Graphical Solution of Linear Inequalities in Two Variables

Let us learn about Graphical Solution of Linear Inequalities in Two Variables
we will discuss graph of a linear inequality in two variables.
We know that a line divides the Cartesian plane into two parts. Each part is
known as a half plane. A vertical line will divide the plane in left and right half planes
and a non-vertical line will divide the plane into lower and upper half planes

A point in the Cartesian plane will either lie on a line or will lie in either of the half
planes I or II. We shall now examine the relationship, if any, of the points in the plane
and the inequalities ax + by <> c.
Let us consider the line
ax + by = c, a ≠ 0,b ≠ 0
There are three possibilities namely:
(i) ax + by = c (ii) ax + by > c (iii) ax + by < c.
Hope the explanation helped you, now me give you examples on Graphical Solution of Linear Inequalities in Two Variables

Solution of a Quadratic Equation by Factorisation

Let us learn about Solution of a Quadratic Equation by Factorisation,
Consider the quadratic equation 2x2 – 3x + 1 = 0. If we replace x by 1 on the
LHS of this equation, we get (2 × 12) – (3 × 1) + 1 = 0 = RHS of the equation.
We say that 1 is a root of the quadratic equation 2x2 – 3x + 1 = 0. This also means that
1 is a zero of the quadratic polynomial 2x2 – 3x + 1.
In general, a real number α is called a root of the quadratic equation
ax2 + bx + c = 0, a ≠ 0 if a α2 + bα + c = 0.

We also say that x = α is a solution of
the quadratic equation, or that α satisfies the quadratic equation. Note that the
zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic
equation ax2 + bx + c = 0 are the same.

Example : Find the roots of the equation 2x2 – 5x + 3 = 0, by factorisation.
Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x) =
6x2 = (2x2) × 3].
So, 2x2 – 5x + 3 = 2x2 – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1)
Now, 2x2 – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0.
So, the values of x for which 2x2 – 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0,
i.e., either 2x – 3 = 0 or x – 1 = 0.
Now, 2x – 3 = 0 gives x = 3/2 and x – 1 = 0 gives x = 1.
So, x = 3/2 and x = 1 are the solutions of the equation.
In other words, 1 and 3/2 are the roots of the equation 2x2 – 5x + 3 = 0.
Verify that these are the roots of the given equation.
Note that we have found the roots of 2x2 – 5x + 3 = 0 by factorising
2x2 – 5x + 3 into two linear factors and equating each factor to zero.
Hope the above explanation was helpful to you, now let me give you more examples on Quadratic Equation by factorization.

QUADRATIC EQUATIONS

Let me help you know about Quadratic equations,

Quadratic equations come up when we deal with many real-life situations. For instance, suppose acharity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall? Suppose the breadth of the hall is x metres. Then, its length should be (2x + 1) metres. We can depict this information pictorially as shown in Figure.
Now, area of the hall = (2x + 1). x m2 = (2x2 + x) m2
So, 2x2 + x = 300 (Given)
Therefore, 2x2 + x – 300 = 0

So, the breadth of the hall should satisfy the equation 2x2 + x – 300 = 0 which is a
quadratic equation.
Hope the above explanation helps you to know what is quadratic equation, now let me give some examples on Quadratic equations.

Composition of Functions and Invertible Function

Let us study the Composition of Functions and Invertible Function,
Consider the set A of all students, who appeared in Class X of a Board
Examination in 2006. Each student appearing in the Board Examination is assigned a
roll number by the Board which is written by the students in the answer script at the
time of examination. In order to have confidentiality, the Board arranges to deface the
roll numbers of students in the answer scripts and assigns a fake code number to each
roll number. Let B ⊂ N be the set of all roll numbers and C ⊂ N be the set of all code numbers. This gives rise to two functions f : A → B and g : B → C given by f (a) = the
roll number assigned to the student a and g(b) = the code number assigned to the roll
number b. In this process each student is assigned a roll number through the function f
and each roll number is assigned a code number through the function g. Thus, by the
combination of these two functions, each student is eventually attached a code number.
This leads to the following definition:
Definition Let f : A → B and g : B → C be two functions. Then the composition of
f and g, denoted by gof, is defined as the function gof : A → C given by
gof (x) = g(f (x)), ∀ x ∈ A.


Hope the above explanation gives you an idea about Composition of Functions and Invertible Function, let me give you more examples,

RELATIONS AND FUNCTIONS

Let me give you some introduction about relations and functions,

The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation
in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are
(i) {(a, b) ∈ A × B: a is brother of b},
(ii) {(a, b) ∈ A × B: a is sister of b},
(iii) {(a, b) ∈ A × B: age of a is greater than age of b},
(iv) {(a, b) ∈ A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination},
(v) {(a, b) ∈ A × B: a lives in the same locality as b}. However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A × B.
If (a, b) ∈ R, we say that a is related to b under the relation R and we write as
a R b. In general, (a, b) ∈ R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of
relations.

Hope the above introduction helped you, now let me explain you the types of relations,

Wave Optics

Wave Optics

The branch of physics dealing with the study of optical phenomena is called optics. This can be divided into two categories, ray optics and wave optics. Wave optics describes the connection between waves and rays of light. The wave theory of light was put forth by Huygen and later modified by Frensel. In this unit we come across the various phenomena related to the wave nature of light.

Here is an example which will help us understand the motion of waves in a simple form.

One of the simplest examples of how Waves are created is when we will drop a small pebble in a pool of water,the moment the pebble will fall in water it creates ripples and hence it disturbes the stagnant position of water.

Let us learn some Types of waves:

Transverse Waves:

A transverse wave is a type of mechanical wave. They travel in a straight line and carry energy and momentum through medium particles from one point to another.

Longitudinal Waves:

A wave motion in which the particles of the medium oscillate about their mean positions in the direction of propagation of the wave, is called longitudinal wave.

Hope you like the above example of Wave Optics.
Please leave your comments, if you have any doubts.

Ray Optics and Optical Instruments

Ray Optics and Optical Instruments

Ray Optics is a very commonly heard term,as it is related to light which is a very important part of our daily life.Optics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it.Ray optics is also called geometrical optics as it uses the geometry of straight-line paths (rays) to explain the optical phenomena. Optics usually describes the behavior of visible, ultraviolet, and infrared light.Ray optics, infact, is the limiting case of wave optics. This means for most practical purposes, we can ignore the deviation from straight-line path as postulated by wave theory.

Optical Instruments: Let us learn about some of the optical Instruments
The first thing that comes to our mind when we say "Optical instruments" is a pair of spectacles.......but it is not true they are other types of optical instruments too,let us learn about the formation of various types of images by a lens for different positions of the object.

Microscopes:

A microscope is an optical device which produces a highly magnified image of very small object such as micro-organisms. Based on the design, there are two types of microscopes. They are, simple microscope and compound microscopes.
Simple Microscope

A simple microscope is nothing but a single biconvex lens. It is referred to as magnifying glass.
Compound Microscope

A compound microscope is an optical instrument which is used to magnify very small objects like blood cells, bacteria which otherwise cannot be seen with the naked eye.
Telescopes

Telescope is an optical instrument which is used for viewing heavenly bodies and distant objects.
Astronomical Telescope

This type of telescope is used to view heavenly bodies like stars, planets and satellites.

We might also come across questions such as:Which instrument is used to measure temperature??

The answer would be a "Thermometer". These are some basic examples of Optical Instruments.


Hope you like the above example of Ray Optics and Optical Instruments
Please leave your comments, if you have any doubts.