Let we have to solve the following system
x+y+z=3
2x-y+z=2
x-2y+3z=2 …..(i)
this system can be put in the form “AX=B” or “A=X-1B” ….(ii)
where A=
[1 1 1]
|2 -1 1|
[1 -2 3]
X=
[x]
|y|
[z]
B=
[3]
|2|
[2]
now |A|=-9

Now we are to find ‘adjA’ and it
adjA=
[A11 A12 A13] [-1-5-3]
|A21 A22 A23|=|-5 2 3|
[A31 A32 A33] [-3 1 -3]
hence
A-1=adjA/|A|
and hence
X=A-1B
[x] [-1-5 -3][2]
|y|=-1/9|-5 2 3||2| =
[z] [ -3 1 -3][3]
[x] [-9]
|y|=-1/9|-9|
[z] [-9]
This gives
x=1, y=1, z=1

Waves can be divided into two parts–
¡)Mechanical
¡¡)Non-mechanical

According to oscillation these are also of two types-
¡)Longitudinal
¡¡)Transverse

=>Mechanical Waves:
Waves that move in a medium which is substance (i.e.solid,liquid,gas) are called ‘mechanical waves’.

->Cathode rays,Cannal rays,a- & B-rays ,Sound waves are mechanical waves.

->SOUND Waves:Based on frequency range these are of three kinds-
i)Infrasonic_ less than 20 Hz
ii)Audible_ 20 to 20000 Hz
iii)Ultrasonic_ over 20000 Hz.

->velocity of sound in air =332m/s, 1483m/s in water,5130m/s in iron.
->Increase in velocity of sound in air when increase in temp is 1^c =0.61m/s
and more about sound learn in the topic “the sound”

=>Non mechanical waves:
These waves can also move in vaccum.These are also called ‘Electromagnetic Waves’.
Properties of E.M.W
i)Velocity of them is constant and is equal to the velocity of Light. i.e. v=c=3.0*E8 m/s

.
ii)E.M.W. have energy and momentum.
iii)Concept of there was given by ‘Mexwell’.
iv)wavelength range is E-14 to E4.
v)these are neutral.
vi)X-rays,Y-rays,UltraViolet,Infrared,Radio rays,drashya radiation are E.M.Ws.

For waves of all type-
v=n.Lamda

QUANTITIES and their REPRESENTATION:
These are two in number- ¡)Scalar ¡¡)Vector
Scalar has only magnitude but Vector has magnitude as well as direction.A Vector is denoted generally by english alphabet a b c with arrow or cap on its head as ‘å’ while scalar by x y z k l m n without arrow.
Here we will denote a Vector by ‘a b c’ and unit vectors by å etc.Magnitude of a b c by l m n and unit vector along axes by i j k while unit vector along normal by ñ.

ADDITION OF VECTOR:-
Let two vectors be a & b as shown in fig_1(a) and R be the resultant of a & b,then R is given by R=a+b and its direction will be as shown in fig_1(b).

SUBSTRACTION OF VECTORS:-
Let R’=a-b
R’=a+(-b)
see in fig_1(c) i.e. the direction of ‘b’ is changed.

Chemistry word is taken from Greek word CHEMIA=black soil .i.e. science related to soil.
But now,
Chemistry is branch of Science in which properties and composition of matter and changes is studied.
There are three main branches of chemisity:- ¡)Inorganic chemistry ¡¡)Organic chemistry ¡¡¡)Physical chemistry

Lavoisier is father of Chemistry.

An equation in which the sum of powers all variables in each term is same is called homogeneous equation. If this sum is ‘n’ then we say that equation is homogeneous of n th degree. E.g.
1>aX+bY=0
This is homo. of 1st degree.
2>aX2+2hXY+bY2=0
This is of 2nd degree.

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Hi This is Ravi.
I’m like your friend.
I’m preparing for UPSC exam.So would like to help you in study. Email me for any quarry at

rksingh1987@indiatimes.com

With all wishes

your
Ravi Kumar Singh
http://www.ravikumar1in.blogspot.com