Addition Rule of Probability

The Addition Rule:

Probability is the probably that event will happen – how likely the event will happen. The addition rule for probability: a statistical property that states the probability of one and/or two events occurring at the same time is equal to the probability of the first event occurring, plus the probability of the second event occurring, minus the probability that both events occur at the same time.
If events A and B are mutually exclusive or disjoint, then P(A U B) = P(A) + P(B)
Otherwise, P(A U B) = P(A) + P(B) – P(A ∩ B).

Quadrants

Quadrants:




Let X'OX and Y'OY perpendicular coplanar lines intersecting each other at O. We refer X'OX as x-axis and Y'OY as y-axis. It is clear from the adjoining figure, that these two lines divide the plane into four equal parts, each part is called a Quadrant.
The four Quadrants are:
XOY - first Quadrant
YOX' - second Quadrant
X'OY' - third Quadrant
Y'OX - fourth Quadrant

Trigonometry Heights and Distances

Trigonometry Heights and Distances:

• The word "Trigonometry" is derived from the two Greek words meaning measurements or solution of triangles. Trigonometry which is a branch of mathematics that deals with the ratio between the sides of a right triangle and its angles.
• Trigonometry is the study about relationships between the sides and angles of a triangle.
• Trigonometry is used in surveying and to determine Heights and Distances, in navigation it is to determine the location and the distances, and in the fields like nondestructive testing for determining things such as the angle for reflection or refraction of an ultrasound wave.

Types of Angles

Types of angles:

Algebra

Algebra is one of the main branches of mathematics. It explains the relations and properties of quantity by means of letters and other symbols. Algebra is used to simplify the long and very complicated statements. The basic algebra has the following subtopics are

1. Variables
2. Expressions
3. Terms
4. Polynomials
5. Equations.

Theorem in Mathematics

Theorem in Mathematics:

Congruent Supplements Theorem: When two angles are supplementary to the congruent angles or to the same angle, then the given angles are congruent.
Vertical Angle Theorem: Two angles are said as congruent even when they are vertical angles.
Alternate Interior Angle Theorem:Pairs of alternative interior angles are said as congruent when two parallel lines are cut transversely.
Consecutive Interior Angle Theorem:Pairs of consecutive interior angle are said as supplementary when two parallel lines are cut transversely.
Alternative Exterior Angle Theorem: Pairs of alternative exterior angle are said as congruent when two parallel lines are cut transversely.

Parabola equation

X intercept and Y intercept for Parabola equation:
The general form of parabola equation is
y = ax2 + bx + c where a,b and c are parts of the parabola.
Here, x intercepts are the roots of the equation of parabola. The x intercepts are the roots of the equation 0 = ax2 + bx + c. The very common methods to solve the equation are by factoring or by quadratic formula. The y intercept is (0 , c) for parabola equation.

Solve the equations Graphically

Solve the equations Graphically:

Check whether the pair of equations x + 3y = 6 and 2x – 3y = 12 is consistent. If so, solve them graphically.
Solution : Let us draw the graphs of the Equations (1) and (2). For this, we find two
solutions of each of the equations, which are given in Table

Plot the points A(0, 2), B(6, 0), P(0, – 4) and Q(3, – 2) on graph paper, and join the points to form the lines AB and PQ as shown in Fig below We observe that there is a point B (6, 0) common to both the lines AB and PQ. So, the solution of the pair of linear equations is x = 6 and y = 0, i.e., the given pair of equations is consistent.

General Equation of a Line

General Equation of a Line:
Equation of a Straight line is also called a Linear Equation.

• A straight line is represented by an equation of the first degree in two variables (x and y). Conversely locus of an equation of the first degree in two variables is a straight line.

• A straight line is completely determined by its slope (direction) and a point is given through which the line must pass.

Graph of the equation Ax + By + C = 0 is always a straight line
Therefore, any equation of the form Ax + By + C = 0, where A and B are not zero
simultaneously is called general linear equation or general equation of a line.

Find the area of a square

Problem:
Find the area of the shaded design in below fig where ABCD is a square of side 10 cm a semicircles are drawn with each side of the square as diameter. (Use π = 3.14).


Solution : Let us mark the four unshaded regions as I, II, III and IV .Area of I + Area of III = Area of ABCD – Areas of two semicircles of each of radius 5 cm
(10×10-2×1/2×π×52)cm2= (100 – 3.14 × 25) cm2
= (100 – 78.5) cm2 = 21.5 cm2
Similarly, Area of II + Area of IV = 21.5 cm2
So, area of the shaded design = Area of ABCD – Area of (I + II + III + IV)
= (100 – 2 × 21.5) cm2 = (100 – 43) cm2 = 57 cm2

Coordinate plane problem

Problem:
Plot the following ordered pairs of number (x, y) as points in the Cartesian
plane. Use the scale 1cm = 1 unit on the axes.

Solution : The pairs of numbers given in the table can be represented by the points
(– 3, 7), (0, –3.5), (– 1, – 3), (4, 4) and (2, – 3). The locations of the points are shown
by dots in Fig.

Triangle Theorem

Triangle Theorem:
The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Proof : Given an arc PQ of a circle subtending angles POQ at the center O and PAQ at a point A on the remaining part of the circle. We need to prove that ∠ POQ = 2 ∠ PAQ.

Consider the three different cases as given in Fig. (i), arc PQ is minor; in (ii),arc PQ is a semicircle and in (iii), arc PQ is major. Let us begin by joining AO and extending it to a point B.
In all the cases, ∠ BOQ = ∠ OAQ + ∠ AQO because an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Also in Δ OAQ, OA = OQ (Radii of a circle) Therefore, ∠ OAQ = ∠ OQA (Theorem 7.5)
This gives ∠ BOQ = 2 ∠ OAQ (1) Similarly, ∠ BOP = 2 ∠ OAP.... (2) From (1) and (2), ∠ BOP + ∠ BOQ = 2(∠ OAP + ∠ OAQ) This is the same as ∠ POQ = 2 ∠ PAQ ...(3) For the case (iii), where PQ is the major arc, (3) is replaced by reflex angle POQ = 2 ∠ PAQ

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Construction of Tangents to a Circle

Construction of Tangents to a Circle:

If a point lies on the circle, then there is only one tangent to the circle at this point and it is perpendicular to the radius through this point. Therefore, if you want to draw a tangent at a point of a circle, simply draw the radius through this point and draw a line perpendicular to this
radius through this point and this will be the required tangent at the point.

We are given a circle with center O and a point P outside it. We have to construct
the two tangents from P to the circle.


Steps of Construction:
1. Join PO and bisect it. Let M be the midpoint of PO.
2. Taking M as center and MO as radius, draw a circle. Let it intersect the given circle at the points Q and R.
3. Join PQ and PR. Then PQ and PR are the required two tangents , now let us see how this construction works. Join OQ. Then ∠ PQO is an angle in the semicircle and, therefore,∠ PQO = 90° Can we say that PQ ⊥ OQ? Since, OQ is a radius of the given circle, PQ has to be a tangent to the circle. Similarly, PR is also a tangent to the circle.

Euclid’s Division Lemma

Euclid’s Division Lemma:
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Euclid’s division algorithm is based on this lemma.

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabrw’al muqabala. A lemma is a proven statement used for proving another statement.

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers.

Equiangular Triangles

Two triangles are similar, if

(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion). Note that if corresponding angles of two triangles are equal, then they are known as equiangular triangles.

A famous Greek mathematician Thales gave an important truth relating to two equiangular triangles which is as follows:
The ratio of any two corresponding sides in two equiangular triangles is always the same. It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same.

Arithmatic Progressions

Arithmetic Progressions:
Arithmetic Progressions was known ever three thousand years ago.
You must have observed that in nature, many things follow a certain pattern, such as
the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the
spirals on a pineapple and on a pine cone etc.
We now look for some patterns which occur in our day-to-day life. Some such
examples are :

(i) Jia applied for a job and got selected. She
has been offered a job with a starting monthly
salary of Rs 8000, with an annual increment of
Rs 500 in her salary. Her salary (in Rs) for the
1st, 2nd, 3rd, . . . years will be, respectively
8000, 8500, 9000, . . . .

(ii) The lengths of the rungs of a ladder decrease
uniformly by 2 cm from bottom to top The bottom rung is 45 cm in
length. The lengths (in cm) of the 1st, 2nd,
3rd, . . ., 8th rung from the bottom to the top
are, respectively
45, 43, 41, 39, 37, 35, 33, 31

(iii) In a savings scheme, the amount becomes 5/4 times of itself after every 3 years.
The maturity amount (in Rs) of an investment of Rs 8000 after 3, 6, 9 and 12
years will be, respectively :10000, 12500, 15625, 19531.25

The number of unit squares in squares with side 1, 2, 3, . . . units (see below Fig)
are, respectively
12, 22, 32, . . . .

Circles

Circles:
The different situations that can arise when a circle and a line
are given in a plane.
So, let us consider a circle and a line PQ. There can be three possibilities given
in Fig. below:

The line PQ and the circle have no common point. In this case,
PQ is called a non-intersecting line with respect to the circle. In above Fig. (ii), there
are two common points A and B that the line PQ and the circle have. In this case, we
call the line PQ a secant of the circle. In Fig. 10.1 (iii), there is only one point A which
is common to the line PQ and the circle. In this case, the line is called a tangent to the
circle.
The tangent to a circle is a special case of the secant, when the two end
points of its corresponding chord coincide.

Matrices

The knowledge of matrices is necessary in various branches of mathematics.

Matrices are one of the most powerful tools in mathematics.
This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use. Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analyzing the results of an
experiment etc. Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices. Matrices
are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management.

Example 1:
Consider the following information regarding the number of men and women
workers in three factories I, II and III
Men workers Women workers
I 30 25
II 25 31
III 27 26
Represent the above information in the form of a 3 × 2 matrix. What does the entry
in the third row and second column represent?

MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:



30 25
A=25 31
21 26


The entry in the third row and second column represents the number of women
workers in factory III.