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Formula book 2009 (3rd Edition)
introduction to physics (phy2018)
University College Lahore
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Mathematics
Important points and formulas
Third Edition (May 2009)
Reference
Content
- 1 Natural numbers No. Topic / sub topic Page
- 2 Whole Numbers
- 3 Integers
- 4 Rational Numbers
- 5 Irrational Numbers
- 6 Terminating Decimals
- 7 Recurring Decimals
- 8 Significant figures
- 9 Decimal Places
- 10 Standard Form
- 11 Conversion Factors
- 12 Time
- 13 Percentages
- 14 Simple Interest
- 15 Compound Interest
- 16 Speed, Distance and Time
- 17 Quadratic Equations
- 18 Expansion of algebraic expressions
- 19 Factorization of algebraic expressions
- 20 Ordering
- 21 Variation
- 22 PYTHAGORAS’ THEOREM
- 23 Area and Perimeter
- 24 Surface Area and Volume
- 25 Angles on a straight line
- 26 Vertically opposite angles
- 27 Different types of triangles
- 28 Parallel Lines
- 29 Types of angles
- 30 Angle properties of triangle
- 31 Congruent Triangles
- 32 Similar Triangles
- 33 Areas of Similar Triangles
- 34 Polygons
- 35 Similar Solids
- 36 CIRCLE
- 37 Chord of a circle
- 38 Tangents to a Circle
- 39 Laws of Indices
- 40 Solving Inequalities
- 41 TRIGONOMETRY
- 42 Bearing
- 43 Cartesian co-ordinates
- 44 Distance – Time Graphs
- 45 Speed – Time Graphs
- 46 Velocity
- 47 Acceleration
- 48 SETS - 49 Loci and construction - 50 Vectors - 51 Column Vectors - 52 Parallel Vectors - 53 Modulus of a Vector - 54 MATRICES - 55 The Inverse of a Matrix - 56 Transformations 16 - - 57 Transformation by Matrices - 58 STATISTICS - 59 Probability - 60 Symmetry
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Conversion Factors:
Length: 1 km = 1000 m ݉݇ means kilometer 1 m = 100 cm ݉ means meter 1 cm = 10 mm ݉ܿ means centimeter ݉݉ means millimeter
Mass: 1 kg = 1000 gm where kg means kilogram 1 gm = 1000 mgm gm means gram 1 tonne = 1000 kg mgm means milligram
Volume: 1 litre = 1000 cm 3 1 m 3 = 1000 litres 1 kilo litre = 1000 litre 1 dozen = 12
Time: 1 hour = 60 minutes = 3600 seconds 1 minute = 60 seconds. 1 day = 24 hours
1 week = 7 days 1 leap year = 366 days 1 light year = 9 × 10 12 km.
1 year = 12 months = 52 weeks = 365 days.
Percentages: Percent means per hundred. To express one quantity as a percentage of another, first write the first quantity as a fraction of the second and then multiply by 100. Profit = S. – C. Loss = C. – S. Profit percentage =
ܲܥ−ܲܵ ܲܥ × 100 Loss percentage = ܲܵ−ܲܥܲܥ × 100
where CP = Cost price and SP = Selling price
Simple Interest: To find the interest: ݅=
ܴܶܲ 100 where P = money invested or borrowed R = rate of interest per annum T = Period of time (in years)
To find the amount: ܣ = ܲ + ܫ where A = amount
Compound Interest: A = 1 + 100 r
n
Where, 𪀀 stands for the amount of money accruing after ݊ year. 𭰀 stands for the principal 𮐀 stands for the rate per cent per annum stands for the number of years for which the money is invested.
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Speed, Distance and Time : Distance = speed x time
Speed =
݁ܿ݊ܽݐݏ݅݀ ݁݉݅ݐ
Time =
݁ܿ݊ܽݐݏ݅݀ ݀݁݁ܵ
Average speed =
݈ܽݐݐ ݁ܿ݊ܽݐݏ݅݀ ݈ܽݐݐ ݁݉݅ݐ
Units of speed: km/hr, m/sec Units of distance: km, m Units of time: hr, sec
km / hr × = m / sec
m / sec × = km / hr
ALGEBRA
Quadratic Equations : An equation in which the highest power of the variable is 2 is called quadratic equation. Thus ax 2 + b x + c = 0 where a , b, c are constants and a ≠ Ϭ is a geŶeƌal eƋuatioŶ. Solving quadratic equations: We can solve quadratic equation by method of,
a) Factorization b) Using the quadratic formula c) Completing the square
(a) Solution by factors: Consider the equation c × d = 0, where c and d are numbers. The product c × d can only be zero if either c or d (or both) is equal to zero. i. c = 0 or d = 0 or c = d = 0.
(b)Solution by formula : The solutions of the quadratic equation ax 2 + b x + c = 0 are given by the formula:
ݔ =
ܾ−± ܾ 2 − 4 ܿܽ
2 ܽ
(c) Completing the square
Make the coefficient of x 2 , i. a = 1 Bring the constant term, i. c to the right side of equation. Divide coefficient of x, i. by 2 and add the square i. ( 2 ܾ) 2 to both sides of the equation. Factorize and simplify answer
Expansion of algebraic expressions ܽ ܾ+ܿ =ܾܽ+ܿܽ ( a + b) 2 = a 2 + 2 a b + b 2
( a – b) 2 = a 2 – 2 a b + b 2
a 2 + b 2 = ( a + b) 2 – 2 a b
a 2 – b 2 = ( a + b)( a – b)
Factorization of algebraic expressions
ܽ 2 + 2ܾܽ+ܾ 2 = (ܽ+ܾ) 2 ܽ 2 − 2 ܾܽ+ܾ 2 = (ܾ− ܽ) 2 ܽ 2 ܾ− 2 = ܽ+ܾ (ܾ− ܽ)
Ordering: = is equal to ≠ is Ŷot eƋual to > is greater than
ш is gƌeateƌ thaŶ oƌ eƋual to < is less than ч is less thaŶ oƌ eƋual to
D
S T
18
5
5
18
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Trapezium
Area = 12 ܽ+ܾ ݄
perimeter = Sum of all sides
Circle
Area = r 2 circumference = 2 r
Semicircle Area = r 2 perimeter = d + d
Sector Area = length of an arc =
Surface Area and Volume: Figure Diagram Surface Area Volume Cylinder Đuƌǀed suƌfaĐe aƌea = Ϯπ rh total suƌfaĐe aƌea = Ϯπ r ( h + r )
Volume =
Cone
nullcurved surface area = ݈ݎ where l = (r 2 h 2 ) total surface area = ݎ(݈ + ݎ)
Volume = 3
1 π r 2 h
Sphere SuƌfaĐe aƌea = κπ r 2
Volume = 3
4 π r 3
Pyramid
nullBase area + area of the shapes in the sides
Volume = 3
1 × base area ×
perpendicular height
Cuboid
Surface area = 2(ܾ݈ + ݄ܾ + ݄݈) Volume = ݈ × ܾ × ݄
Cube Surface area = 6 ݈ 2 Volume = ݈ 3
Hemisphere Curved surface area = = 2 r 2 Volume =
2
1
2
1
360
r 2 360
2
r
r 2 h
3 3
2
r
r
r
ݎ
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GEOMETRY
(a) Angles on a straight line The angles on a straight line add up to 180o.
x + y + z = 180 o
(b) Angle at a point
The angles at a point add up to 360o. a + b + c + d = 360 o
(c) Vertically opposite angles If two straight line intersect, then ܽ=ܿ ܾ=݀ (Vert,opp.ݏ∠)
Parallel Lines: When lines never meet, no matter how far they are extended, they are said to be parallel.
Vertically opposite angles are equal. a = c; b = d; p = s and q =r
Corresponding angles are equal. ܽ= ݍ; ܾ = ; ܿ = ݎ and ݀ = ݏ Alternate angles are equal. c= q and d = p. Sum of the angles of a triangle is 180o. Sum of the angles of a quadrilateral is 360o.
Types of angles Given an angle , if < 90 ° , then is an acute angle
90 ° << 180 ° , then is an obtuse angle
180 ° << 360 ° , then is an reflex angle
Triangles
Different types of triangles:
An isosceles triangle has 2 sides and 2 angles the same. AB = AC ABC = BCA
An equilateral triangle has 3 sides and 3 angles the same.
AB = BC = CA and ABC = BCA = CAB
- A triangle in which one angle is a right angle is called the right angled triangle.
ABC = 90o
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iii) The ratios of the corresponding sides are equal and the angles between them are equal.
∆ PQ‘ is siŵilaƌ to ∆ XY) ;if, foƌ eg: P = X and XY
P Q=
XZ
P R)
Areas of Similar Triangles: The ratio of the areas of similar triangles is equal to the ratio of the square on corresponding sides.
a rea of PQR
a rea of ABC
= 2
2
P Q
AB =
2
2
QR
BC =
2
2
P R
AC
Polygons: i) The exterior angles of a polygon add up to 360o. ii) The sum of the interior angles of a polygon is (݊ – 2) × 180o where ݊ is the number of sides of the polygon. iii) A regular polygon has equal sides and equal angles. iv) If the polygon is regular and has ݊ sides, then each exterior angle = 360 ݊ v) 3 sides = triangle 4 sides = quadrilateral 5 sides = pentagon 6 sides = hexagon 7 sides = heptagon 8 sides = octagon 9 sides = nonagon 10 sides = decagon
Similar Solids: If two objects are similar and the ratio of corresponding sides is k, then the ratio of their areas is ݇ 2. the ratio of their volumes is ݇ 3.
Length Area Volume ݈ 1 ݈ 2
=
ݎ 1
ݎ 2
=
݄ 1
݄ 2
2
1 A
A
= 2
2
2 1 r
r = 2 2
2 1 h
h 2
1 V
V
= 3
2
3 1 r
r = 3 2
3 1 h
h
Z
X Y
R
P Q
R
P Q
C
A B
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CIRCLE
The angle subtended by an arc at the centre is twice the angle subtended at the circumference
Angles subtended by an arc in the same segment of a circle are equal.
The angle in a semi-circle is a right angle. [or if a triangle is inscribed in a semi-circle the angle opposite the diameter is a right angle]. ܤܲܣ∠= 90 °
Opposite angles of a cyclic quadrilateral add up to 180 o (supplementary). The corners touch the circle. A+C = 180o, B+D 180o
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.(ܾ=)
Chord of a circle: A line joining two points on a circle is called a chord. The area of a circle cut off by a chord is called a segment. AXB is the minor arc and AYB is the major arc.
a) The line from the centre of a circle to the mid-point of a chord bisects the chord at right angles. b) The line from the centre of a circle to the mid-point of a chord bisects the angle subtended by the chord at the centre of the circle.
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Solving Inequalities : When we multiply or divide by a negative number the inequality is reversed. Eg: 4 > - By multiplying by -2 [4 (-2) < (-2)(-2) ] -8 < +
TRIGONOMETRY
Let ABC be a right angled triangle, where B = 90o
Sin =
݁ݐ݅ݏܱ ݁݀݅ܵ ݁ݏݑ݊݁ݐݕܪ =
ܱ ܪ
Cos =
ݐ݆݊݁ܿܽ݀ܣ ݁݀݅ܵ ݁ݏݑ݊݁ݐݕܪ =
ܣ ܪ
Tan =
݁ݐ݅ݏܱ ݁݀݅ݏ ݐ݆݊݁ܿܽ݀ܣ ݁݀݅ܵ =
ܱ ܣ
Sine Rule:
A
a sin
=
B
b sin
=
C
c sin
Cosine Rule:
To find the length of a side: a 2 = b 2 + c 2 - 2bc cosA
b 2 = a 2 + c 2 – 2ac cos B
c 2 = a 2 + b 2 – 2ab cos C
To find an angle when all the three sides are given:
cos A = bc
b c a 2
2 2 2
cos B = a c
a c b 2
2 2 2
cos C = a b
a b c 2
2 2 2
Bearing The bearing of a point B from another point A is; (a) an angle measured from the north at A. (b) In a clockwise direction. (c) Written as three-figure number (i. from 000 ° to 360°) Eg: The bearing of B from A is 050 °.
SOH CAH TOA
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Cartesian co-ordinates
Gradient and equation of a straight line The gradient of the straight line joining any two given points xA 1 ,( y 1 )and B(x 2 ,y 2 ) is;
2 1
2 1 x x
y y m
The gradient/intercept form of the equation of a straight line is ymxc, where mgradient and c intercept on y – axis.
The midpoint of the line joining two points xA 1 ,( y 1 )and B(x 2 ,y 2 ) is; 𠀀= ݔ 1 + 2 ݔ 2 ,ݕ 1 + 2 ݕ 2
The distance between two points xA 1 ,( y 1 )and B(x 2 ,y 2 ) is; ܤܣ= ݔ 2 ݔ− 1 2 + ݕ 2 ݕ− 1 2 Parallel lines have the same gradient.
In a graph, gradient = Hor izonta lheight
Vertica lheight or ݕݔ
Distance – Time Graphs:
From O to A : Uniform speed From B to C : uniform speed From A to B : Stationery (speed = 0) The gradient of the graph of a distance-time graph gives the speed of the moving body.
Speed – Time Graphs :
From O to A : Uniform speed From A to B : Constant speed (acceleration = 0) From B to C : Uniform deceleration / retardation
The area under a speed –time graph represents the distance travelled. The gradient of the graph is the acceleration. If the acceleration is negative, it is called deceleration or retardation. (The moving body is slowing down.)
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Loci and construction The locus of a point is a set of points satisfying a given set of conditions.
(a) Locus of points at a distance x from a given point, O.
Locus: The circumference of a circle centre O, radius x.
(b) Locus of a points at a distance x from a straight line AB
Locus: A pair of parallel lines to the given line AB.
(c) Locus of points equidistance between 2 points.
Locus: Perpendicular bisector of the two points.
(d) Locus of points equidistant from two given lines AB and AC
Locus: Angle bisector of ܥܣܤ∠
Vectors: A vector quantity has both magnitude and direction.
Vectors a and b represented by the line segments can be added using the parallelogram rule or the nose- to- tail method.
A scalar quantity has a magnitude but no direction. Ordinary numbers are scalars. The negative sign reverses the direction of the vector. The result of a – b is a + -b i. subtracting b is equivalent to adding the negative of b.
Addition and subtraction of vectors
ܣܱ +ܥܣ = ܥܱ (Triangular law of addition)
ܤܱ +ܣܱ = ܥܱ ( parallelogram law of addition)
b
a
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Column Vectors: The top number is the horizontal component and the bottom number is the vertical component
y
x
Parallel Vectors: Vectors are parallel if they have the same direction. Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.
In general the vector k
b
a is parallel to
b
a
Modulus of a Vector : The modulus of a vector a, is written as a and represents the length (or magnitude) of the vector.
In general, if x =
n
m , x = (m 2 n 2 )
MATRICES:
Addition and Subtraction: Matrices of the same order are added (or subtracted) by adding (or subtracting) the corresponding elements in each matrix.
c d
a b +
r s
p q =
c r d s
a p b q
c d
a b -
r s
p q =
c r d s
a p b q
Multiplication by a Number: Each element of a matrix is multiplied by the multiplying number.
k ×
c d
a b =
kc kd
ka kb
Multiplication by another Matrix: Matrices may be multiplied only if they are compatible. The number of columns in the left-hand matrix must equal the number of rows in the right-hand matrix.
c d
a b ×
r s
p q =
cp dr cq ds
a p br a q bs
In matrices A 2 means A × A. [you must multiply the matrices together]
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Repeated Transformations: XT;PͿ ŵeaŶs ͚peƌfoƌŵ tƌaŶsfoƌŵatioŶ T oŶ P aŶd theŶ peƌfoƌŵ X oŶ the iŵage.͛ XX(P) may be written X 2 (P).
Inverse Transformations : The inverse of a transformation is the transformation which takes the image back to the object.
If translation T has a vector
y
x , then the translation which ahs the opposite effect has vector
y
x .
This is written as T-1. If rotation R denotes 90o clockwise rotation about (0, 0), then R-1 denotes 90o anticlockwise rotation about (0, 0). For all reflections, the inverse is the same reflection. Base vectors
The base vectors are considered as I =
0
1
and J =
1
0
The columns of a matrix give us the images of I and J after the transformation.
Shear:
Shear factor =
݁ܿ݊ܽݐݏ݅ܦ ܽ ݐ݊݅ ݏ݁ݒ݉ ݁ݑ݀ ݐ ݄݁ݐ ݎ݄ܽ݁ݏ ݎ݈ܽݑܿ݅݀݊݁ܽ݁ܲ ݁ܿ݊ܽݐݏ݅݀ ݂ ݄݁ݐ ݐ݊݅ ݉ݎ݂ ݄݁ݐ ݀݁ݔ݂݅ ݈݁݊݅ =
ܽ
ܾ
[The shear factor will be the same calculated from any point on the object with the exception of those
on the invariant line]
Stretch:
To describe a stretch, state;
i. the stretch factor, p ii. the invariant line, iii. the direction of the stretch (always perpendicular to the invariant line)
Scale factor =
ݎ݈ܽݑܿ݅݀݊݁ݎ݁ܲ ݁ܿ݊ܽݐݏ݅݀ ݂ ܥ ′ ݉ݎ݂ ܤܣ ݎ݈ܽݑܿ݅݀݊݁ܽ݁ܲ ݁ܿ݊ܽݐݏ݅݀ ݂ ܥ ݉ݎ݂ ܤܣ
Area of image = × Area of object Where, P is the stretch factor
Area of image = Area of object
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Transformation by Matrices Reflection
Rotation Matrix Angle Direction centre
01 − 01 90 ° anticlockwise (0, 0)
0 1
− 1 0
90 ° clockwise (0, 0)
− 10
0 − 1
180 ° Clockwise/ anticlockwise (0, 0)
Enlargement
݇ 0
0 ݇
where ݇= scale factor and centre of enlargement = (0, 0)
Stretch Matrix Stretch factor Invariant line Direction
݇ 0 01 ݇ y-axis Parallel to x - axis
1 0
0 ݇
݇ x - axis Parallel to y - axis
Shear
Matrix Transformation
1 0 0 − 1
Reflection in the x-axis
− 01 01 Reflection in the y-axis
0 1
1 0
Reflection in the line y = x
− 01 − 01 Reflection in the line y = - x
Matrix Shear factor Invariant line Direction
1 ݇
0 1
݇ x-axis Parallel to x - axis
݇ 101 ݇ y - axis Parallel to y - axis
Formula book 2009 (3rd Edition)
Course: introduction to physics (phy2018)
University: University College Lahore
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