MyMaths MyLife

When a straight line XY cuts two parallel line PQ and RS [as shown in figure], the following are the relationships between various angles that are formed. [M and N are the points of intersection of XY with PQ and RS respectively].

a) alternate angles are equal
i.e, angle PMN = angle MNS
angle QMN = angle MNR
b) corresponding angles are equal
i.e, angle XMQ = angle MNS
angle QMN = angle SNY
angle XMP = angle MNR
angle PMN = angle RNY
c) Sum of interior angles on the same side of cutting line is equal to 180⁰
i.e, angle QMN + angle MNS = 180⁰
angle PMN + angle MNR = 180⁰
d) Sum of exterior angles on the same side of transversal line is equal to 180⁰
i.e, angle XMQ + angle SNY = 180⁰
angle XMP + angle RNY = 180⁰
If three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on any other transversal as well.

Important Facts:

Direct Proportion: Two Quantities are said to be directly
proportional, if on the increase (or decrease) of the one, the
other increases(or decreases) to the same extent.

Ex:(i) Cost is directly proportional to the number of articles.
(More articles, More cost).

(ii) Work done is directly proportional to the number of men
working on it. (More men, more work).

Indirect Proportion: Two Quantities are said to be
indirectly proportional,if on the increase of the one , the other
decreases to the same extent and vice-versa.

Ex:(i) The time taken by a car covering a certain distance is
inversely proportional to the speed of the car.(More speed,
less is the time taken to cover the distance).

(ii) Time taken to finish a work is inversely proportional to
the number of persons working at it.
(More persons, less is the time taken to finish a job).

Note: In solving Questions by chain rule, we compare every
item with the term to be found out.

We all know about cm, mm, Km ..................., but we dont know much about inches, yard .............
you might have got a question that " what is the relationship between inch and cm, inch and yard"...........
This is going to list all the conversions of such units

LINEAR

Inch=Basic Unit

1 hand = 4 inches
1 link = 7.92 inches
1 span = 9 inches
1 foot = 12 inches
1 yard = 3 feet
1 fathom = 2 yards
1 rod = 5.5 yards
1 chain = 100 links=22 yards
1 furlong = 20 yards
1 mile = 1760 yards
1 knot mile= 6076.1155 feet
1 league = 3 miles
1 inch = 2.54 cm

Cartesian Product:

Let A and B be any two sets. Then the Cartesian product of A and B is the set of all ordered pairs of the form (a, b), where aЄA and bЄB
The product is denoted by A×B
A×B = {(a, b)/ aЄA, bЄB }

Example
A = {a, b, c} and B={1,2}, then
A×B = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2)}
B×A = {(1, a), (2, a), (1, b), (2, b), (1, c), (2, c)}

Mathematical concept means just about anything with a mathematical name. For example, some of the mathematical concepts we learn in high school are: constant, variable, polynomial, factor, factoring, equation, solving an equation, logarithm, sine, cosine, tangent, etc., point, line, triangle, square, and other geometric figures, area, perimeter of a geometric figure, etc., and many others. Among the mathematical concepts we learn in our first years of college mathematics are: set, operation, limit, function, and, specifically, continuous function, derivative, integral, theorem, proof, countable infinity, uncountable infinity, algebra, linear algebra, vector space, group, ring, field, and many others.

Now one thing that makes the understanding of these concepts difficult is that they are defined in terms of other concepts.

Thus, e.g., a vector space is defined in terms of the concepts of vector, set, function, abelian group, field, and others. How does the typical mathematics textbook, and mathematics course, deal with this fact? It attempts to teach the concepts in logical order, i.e., it assumes that, e.g., when you begin your study of vector spaces, you will already know — through having remembered what you learned in previous courses — the meaning of each of the concepts in terms of which a vector space is defined. And, indeed, one of the things that makes mathematics such a frightening subject to many students, is the grandiose manner with which these assumptions are set forth in the list of prerequisites for the course.

Combinations:
Each of different groups or selections which can be formed by
taking some or all of a number of objects,is called a combination.
eg:- Suppose we want to select two out of three boys A,B,C .
then ,possible selection are AB,BC & CA.
Note that AB and BA represent the same selection.

Number of Combination:
The number of all combination of n things taken r at a time is:
nCr = n! / (r!)(n-r)!
= n(n-1)(n-2). . . . . . . tor factors / r!
Note: nCn = 1 and nC0 =1

An Important Result:
nCr = nC(n-r)

For problems click Here.

An equation which has the unknown quantity raised only to powers which are whole numbers and the highest power being the square of the unknown quantity, is called a quadratic equation.
The most general form of a quadratic equation is ax^2 + bx + c = 0.
There are two values of x that satisfy such a quadratic equation. These values are called the roots of the quadratic equation.

The roots of the above quadratic equation are given by (-b±√(b^2-4ac))/2a

For ax^2 + bx + c = 0, sum of the roots = -b/a; Product of the roots = c/a