CHE142) (EH110) (Diploma in Chemical Engineering UiTM)
- Lab Report 1 (Electrolysis) (CHE142) (EH110) (Diploma in Chemical Engineering UiTM)
- Guidelines For Siting and Zoning of Industry and Residental Areas 2012
- A+Step+by+Step+Guide+to+a+More+Strategic+Site+Selection+Approach
- Water Analysis Group 5
- Liquid Liquid Extraction
Preview text
NOTES AND FORMULAE
SPM MATHEMATICS
FORM 1 – 3 NOTES
1. SOLID GEOMETRY
(a) Area and perimeter Triangle
A = 21 base height
= 21 bh
Trapezium
A = 21 (sum of two parallel sides) height = 21 ( a + b) h
Circle
Area = r 2 Circumference = 2 r
Sector Area of sector =
360
r 2 Length of arc =
360
2 r
Cylinder
Curve surface area = 2 rh
Sphere
Curve surface area = 4 r 2
(b) Solid and Volume Cube:
V = x x x = x 3
Cuboid:
V = l b h = lbh
Cylinder
V = r 2 h
Cone
V = 31 r 2 h
Sphere
V = 34 r 3
Pyramid
V = 31 base area height
Prism
V = Area of cross section length
2. CIRCLE THEOREM
Angle at the centre = 2 × angle at the circumference x = 2y
Angles in the same segment are equal x = y
Angle in a semicircle
ACB = 90o
Sum of opposite angles of a cyclic quadrilateral = 180o
a + b = 180o
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. b = a
Angle between a tangent and a radius = 90o
OPQ = 90o
The angle between a tangent and a chord is equal to the angle in the alternate segment. x = y
If PT and PS are tangents to a circle, PT = PS TPO = SPO TOP = SOP
3. POLYGON
(a) The sum of the interior angles of a n sided polygon = ( n – 2) 180 o
(b) Sum of exterior angles of a polygon = 360o
(c) Each exterior angle of a regular n sided polygon =
n
0
360
(d) Regular pentagon
Each exterior angle = 72o Each interior angle = 108o
(e) Regular hexagon
Each exterior angle = 60o
Each interior angle = 120o
(f) Regular octagon
Each exterior angle = 45o
Each interior angle = 135o
4. FACTORISATION
(a) xy + xz = x(y + z)
(b) x 2 – y 2 = (x – y)(x + y)
(c) xy + xz + ay + az = x (y + z) + a (y + z) = (y + z)(x + a)
(d) x 2 + 4 x + 3 = ( x + 3)( x + 1)
5. EXPANSION OF ALGERBRAIC
EXPRESSIONS
(a)
2x 2 – 6x + x – 3 = 2x 2 – 5x − 3
(b) (x + 3) 2 = x 2 + 2 × 3 × x + 3 2 = x 2 + 6x + 9 (c) (x – y)(x + y) = x 2 + xy – xy – y 2 = x 2 – y 2
6. LAW OF INDICES
(a) x m x n = x m + n
(b) x m x n = x m – n
(c) ( x m)n = x m n
(d) x- n = n
x
1
(e) xn nx
1
(f) n m
xn x
m
( )
(g) x 0 = 1
7. ALGEBRAIC FRACTION
Express 2
1 10
2 6
k k k
as a fraction in its simplest
form. Solution:
22
1 10 1 3 (10 )
266
k k k k kk
=
2 2 2 2
3 10 4 10 2( 5) 5
6 6 6 3
k k k k k k k k k
8. LINEAR EQUATION
Given that
1
5
(3n + 2) = n – 2, calculate the value
of n. Solution:
1
5
(3n + 2) = n – 2
5 ×
1
5
(3n + 2) = 5(n – 2)
3n + 2 = 5n – 10 2 + 10 = 5n – 3n 2n = 12 n = 6
- SIMULTANEOUS LINEAR EQUATIONS (a) Substitution Method: y = 2x – 5 --------(1) 2x + y = 7 --------(2) Substitute (1) into (2) 2x + 2x – 5 = 7 4x = 12 x = 3 Substitute x = 3 into (1), y = 6 – 5 = 1 (b) Elimination Method: Solve: 3x + 2y = 5 ----------(1) x – 2y = 7 ----------(2) (1) + (2), 4x = 12, x = 3 Substitute into (1) 9 + 2y = 5 2y = 5 – 9 = −
n(A ) –number of element in set A. A – Complement of set A.
(b) Venn Diagram
A B
A B
A
Example:
n ( A ) = 7 + 6 = 13
n ( B ) = 6 + 10 = 16 n ( A B ) = 6 n ( A B ) = 7 + 6 + 10 = 23 n ( A B ‟) = 7 n ( A ‟ B ) = 10 n ( A B ) = 7 + 10 + 2 = 19 n ( A B ) = 2
4. MATHEMATICAL REASONING
(a) Statement A mathematical sentence which is either true or false but not both.
(b) Implication If a , then b a – antecedent b – consequent
„ p if and only if q ‟ can be written in two implications: If p , then q If q , then p
(c) Argument Three types of argument: Type I Premise 1: All A are B Premise 2 : C is A Conclusion: C is B
Type II Premise 1: If A , then B Premise 2: A is true Conclusion: B is true.
Type III Premise 1: If A , then B Premise 2: Not B is true. Conclusion: Not A is true.
5. THE STRAIGHT LINE
(a) Gradient
Gradient of AB =
m = 2 1
2 1
x x
y y
(b) Equation of a straight line
Gradient Form:
y = mx + c
m = gradient c = y-intercept
Intercept Form:
1
b
y
a
x
a = x− intercept b = y− intercept
Gradient of straight line m =
-int ercept
-intercept
y
x
=
a
b
6. STATISTICS
(a) Class, Modal Class, Class Interval Size, Midpoint, Cumulative frequency, Ogive Example : The table below shows the time taken by 80 students to type a document.
Time (min) Frequency 10 - 15 -
1
7
20 -
25 -
30 -
35 -
40 -
45 -
12
21
19
12
6
2
For the class 10 – 14 : Lower limit = 10 min Upper limit = 14 min
Lower boundary = 9 min Upper boundary = 14 min
Class interval size = Upper boundary – lower boundary = 14 – 9 = 5 min
Modal class = 25 – 29 min
Midpoint of modal class =
2
25 29 = 27
To draw an ogive, a table of upper boundary and cumulative frequency has to be constructed. Time (min) Frequency
Upper boundary
Cumulative frequency 5- 10 - 15 - 20 - 25 - 30 - 35 - 40 - 45 -
0
1
7
12
21
19
12
6
2
9.
14.
19.
24.
29.
34.
39.
44.
49.
0
1
8
20
42
60
72
78
80
From the ogive : Median = 29 min First quartile = 24. 5 min Third quartile = 34 min Interquartile range = 34 – 24. 5 = 9 min.
(b) Histogram, Frequency Polygon Example: The table shows the marks obtained by a group of students in a test.
Marks Frequency 1 – 10 11 – 20 21 – 30 31 – 40 41 – 50
2
8
16
20
4
7. TRIGONOMETRY
sin o = Opposite hypotenuse
AB AC
cos o = adjacent BC hypotenuse AC
tan o = opposite adjacent
AB BC
Acronym: “Add Sugar To Coffee”
Trigonometric Graphs
y = sin x
y = cos x
y = tan x
8. ANGLE OF ELEVATION AND DEPRESSION
(a) Angle of Elevation
458 = 100 101 2
11. GRAPHS OF FUNCTIONS
(a) Linear Graph y = mx + c
(b) Quadratic Graph
y = ax 2 + bx + c
(c) Cubic Graph y = ax 3 + c
null(d) Reciprocal Graph
x
a
y
12. TRANSFORMATION
(a) Translastion
Description: Translastion
k
h
Example : Translastion
3
4
(b) Reflection Description: Reflection in the line __________
Example: Reflection in the line y = x.
(c) Rotation Description: Direction ______rotation of angle______about the centre _______.
Example: A clockwise rotation of 90o about the centre (5, 4).
(d) Enlargement Description: Enlargement of scale factor ______, with the centre ______.
Example : Enlargement of scale factor 2 with the centre at the origin.
2
Area of image
Area of object
k
k = scale factor
(e) Combined Transformtions Transformation V followed by transformation W is written as WV.
13. MATRICES
(a)
b d
a c
d
c
b
a
(b)
kb
ka b
a k
(c)
ce dg cf dh
ae bg af bh g h
e f c d
a b
(d) If M =
c d
a b
, then
M -1 =
c a
d b
ad bc
1
(e) If ax + by = h cx + dy = k
k
h
y
x
c d
a b
k
h c a
d b y ad bc
x 1
(f) Matrix
ac
bd
has no inverse if ad – bc = 0
14. VARIATIONS
(a) Direct Variation If y varies directly as x , Writtn in mathematical form: y x , Written in equation form: y = kx , k is a constant.
(b) Inverse Variation If y varies inversely as x ,
Written in mathematical form:
1
y
x
Written in equation form:
x
k
y , k is a constant.
(c) Joint Variation If y varies directly as x and inversely as z ,
Written in mathematical form:
x
y
z
,
Written in equation form:
z
kx
y , k is a
constant.
15. GRADIENT AND AREA UNDER A GRAPH
(a) Distance-Time Graph
Gradient =
distance
time
= speed
Average speed =
Total distance
Total time
(b) Speed-Time Graph
Gradient = Rate of change of speed
=
t
v u
= acceleration
Distance = Area below speed-time graph
16. PROBABILITY
(a) Definition of Probability Probability that event A happen,
()
()
()
nA
PA
nS
S = sample space
(b) Complementary Event P ( A ) = 1 – P ( A )
(c) Probability of Combined Events
(i) P(A or B ) = P ( A B )
(ii) P(A and B) = P(A B)
- BEARING Bearing Bearing of point B from A is the angle measured clockwise from the north direction at A to the line joining B to A. Bearing is written in 3 digits.
Example : Bearing B from A is 060 o
18. THE EARTH AS A SPHERE
(a) Nautical Miles 1 nautical mile is the length of the arc on a great circle which subtends an angle of 1 at the centre of the earth.
(b) Distance Between Two Points on a Great Circle.
Distance = 60 nautical miles = angle between the parallels of latitude measured along a meridian of longitude.
Notes and formula for easy mathematics
Course: Diploma in Chemical Engineering (EH110)
University: Universiti Teknologi MARA
- Discover more from:
Recommended for you
Students also viewed
Related documents
- Lab Report 2 (Titration) (CHE142) (EH110) (Diploma in Chemical Engineering UiTM)
- Lab Report 1 (Electrolysis) (CHE142) (EH110) (Diploma in Chemical Engineering UiTM)
- Guidelines For Siting and Zoning of Industry and Residental Areas 2012
- A+Step+by+Step+Guide+to+a+More+Strategic+Site+Selection+Approach
- Water Analysis Group 5
- Liquid Liquid Extraction