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True Discount and Banker's Discount problems tricks in Hindi | fast track formulae for problem solving.

TRUE DISCOUNT AND BANKER'S DISCOUNT TRICKS IN HINDI

TRUE DISCOUNT 1). वास्तविक बट्टा = मिश्रधन - वर्तमान धन
2). यदि ब्याज की दर r% वार्षिक , समय t व वर्तमान धन (pw) है, तो वास्तविक बट्टा = PW×r×t/100
3). यदि r% तथा समय t के बाद देय धन A है, तो तत्काल धन pw = 100×A/(100+r.t)
4). यदि देय धन A पर r% व समय t दिए गए है, तो वास्तविक बट्टा = A.r.t /(100+r.t)
5). यदि किसी निश्चित समय के पश्चात निश्चित वार्षिक दर पर, देय धन पर वास्तविक बट्टा (TD) व समान समय व दर के लिए साधारण ब्याज (SI) हैं, तो देय धन A =      SI × TD/(SI - TD)
6). यदि t समय पश्चात r% वार्षिक दर पर, देय धन पर वास्तविक बट्टा (TD) व साधारण ब्याज (SI) है , तो -  SI - TD = TD×r×t/100
7). t वर्ष बाद r% चक्रवृद्धि दर पर देय धन A का तत्काल धन (PW) = A/(1+r/100)^t   व वास्तविक बट्टा = A - PW
BANKER'S DISCOUNT1). महाजनी बट्टा = शेष समय (समाप्त न हुए समय) के लिए बिल पर ब्याज = बिल की राशि × दर × शेष समय /100
2). महाजनी लाभ = महाजनी बट्टा - वास्तविक बट्टा
3). यदि बिल का मान / अंकित मूल्य A है, समय t व दर r% है, तो महाजनी बट्टा = A×r×t/100
4).…

The concept of projectiles | Derivations for time of flight , maximum height , horizontal range , trajectory of a projectile.

THE PROJECTILE MOTION
It is the type of motion in which if any object is fired with some initial velocity near the earth surface , then it's motion follows a curved path called projectile motion. This curved path by the objects is called the trajectory of the projectile. Any missiles which follows these path called ballistic missiles.

SOME EXAMPLES

*. When we throw a ball.
* Firing of a missile.
*. Path of the water which is flowing from the hole of a tank.
*. Motion of a bullet. etc

SOME ASSUMPTIONS IN THE STUDY OF PROJECTILE MOTION

*. Zero air resistance.
*. Effect of earth rotation on the object is negligible.
*. Constant acceleration due to gravity.

EQUATION OF THE TRAJECTORY

MOTION IN HORIZONTAL DIRECTION

The velocity of the object in horizontal direction is constant, so it's acceleration is zero.
The position of the object at any time t in the horizontal direction is-

 x = x0 + (ux)t + 1/2(ax)t^2 , 
[ux and ax is horizontal velocity and acceleration]
But x0 = 0 , ux = ucosθ , ax = 0 , and t=t
Substituting these values in the above equation, we get.
                      x = ucosθ.t
                      t = x/ucosθ

MOTION IN VERTICAL DIRECTION

In the vertical direction object velocity starts decreasing due to acceleration due to gravity. So acceleration due to gravity is taken as (ay) = -g
The position of the object at any time t in the vertical direction is given as;
          y = y0 + (uy)t + 1/2(ay)t^2
But y0 = 0 , uy = usinθ , ay = -g, and t = t

             y = usinθ.t - 1/2gt^2
Substituting the value of t = x/ucosθ
  y = usinθ( x/ucosθ) - 1/2g(x/ucosθ)^2
     y = xtanθ - (gx^2/2.u^2cos^2.θ)
This is the trajectory of the object who is performing projectile motion.

*. Velocity of the object at any time t is given by-
    v = √[u^2+(gt)^2 - 2ugtsinθ]

TIME OF FLIGHT

It is the total time for which a projectile remains in air. It is denoted by T.
Time of flight can be split in the two time format , one is time of ascent from O to H and second is time of descent from H to R.
Total time can be expressed as:
  T = t+t = 2t , t = T/2
Takes a vertical upward motion of an object from O to H.
uy = usinθ , ay = -g , t = T/2 , and vy = 0, [vy become 0 at the maximum height of flight]

                   vy = uy + ay.t
                 0 = usinθ -g(T/2)
 Time of flight T = 2usinθ/g

MAXIMUM HEIGHT

It is the maximum vertical height attained by the object during its flight.
Takes a vertical upward motion of an object from O to H.
 We have, 
uy = usinθ , ay = -g , y0 = 0 , y = H , t = T/2 

Using second equation of motion.
    y = y0 + (uy)t + 1/2(ay)t^2
Substituting above values; we get,
H = 0 + usinθ.(usinθ/g) + 1/2(-g).(usinθ/g)^2
H = (u^2.sin^2.θ/g) - 1/2(u^2.sin^2.θ/g)
        Then H = u^2.sin^2.θ/2g

HORIZONTAL RANGE

It is the horizontal distance covered by the object during its flight. It is denoted by R.
If the uniform horizontal velocity of the object is ucosθ and it's time of flight is T , then distance covered is-

  R = ucosθ ×T = ucosθ×2usinθ/g = u^2.sin2θ/g
Range is maximum if sin2θ = 1 , sin(π/2) =1
  It means , 2θ = π/2 , θ  = π/4 = 45°

[NOTE: here θ is the angle between the velocity and the horizontal range, if the object is fired at the angle θ with the vertical, then use (β=90° - θ) angle with horizontal.]

MOTION OF THE PROJECTILE ON INCLINED PLANE


Take a plane of inclination α , and if projectile is fixed with angle θ with the horizontal range , then (θ - σ) is the angle of projectile with the inclined plane.

*. TIME OF FLIGHT
  
   T' = 2usin(θ - α)/gcosα

*. RANGE

R' = 2u^2.sin(θ - α). cosθ/g.cos^2α
R' = u^2[sin(2θ - α) - sinα]/gcos^2.α

Range is maximum if 2θ - α = π/2 or
  θ - α = π/2 - θ
Then 
R'(max) = u^2[1 - sinα]/gcos^2.α

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Boat and stream short-tricks in Hindi | Fast track arithmetic formulae for competitive examination.

BOAT AND STREAM (नाव एवं धारा)
1). यदि शांत जल में नाव या तैराक की चाल x किमी/घंटा व धारा की चाल y किमी/घंटा है, तो धारा के अनुकूल नाव अथवा तैराक की चाल = (x+y) किमी/घंटा
2). धारा के प्रतिकूल नाव अथवा तैराक की चाल = (x-y) किमी /घंटा
3). नाव की चाल = (अनुप्रवाह चाल + उद्धर्वप्रवाह चाल)/2
4). धारा की चाल =  (अनुप्रवाह चाल - उद्धर्वप्रवाह चाल)/2
5). यदि धारा की चाल a किमी/घंटा है, तथा किसी नाव अथवा तैराक को उद्धर्वप्रवाह जाने में अनुप्रवाह जाने के समय का n गुना समय लगता है,(समान दूरी के लिए), तो शांत जल में नाव की चाल = a(n+1)/(n-1) किमी/घंटा
6). शांत जल में किसी नाव की चाल x किमी/घंटा व धारा की चाल y किमी/घंटा है, यदि नाव द्वारा एक स्थान से दूसरे स्थान तक आने व जाने में T समय लगता है, तो दोनो स्थानों के बीच की दूरी = T(x^2 - y^2)/2x km
7). कोई नाव अनुप्रवाह में कोई दूरी a घंटे में तय करती है, तथा वापस आने में b घंटे लेती है, यदि नाव कि चाल c किमी/घंटा है, तो शांत जल में नाव की चाल = c(a+b)/(b-a) km/h
8). यदि शांत जल में नाव की चाल a किमी/घंटा है, तथा वह b किमी/घंटा की चाल से बहती हुई नदी में गत…

Emergence of British East India Company as an Imperialist Political Power in India

EMERGENCE OF BRITISH EAST INDIA COMPANY AS AN IMPERIALIST POLITICAL POWER IN INDIA
Dynamically changing India during early eighteenth century had a substantially growing economy under the authority of Mughal emperor Aurangzeb. But after his demise in 1707, several Mughal governors established their control over many regional kingdoms by exerting their authority. By the second half of eighteen century, British East India Company emerged as a political power in India after deposing regional powers and dominating over Mughal rulers. The present article attempts to analyze the reasons for emergence of British East India Company as an imperial political power in India and their diplomatic policies of territorial expansion. In addition to this, I briefly highlighted the Charter Acts (1773, 1793, 1813, 1833 and 1853) to trace its impact on the working process of Company.Establishment of East India Company in India

In 1600, British East India Company received royal charter or exclusive license…

Three dimensional geometry (part-1) | study material for IIT JEE | concept booster , chapter highlights.

THREE DIMENSIONAL GEOMETRY

ORIGIN
In the following diagram X'OX , Y'OY and Z'OZ are three mutually perpendicular lines , which intersect at point O. Then the point O is called origin.
COORDINATE AXES 
In the above diagram X'OX is called the X axes, Y'OY is called the Y axes and Z'OZ is called the Z axes.
COORDINATE PLANES 
1). XOY is called the XY plane. 2). YOZ is called the YZ plane. 3). ZOX is called the ZX plane.
If all these three are taken together then it is called the coordinate planes. These coordinates planes divides the space into 8 parts and these parts are called octants.
COORDINATES  Let's take a any point P in the space. Draw PL , PM and PN perpendicularly to the XY, YZ and ZX planes, then
1). LP is called the X - coordinate of point P. 2). MP is called the Y - coordinate of point P. 3). NP is called the Z - coordinate of the point P.
When these three coordinates are taken together, then it is called coordinates of the point P.
SIGN CONVENTI…

A detailed unit conversion table in Hindi.

UNITS CONVERSION TABLE
CENTIMETRE GRAM SECOND SYSTEM (CGS)1). MEASUREMENT OF LENGTH (लंबाई के माप) 10 millimeter = 1 centimetres10 centimetre = 1 decimetres  10 decimetre = 1 metres 10 metre = 1 decametres 10 decametres = 1 hectometres 10 hectometres = 1 kilometres 10 kilometres = 1 miriametresMEASUREMENTS OF AREAS ( क्षेत्रफल की माप )  100 millimetre sq. = 1 centimetre sq.
 100 centimetre sq. = 1 decimetres sq. 100 decimetres sq. = 1 metre sq. 100 metre sq. = 1 decametres sq  100 decametres sq. = 1 hectometres sq. 100 hectometres sq. = 1 kilometres sq. 100 kilometres sq. = 1 miriametres sq.
MEASUREMENTS OF VOLUME ( आयतन की माप) 1000 millimetre cube. = 1 centimetre cube.
 1000 centimetre cube. = 1 decimetres cube. 1000 decimetres cube. = 1 metre cube. 1000 metre cube. = 1 decametres cube. 1000 decametres cube. = 1 hectometres cube. 1000 hectometres cube. = 1 kilometres cube. 1000 kilometres cube. = 1 miriametres cube.
MEASUREMENTS OF VOLUME OF LIQUIDS  (द्रव्य के आयतन का माप) 10 millilitre=…

THE GENERAL THEORY OF RELATIVITY | A Unique way to explain gravitational phenomenon.

Today we are going to talk about a very important and revolutionary concept that is THE GENERAL THEORY OF RELATIVITY.
This theory came into existence after 10 years of special theory of relativity (1905), and published by Albert Einstein in 1915.
This theory generalise the special theory of relativity and refines the Newton's laws of universal gravitation.
After coming this theory people's perspective about space and time has been changed completely. And this theory give a new vision to understand the spacetime geometry.
This theory gives a unified description of gravity as a geometrical properties of space and time.
This theory helps us to explain some cosmological phenomenon that is ,

* why small planets revolve around the big stars?
* Why everything in this universe is keep moving?
* Why mostly planets and stars are spherical in shape?
* Why does gravity create?
* Why does time become slow near the higher gravitating mass. Ie. Gravitational time dilation.
And gravitational…