basic math formula
SETS
AU(B C)=(AUB)∩(AUC): Union of Sets is distributive over intersection of sets.
A∩(BUC)=(A∩B)U(A∩C): Intersection of sets is distributive over union of sets.
DEMORGAN’S LAWS:
(AUB)l=Al∩Bl The complement of union of sets in the intersection of their complements.
(A∩B)l=AIUBI The complement of intersection of sets is the union of their complements.
A={Prime numbers < 12}, i.e. A= {2,3,5,7,11}
B={x/x Є N 2 ≤ 5}, i. e. B = {2,3,4,5}
C={Perfect square number < 16}, i.e. C={1,4,9}
In Disjoint sets: A∩B ={} or ∅ and n (A∩B) = 0
A.P.
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G.P.
|
H.P.
| |
General form
nth term
Mean
Sum of nth terms
|
A, a+d, a+2d…..
Tn = a+(n-1)d
A=
Sn=
Sn=
Σn =
|
A, ar, ar2………
Tn = arn-1
G =
If r > 1, Sn=
r < 1, Sn =
S∞ =
|
d= , d= Tn = Sn – Sn-1 = + 1, =
nth term of even natural number = 2n
nth term of odd natural number = 2n-1
Sum of first ‘n’ odd natural numbers = n2
A ≥G ≥ H, A G & H are in G.P
Statistics is the science of collection, classification, tabulation, analysis and interpretation of numerical data. Measures of central tendency give us a bird’s eye view of the huge mass of statistical data. Measures of dispersion give an idea about the homogeneity or heterogeneity of the distribution. The measures of dispersion, which are in common use, are: a) Range b)Quartile deviation c) Mean Deviation d) Standard Deviation.
· Range = H – L, where H-Highest Frequency, L=Lowest Frequency
· Co-efficient of range = (H – L) / (H + L)
· Quartile Deviation (Q.D.) =
Statistical data is the set of observations of characteristics of age, height, weight, income, marks scored, etc. these characteristics are generally are called variables. The values of the variables may be close to the arithmetical average or scattered away from the average. The values vary from the mean and the measure of such variation is called the variance of distribution. The square root of variance is the standard Deviation of distribution. The standard deviation is conventionally represented by the Greek letter ‘Sigma’. Standard Deviation is the square root of the arithmetic average of the squares of the deviations from the mean.
When individual observations are given: 1. The arithmetic mean is computed. 2. The deviations of the individual scores from the arithmetic mean are obtained. 3. The squares of the deviations are calculated. 6. The positive square root of variance is the Standard Deviation.
The calculation of Standard Deviation for a grouped data: 1. The average of class intervals are represented by the middle points. 2. The Deviations from the arithmetic mean are obtained. 3. The squares of these deviations are multiplied by the respective frequencies. 4. The total of these products is divided by the total of the frequencies and the result is the variance. 5. The positive square root of the Variance is the Standard Deviation of the group.
Co-efficient of variation is a relative measure of dispersion. It is based on the arithmetic mean and standard deviation of a frequency distribution. Co-efficient of Variation is generally expressed as a percentage. It is independent of units. Consistency or variability is determined by the co-efficient of variation.
H.C.F AND L.C.M
If the expressions cannot be easily resolved into factors, their H.C.F. may be found by division method. Steps to be followed while finding H.C.F. by Division method: 1. Arrange the terms of the expressions in descending order of their powers. 2. If the powers of the first terms in the expressions are same, consider the form which has a smaller co-efficient as divisor. 3. Continue the division process till the remainder of one of the expressions reduces to zero. 4. The last divisor is the H.C.F. of two expressions. It the last remainder is a constant and not zero, then the H.C.F. of two expressions is 1. In other words, the two expressions are prime to each other. The product of H.C.F and L.C.M. of two expressions is equal to the product of the expressions. A x B = H x l
An expression in three variables a, b, c is said to possess “Cyclical Symmetry” if we get back the original expression by changing a to b, b to c and c to a in order. Then the expression is called a Cyclic Symmetric Expression in a, b, c
An identity is a statement which is true for all values of the unknown. ≡ is used to denote an algebraic identity. The identities which are true for all values of the variables involved, subject to the conditions mentioned, are called conditional identities.
SURDS
If orders and radicands are the same in their simplest form, then such Surds are called Like Surds. A group of surds of different order or different radicand in their simplest form are called Unlike Surds. Sum or difference of the reduced surds can be obtained by adding or subtracting the co-efficient. Only like surds can be added or subtracted. If the surds of different order into same order: 1. Find the L.C.M. of orders of the given surds. 2. Convert each surd into surds of equal order (order equal to the L.C.M). 3. Then, multiply the surds by using the formula: The process of multiplying a surd by another surd to get a rational number is called Rationalization. Then, each surd is a rationalizing factor (R.F.) of the other. A Binomial Surd is the algebraic sum of two surds or the algebraic sum of rational quantity and a monomial surd. While rationalizing the denominator, both the numerator and the denominator must be multiplied by the rationalizing factor of the denominator. When the denominators are conjugate to each other, it can be simplified by taking I. C.M.
An equation involving a variable of degree one is a Linear Equation. A linear Equation has only one root. An equation involving a variable of degree 2 is Quadratic Equation. A Quadratic Equation has only two roots. Quadratic equation involving a variable only in second degree is a “Pure Quadratic equation”.
An equation that can be expressed in the form –
ax2 + c = 0, where a and c are real numbers a = 0 is a Pure Quadratic Equation. Quadratic equation has only two roots. Quadratic equation involving variable in second degree as well as in the first degree is an “Adfected Quadratic Equation”.
ax2 + bx + c = 0 is the standard form of a quadratic equation where a, b and c are variables and a=0. If mn=0, then either
m=0 or n=0. If ‘m’ and ‘n’ are roots, then the standard form of the equation is x2-(Sum of roots) x+ Product of the roots =0. The graph of a quadratic polynomial is a curve called ‘Parabola’.
MODULAR ARITHMETIC
Note that the first day of the week is Monday. Eighth day is again Monday. And fifteenth day is also Monday. This is true because repetitions occur after every seven days. This system is ‘Modulo 7’. This relation is denoted by 1 ≡ 1 (mod 7), 8 ≡ 1 (mod 7), 15 ≡ 1 (mode 7). Here, we have 15-8 ≡ 7 and 15-1 ≡ 14. Here all the differences are multiples of 7. Caley’s table is the representation of modular arithmetic system.
CIRCLE
Circle: The locus of a point moving on a plane, such that it is always at a constant distance from a fixed point i.e., Centre.
Radius: The line segment joining the centre and any point on the circle.
Chord: The line segment joining any two points on the circle
Diameter: A chord that passes through the centre of the circle.
Arc: Any part of the circumference of a circle.
· Number of radii that can be drawn in a circle is infinite.
· Number of chords that can be drawn in a circle is infinite.
· Number of diameters that can be drawn in a circle is innumerable or infinite.
· All radii of circles are equal.
· All chords of a circle need not be equal. # All diameters of circle are equal.
· We cannot construct a circle in space.
· A common circle cannot be constructed in two different planes.
· Examples of circle: Coins, Bangles, Wheels, etc.
· The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
· Equal chords of a circle are equidistant from the centre.
· The greater chord is nearer the centre of the circle.
· As the length of the chord increases, the perpendicular distance decreases.
· As the length of the chord decreases, the perpendicular distance increases.
· The biggest chord in a circle is the diameter. It passes through the centre. The perpendicular distance is zero.
Segment of a Circle: The region bounded by the chord and an arc is called a segment of the circle.
Major Segment: The segment bounded by the chord and the major arc is called the major segment.
Minor Segment: The segment bounded by the chord and the minor arc is called the minor segment.
# Minor arc subtends acute angles. # Semi-circle subtends right angles. # Major arc subtends obtuse angles.
Concentric Circles: Circles having the same centre but different radii are concentric circles.
Congruent Circles: Circles having equal radii are congruent circles.
Secant: A straight line, which cuts the circle at two distinct points, is a secant.
Tangent: A straight line that meets the circle at one and only one point is a tangent. The point where the tangent touches the circle is the point of segment. A Chord cannot become a tangent. In any circle, the radius drawn at the point of contact is perpendicular to the tangent. If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent. If the centres of the circle lie on either side of the common tangent, then it is called a transverse common tangent.
TWO POLYGONS
Two Polygons having the same number of sides are similar, if and only if: 1) The angles of one are equal to the corresponding angles of the other, and 2) the sides of one are proportional to the corresponding sides of the other. If two ratios are equal, then they are said to be in proportion. If a: b=c: d or a/b = c/d, then a,b,c and d are in proportion. Two right angled triangles will be similar, when their acute angles are equal to one another. To prove the similarity of triangle, it is important to identify the corresponding sides and corresponding angles.
Basic Proportionality Theorem or Thales Theorem:
A Straight line drawn parallel to a side of a triangle, divides the other two sides proportionately.
Converse of Thales Theorem: If a line divides the two sides of a triangle in proportion, then the line is parallel to the third side of the triangle.
Corollary of Thales Theorem: If a line is drawn parallel to a side of a triangle, then the sides of the new triangle formed are proportional to the sides of the given triangle.
THEOREMS ON SIMILAR TRIANGLES:
Theorem 1: If two triangles are equiangular, then their corresponding sides are proportional.
Converse of Theorem 1: If the corresponding sides of two triangles are proportional, then the triangles are equiangular.
Theorem 2: The areas of similar triangles are proportional to the squares of the corresponding sides.
THEOREMS ON RIGHT ANGLED TRIANGLES:
Baudhayana Theorem: “The diagonal of the rectangle produces both areas, which its length and breadth produce separately”.
Pythagoras Theorem: “In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining sides.”
Converse of Pythagoras Theorem: If the square on one side of a triangle is equal to the sum of the squares on the other two sides contain a right angle.
Pythagorean Triplets: The sides of a right angled triangle have a special relationship among them. This relation is widely used in many branches of Mathematics such as Mensuration and Trigonometry.
TOUCHING CIRCLES:
Theorem 4: If two circles touch each other, the point of contact and the centres of the circles are collinear.
If two circles touch each other externally, the distance between their centres is equal to the sum of their radii (d=R-r)
If two circles touch each other internally, the distance between their centres is equal to the difference of their radii. (d=R-r) Note: Radii of circles are denoted by R and r.: The tangents drawn to a circle from an external point are: a) equal, b) equally inclined to the line joining the external point and the centre, c) subtend equal angles at the centre.
MENSURATION
It is a branch of mathematics which deals with the measurements of lengths of lines, areas of surfaces and volumes of solids. Mensuration may be divided into two parts: a) Plane Mensuration and b) Solid Mensuration
Plane Mensuration deals with perimeter, length of sides and areas and areas of two dimensional figures and shapes.
Solid Mensuration deals with areas and volumes of solids.
Right circular Cylinder: Wheels of a road roller, a circular based storage tank, etc. are example of a Right Circular Cylinder. It is a solid described by revolution of a rectangle about one of its sides which remains shape. 2) The curved surface joining the plane surfaces is the lateral surface of the cylinder. 3) The two circular planes are parallel to each other and also congruent. 4) The line joining the centres of the circular planes is the axis of the cylinder.
Circular plane is the radius of the cylinder. The two types of Cylinders are: 1) Hollow Cylinder and 2)Solid Cylinder. A hollow cylinder is formed by the lateral surface only. Eg: A Pipe. A solid Cylinder is the region bounded by two circular plane surfaces and also the lateral surface. Ex: A garden roller.
Mensuration
Solid
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C.S.A (sq. units)
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T.S.A. (sq. units)
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Volume (Cubic units)
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Cylinder
Cone
Hemisphere
Sphere
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2 π r h
π r l
2 π
4 π
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2π r (r+h)
π r (r+l)
3 π
4 π
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π
1/3 π
2/3 π
4/3 π
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Area of a circle = π r2
Area of the base of a cylinder/cone/hemisphere = π r2
Circumference of the base of a cylinder/cone/hemisphere = 2 πr
T.S.A. of a cylinder with one side open = π r2 + 2 πrh
Revolution of
(a) Semicircle (along with diameter) → Sphere
(b) Right angled triangle → Cone
(c) Rectangle → Cylinder
SCALE DRAWING
Area of a triangle = x base x height = bh
# Area of rectangle = length x breadth = Ib
# Area of a trapezium = x height x (sum of two parallel sides) = h (a+b)
Measurement of the area of a land: 1) irregular shaped field is divided into known geometrical shaped fragments. 2) Measurements are recorded and a sketch is drawn to the scale. 3) Measurements are recorded in the surveyor’s field book. 4)
Total area of the land is the sum of the areas of all right angled triangles and the trapezium. Area of a land is expressed in hectares. 1 hectare = 10,000 sq.mts.
POLYHEDRA AND NETWORKS
A polyhedron has only area but no volume.
Polygon: A closed figure bounded by straight line segments. The region bounded by a polygon is polygonal region.
Regular Polygon: It is a polygon having equal sides and equal angles. All regular polygons are cyclic.
Polyhedron: A closed figure bounded by polyhedron is a polyhedron. It divides the space into 2 regions, within the polyhedron and outside it. Polyhedron is the plural of Polyhedron.
Polyhedral Solid: A solid bounded by a polyhedron is a polyhedron solid. We denote the number of faces by F edges by E and vertices by V.
Regular Polyhedra: A polyhedron is called a regular polyhedron, if its faces are congruent regular polygons. There are only five types of regular polyhedral:
1) Tetrahedron 2) Hexahedron 3) Octahedron 4) Dodecahedron
Icosohedron: These five polyhedra are known as ‘Platonic Solids’. The names of these solids are based on the number of the faces.
Polyhedra & Network
Regular Polyhedra
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F
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V
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E
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Shape of each face
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Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosohedron
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4
6
8
12
20
|
4
8
6
20
12
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6
12
12
30
30
|
Equilateral Triangle
Square Triangle
Equilateral Triangle
Regular pentagon
Equilateral Triangle
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Name of some pyramids
Name of polyhedra
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F = n+1
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V=n+1
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E=2n
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Square based pyramid (n=4)
Pentagonal based pyramid (n=5)
Hexagonal based pyramid (n=6)
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5
6
7
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5
6
7
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8
10
12
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Name of some prisms
Name of prisms
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F=n+2
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V=2n
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E=3n
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Triangle based prism (n=3)
Square based prism (n=4)
Pentagonal based prism (n=5)
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5
6
7
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6
8
10
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9
12
15
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Euler’s formula for
a) Polyhedra: F+V=E+2
b) Network of graph: N+R=A+2
GRAPH THEORY
It is a branch of mathematics wherein graphs are studied. Graph theory has its application in various fields such as electronics, electrical engineering, network analysis and graphs, etc.
Graph: A set of points together with line segments joining the points in pairs is a graph or network.
Nodes in a Graph: A point is a node, if there is atleast one path (line) starting from it or reaching it. Nodes are named by capital letters of English Alphabet. However, the number of nodes is denoted by ‘N’.
Arcs in a Graph: The line segment (path) joining two nodes is an arc. Number of arcs in a graph is denoted by A. In a network, an arc may be a straight line or a curved line.
Region: An area bounded by arcs (including outside) is called a Region. Number of regions is denoted by R.
#Traversable Graph: A graph is said to be traversable, if it can be drawn in one sweep without lifting the pencil from the paper and tracing the same are twice. It can pass through the nodes several times.
# Euler discovered that, a graph is:
Traversable, if it has only even nodes.
Traversable, if it has only two odd nodes. 3) not traversable, if it has more than two odd nodes.
# Euler’s analysis of seven bridges problem was the first hint for a new branch of mathematics.
Topology, which reached its highest development in the 20th century.
# The information regarding the number of arcs, connecting nodes can be displayed by a matrix.
# If a node is not connected to itself or not to any other node, then it is indicated by zero.
# Order of Node in a Graph: The order of node in a graph is the number of paths starting from it or reaching it.
# Loop of a Node: A single arc connecting a node to itself is called lop at a node.
# Order of a Loop at a Node: A loop can be traced in the clockwise or anti-clockwise directions. Therefore, the order of a node with a loop is 2.
# Even Node: A node is called an even node, if its order is an even number.
# Odd Node: A node is called an odd node. If its order is an odd number
# The sum of the elements in the matrix is the sum of the orders of the nodes, which is equal to twice the total number of arcs in the graph.
# The Formula N+R=A+2 is known as Euler’s formula for graph or net work.
# The formula F+V=E+2 is known as Euler’s Formula for Polyhedron.
SQUARES AND CUBES
Numbers
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Square
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Number
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Square
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1
|
1
|
11
|
121
|
2
|
4
|
12
|
144
|
3
|
9
|
13
|
169
|
4
|
16
|
14
|
196
|
5
|
25
|
15
|
225
|
6
|
36
|
16
|
256
|
7
|
49
|
17
|
289
|
8
|
64
|
18
|
324
|
9
|
81
|
19
|
361
|
10
|
100
|
20
|
400
|
Numbers
|
Cube
|
Number
|
Cube
|
1
|
1
|
11
|
1331
|
2
|
8
|
12
|
1728
|
3
|
27
|
13
|
2197
|
4
|
64
|
14
|
2744
|
5
|
125
|
15
|
3375
|
6
|
216
|
16
|
4096
|
7
|
343
|
17
|
4913
|
8
|
512
|
18
|
5832
|
9
|
729
|
19
|
6859
|
10
|
1000
|
20
|
8000
|
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