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Three Dimensional Geometry (part - 2) , The Planes | study material for IIT JEE | concept booster, chapter highlights/

By Laws Of Nature
THREE DIMENSIONAL GEOMETRY (PART - 2), THE PLANES










THE PLANESPlane is the surface such that if any two points are taken on it, then the line joining the two points lies on it.
General equation of plane is given as -  ax + by + cz + d = 0 , where a ,b and c is not equal to zero.
           POINTS TO BE REMEMBERED 1). a, b and c are the directions ratios of the normal to the plane ax + by + cz + d = 0.

2). Equation of the yz - plane is x = 0

3). Equation of the zx - plane is y = 0

 4). Equation of the xy - plane is z = 0

5). Equation of the any plane parallel to the xy - plane is z = c. Similarly for Planes parallel to yz and zx is x = c and y = c
EQUATION OF THE PLANE IN NORMAL FORMVECTOR FORMIf n̂ be a unit vector normal to a given plane and d be the length of the perpendicular from the origin to the plane, then the equation of the plane is given by -                           r.n̂ = d CARTESIAN FORMIf l, m, n are the directions cosines of the normal to the plane and d is the perpendicular distan…
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Inverse trigonometrc function IIT JEE study material | ITF concept booster | ITF notes | ITF important formulae.

By Laws Of Nature
INVERSE TRIGONOMETRY FUNCTION

INVERSE FUNCTION
If f:X⟶ Y is a function which is both one - one and onto, then it's inverse function  f^-1:Y⟶X is defined as: y = f(x) ⟺ f^-1(y)=x
Such that , ∀x∈X , ∀y∈Y
INVERSE TRIGONOMETRY FUNCTION
Let's take a sine function, whose domain is R and range [-1,1]. We see that this function is many -one and onto, so , it's inverse doesn't exist, but if we restrict the domain of the sine function to the interval [-π/2 , π/2] , then the function is:
sin: [-π/2 , π/2] ⟶  [-1,1] and given by sinθ = x Which is one-one and onto and therefore it's inverse exist. And the Inverse of sine function is defined as Sin-1: [-1,1] ⟶ [-π/2 , π/2], such that sin-1x = θ From this we can say that if x is a real numbers between -1 and 1 then θ will be in -π/2 and π/2.
Sin-1x = θ then x = sinθ , where, -π/2⪯ θ⪯  π/2 and -1 ⪯x⪯ 1
So , the least numerical value among all the values of the angle whose sine is x , is called the principal value of sin-1x. So we …
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what is Roche limit| Roche limit and the formation of the rings |

By Laws Of Nature
Have you ever think, why did Earth have no rings? Why did only some planets have rings? Who initiates the rings formation? And how did rings form?
If you want to know the answers of these questions then stay with us till end. Before giving the explanation of these questions, first of all, we have to know about the Roche limit. Who is responsible for these phenomena. So let's starts:
ROCHE LIMIT
Then what is Roche limit? And how is it responsible for the formation of the rings? We all know that, in the starting of the universe , various planets, stars and satellites and other celestial bodies formed by applying the Gravitational force to the nearby small rocks, gaseous matter and particles and get integrated to a bigger one. But sometimes, This integration of the celestial bodies get hindered when a comparatively larger body came nearer to the integrating celestial body. This hindrance occurred due the Roche limit of the larger celestial body. So Roche limit(also called Roche radius…
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ALTERNATING CURRENT fast revision notes for IIT JEE | quick concept booster | Laws of nature.

By Laws Of Nature
ALTERNATING CURRENT
*. It is the current which varies in magnitude continuously and changes its direction alternatively and periodically. It is represented as , I = I0sinωt , where I0 is the maximum value of AC current called as peak current. I is called the instantaneous value of alternating current. ω is called angular frequency of alternating current. Which is given as-   ω  = 2π/T = 2πυ, T is time period of A.C and υ is frequency of A.C. 
*. It is the type of sinusoidal waves , which are represented by sine and cosine angle.
*. Alternating voltage or EMF can be represented as. E = E0sinωt , where E is instantaneous value of alternating voltage , E0 is the peak voltage.
MEAN VALUE OF A.C
*. It is the value of alternating current , which send same amount of electric charge through the circuit in half cycle as it sent by steady current in the same time. It is denoted by Im. Derivations for mean value of alternating current over half cycle is as below.

*.  Mean value ofAlternating curr…
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ELECTROMAGNETIC INDUCTION notes for IIT JEE/ NEET , AIIMS | chapters on your finger tips | laws of nature.

By Laws Of Nature
ELECTROMAGNETIC INDUCTION
Electromagnetic induction is a very important phenomenon of electromagnetism. It is a phenomenon in which EMF is produced in the coil, whenever magnetic flux linked with the coil changes. The produced EMF is called induced EMF and current so produced due to induced EMF is called induced current.
MAGNETIC FLUX (Φ)
*.It is the total no. of magnetic field lines passing perpendicularly through the surface , when it is placed in the magnetic field(B). Φ = B.A = BAcosθ B is the magnetic field , A is the surface area and θ is the angle between the direction of the magnetic field and the normal vector of the surface area.
*. The SI units of magnetic flux(φ) is weber (Wb).
*. Any one can change the magnetic flux by-    -  changing the intensity of magnetic field.    -  changing the coil orientation wrt to magnetic field.   - changing the surface area.
FARADAY'S LAWS OF ELECTROMAGNETIC INDUCTION
FARADAY'S FIRST LAW - whenever magnetic flux linked with the coil or …
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RATIO & PROPORTION important arithmetic formulae | Ratio and proportion short-tricks | list of all formulae.

By Laws Of Nature
RATIO AND PROPORTION



 YOU MAY ALSO LIKE Important arithmetic formulae on PERCENTAGE Important arithmetic formulae on LCM and HCF Important arithmetic formulae on AVERAGE Important arithmetic formulae on NUMBER SYSTEM Important arithmetic formulae on cube , cuboid etc.

* SOME IMPORTANT SUTRAS
1). If the ratio of any three quantities P , Q and R is a:b:c , then P = aK , Q = bK and R = cK. Where K is any constant.
2). Inverse ratio of x , y and z is = 1/x : 1/y : 1/z =          yz : zx : xy
3). If x is to be divided in the ratio of a:b:c , then first part = ax/a+b+c , second part = bx/a+b+c and third part = cx/a+b+c.
4). If x is the mean proportion between numbers a and b , then x = √(ab).
5). If x and y are the two numbers then their duplicate ratio is = x² : y²
6). If x and y is the numbers then their sub duplicate ratio is = √x : √y
7). If x and y is the two numbers then their triplicate ratio is = x³ : y³
8). If x and y is the two numbers then their sub triplicate ratio = x^1/3 : y^1/3
9). I…
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Sphere , Hemisphere , prism and pyramid important arithmetic formulae | mensuration short tricks | list of all formulae | laws of nature.

By Laws Of Nature
Today we are going to talk about some important arithmetic formulae on sphere , hemisphere , prism and pyramid.
Students are advised to learn all these formulae and keep practicing with it.
SOME IMPORTANT SUTRAS
*SPHERE
1). Volume of solid sphere = (4/3).πr^3= πd^3/6 2). Total surface area of solid sphere = 4πr^2 = πd^2 3). Volume of hollow sphere = (4/3).π(R^3-r^3) 4). Total surface area of hollow sphere =       4π(R^2-r^2) 5). Radius of sphere = (3V/4π)^1/3 6). Diameter of sphere = (6V/π)^1/3 7). Radius of sphere = √TSA/4π 8). Diameter of sphere = √TSA/π 9). t is the thickness of the material of the hollow sphere , if R is larger radius then smaller radius is= R-t 10). Volume of capsule = π r^2.[(4r/3)+h] 11). Total surface area of capsule =  2πr.(2r+h) 12). Volume of sphere = (r/3). surface area 13). Surface area of sphere = 3V/r 14). Radius of largest sphere in cube = a/2 15). Volume of largest sphere in cube = πa^3/6 =         11a^3/21 16). Surface area of largest sphere in cube =…
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cube , cuboid , right circular cylinder and frustum/bucket |mensuration short tricks | important arithmetic formulae | LAWS OF NATURE.

By Laws Of Nature
Today we are going to talk about some important arithmetic formulae on cube , cuboid , Right circular cylinder and frustum/bucket.Students are advised to learn all the formulae on Keep practicing with it.
SOME IMPORTANT SUTRAS
* CUBE
1). Volume of cube = a^3 = d^3/3√3 2). Diagonal of the cube = a.√3 3). Surface area of cube = 6a^2 4). Side of cube = (V)^1/3 = d/√3 5). Surface area of cube = 2d^2 5). From the three small cube having volume V1 , V2  and V3 , a big cube are formed then side of the big cube = (V1+V2+V3)^1/3 6). Lateral surface area of cube = 4a^2 8). No of y sided cubes formed from the cube of  side x is = (x/y)^3
* CUBOID
1). Volume of cuboid = lbh 2). Length of cuboid = V/bh 3). Breadth of the cuboid = V/lh 4). Height of the cuboid = V/lb 5). Surface area of cuboid = 2(lb+bh+hl) 6). Lateral surface area of cuboid = 2(l+b)h 7). If surface area of cuboid is x , y and z respectively , then xyz = V^2 8). Diagonal of cuboid = √(l^2+b^2+h^2)
* RIGHT CIRCULAR CYLINDER
1). Curved surface area of cy…
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