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

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…

### NUMBER SYSTEM Shortcut and tricks for all competitive examination.

Today we are going to talk about some important arithmetic formulae on the chapter "number system". Which are very useful to all types of competitive examination. Readers are advised to learn all the formulae and keep practicing with it.

SOME IMPORTANT SUTRAS

1). Dividend = (divisor × quotient) + remainder
2). Quotient = (dividend - remainder)/divisor
3). Divisor = (dividend - remainder)/quotient
4). Remainder = dividend -(divisor × quotient)
5). Sum of n consecutive natural number=          n(n+1)/2

6). Sum of squares of n even numbers =    [n(n+1)/2]^2

7). Sum of multiples of x till to n = x.n(n+1)/2

8). Sum of squares of n odd numbers=
n(2n-1)(2n+1)/3
9). Sum of squares of consecutive natural numbers till n =  n(n+1)(2n+1)/6

10). Sum of cubes of consecutive natural numbers till n = [n(n+1)/2]^2

11). Sum of consecutive even numbers till n =        n(n+1)/4

12). Sum of consecutive n odd numbers = n^2

13). Sum of consecutive odd numbers till n =
(n+1/2)^2
14). Sum of n consecutive even numbers= n(n+1)

15). If any number is multiplied by any other number x and in the obtained results , when x is added then it is divisible by any other number y, then the smallest divisible number = [x(y-1)+x]

16). If any number is multiplied by any other number x and in the obtained results , when x is substracted then it is divisible by any other number y, then the smallest divisible number = [x(y+1)-x]

17). If x is the sum of a number of two digit and the number obtained by interchanging their digit , then the sum of the digit is = x/11

18). If y is the difference of a number of two digit and the number obtained by interchanging their digit , then the difference of the digit is = y/9

19). m the sum of the digit of any two digit number, if their digit are interchanged then the number become smaller by n from the original number. Then the original number(x,y) is =
x = m -[1/2({9m-n}/9)] , y =1/2({9m-n}/9)

*. If number become greater by n then the original number (x,y) =
x = m -[1/2({9m+n}/9)] , y =1/2({9m+n}/9)
20). If the sum of two numbers is x and their differences is y then their products is=(x^2-y^2)/4

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21). If the sum of the number and its square is x then the number is = [√(1+4x) - 1]/2

22). If x% of a number is n , then y% or z% of that number is = (yzn/x)×100

23). If the ratio of sum and difference of two numbers is a:b , then the ratio of that two numbers = a+b/a-b

24). If the difference of the original number and and the number obtained by interchanging their digit is x ,  then the difference of digit is = x/9

25). If the ratio between the two digit number and the sum of their digit is a:b , if unit place digit is n more than the digit in that of tens place. Then the number is given by = (9a/11b-2a)n
*. Unit place digit = (10b-a/11b-2a)n
*. Tens place digit = (a-b/11b-2a)n

26). If f(x) = x^n +K , is divided by x-1 , then the remainder is =
*. Remainder = K+1, if K<(x-1)
*. Remainder = (1+ remainder obtained when K is divided by x-1) , if K>(x-1)

27). When a number is divided by x1 and x2 separately , it leaves remainder r1 and r2 respectively. If that number is multiplied by x1×x2, then the remainder is = (x1×r2 + r1)

28). The numbers of zeros at the end of the product is given by;
*. Number of zeros = 2^m×5^n if m>n
*. Number of zeros = m if m<n

29). When a number is multiplied by x , then it increased by y , then the number is = y/(x-1)

30). If N is a composite number and it can be expressed as N = a^p.b^q.c^r.... , where a , b and C is different prime numbers and p ,q and r is positive integers, then number of divisor are =  (p+1).(q+1).(r+1)...

31). If a number is divided by N1 then it leaves remainder R , if same number is divided by N2 , then the remainder is = R/N2 ,
[ Note: N1>N2 , and N1 is divisible by N2]

32). If the sum of two numbers is x and their differences is y , then the difference in their squares is = xy

33). The difference of the squares of two consecutive numbers X and Y is = X+ Y

34). If the sum two numbers is X and the sum of their squares is Y , then
*. The product of the numbers = X^2-Y/2
*. The numbers are = [X-√(2Y-X^2)]/2 and        [X +√(2Y-X^2)]/2

35). If the sum of the squares of two numbers is x and the squares of their differences is y , then the product of that two numbers is = (x-y)/2

36). If the product of two numbers is x and the sum of their squares is y , then the sum of the two numbers is √(y+2x) and the difference of two numbers is √(y-2x).

37). If the denominator of a rational number is d more than to its numerator. If numerator is increased by x , and denominator is decreased by y. Then we obtained P. Then the rational number is =
[x-P(d-y)/x+(yP-d)]

38). When A is added to another number B then total (A+B) becomes p% of number B. Then the ratio between A and B is =
(p-100/100)

39). When the two different number is divided by the same divisor , they leave remainder x and y respectively. If their sum is divided by the same divisor then remainder is z , then the divisor is = (x+y-z)

40). If the product of two numbers is x and their sum is y , then the two numbers is =
[y+√(y^2-4x)]/2 and   [y-√(y^2-4x)]/2

41). If the product of two numbers is x and their differences is y , then the two numbers is = √(y^2+4x)+y/2 and  √(y^2+4x)-y/2

### The Revolt of 1857: The Broader Manifestation of Emerging Mass Nationalism in India ๐ฎ๐ณ

The Revolt of 1857: The Broader Manifestation of Emerging Mass Nationalism in India ๐ฎ๐ณเคธिंเคนाเคธเคจ เคนिเคฒ เคเค े เคฐाเคเคตंเคถों เคจे เคญृเคुเคी เคคाเคจी เคฅी,
เคฌूเคข़े เคญाเคฐเคค เคฎें เคเค เคซिเคฐ เคธे เคจเคฏी เคเคตाเคจी เคฅी,
เคुเคฎी เคนुเค เคเค़ाเคฆी เคी เคीเคฎเคค เคธเคฌเคจे เคชเคนเคाเคจी เคฅी,
เคฆूเคฐ เคซिเคฐंเคी เคो เคเคฐเคจे เคी เคธเคฌเคจे เคฎเคจ เคฎें เค ाเคจी เคฅी।
เคเคฎเค เคเค ी เคธเคจ เคธเคค्เคคाเคตเคจ เคฎें, เคตเคน เคคเคฒเคตाเคฐ เคชुเคฐाเคจी เคฅी,
เคฌुंเคฆेเคฒे เคนเคฐเคฌोเคฒों เคे เคฎुँเคน เคนเคฎเคจे เคธुเคจी เคเคนाเคจी เคฅी,
เคूเคฌ เคฒเคก़ी เคฎเคฐ्เคฆाเคจी เคตเคน เคคो เคाँเคธी เคตाเคฒी เคฐाเคจी เคฅी।। - เคธुเคญเคฆ्เคฐा เคुเคฎाเคฐी เคौเคนाเคจ (เคญाเคฐเคคीเคฏ เคเคตि)
Around 3000 rebellious Indian soldiers erupted with excessive outrage on 10 May 1857 against the oppressive colonial system in the garrison town of Merrut. Crossing Jamuna river, the revolutionaries entered Delhi, attacked British cavalry posts, police stations, killed European officials and ascertained their control over Red Fort and Salimgarh Fort. The upsurging rebellion gained the political support of Nana Saheb- the adopted son of late Peshwa Baji Rao, the queen of Jhansi Rani Laxmibai, Mughal Emperor Bahadur Shah ZafarTantia Tope, Rani Awantibai. Identifying with the ong…

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

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…

### " THE LAWS OF NATURE" เคช्เคฐเคृเคคि เคे เคจिเคฏเคฎ, เคिเคธเคธे เคोเค เคญी เคฌเค เคจเคนीं เคธเคเคคा, เคเคช เคญी เคจเคนीं | เคช्เคฐเคृเคคि เคे เคคीเคจ เคुเคฃ เค्เคฏा เคนै?|

"เคฒॉเค เคเคซ เคจेเคเคฐ" เคเคนเคคा เคนै -

เคช्เคฐเคृเคคि เค्เคฏा เคนै?
เคिเคธी เคฐाเคท्เค्เคฐ เคฏा เคฆेเคถ เคो เคเคฆเคฐ्เคถ เคฐाเคท्เค्เคฐ เคฏा เคฆेเคถ เคฌเคจाเคจे เคे เคฒिเค เคจिःเคธंเคฆेเคน เคเค เคเคฆเคฐ्เคถ เคाเคจूเคจ เคต्เคฏเคตเคธ्เคฅा เคी เคเคตเคถ्เคฏเคเคคा เคนोเคคी เคนै, เคिเคธเคे เคจเคเคฐ เคฎें เคเคธ เคฆेเคถ เคฎें เคฐเคนเคจे เคตाเคฒा เคธूเค्เคท्เคฎ เคीเคต เคธे เคฒेเคเคฐ เคตिเคถाเคฒเคाเคฏ เคीเคต เคคเค เคธเคญी เคเค เคธเคฎाเคจ เคนोเคคे เคนै। เคिเคธी เคฆेเคถ เคा เคธ्เคตाเคฎी เคเค เคฎเคจुเคท्เคฏ เคนो เคธเคเคคा เคนै, เคเคธ เคชृเคฅ्เคตी เคा เคธ्เคตाเคฎी เคญी เคเค เคฎเคจुเคท्เคฏ เคนो เคธเคเคคा เคนै, เคिเคจ्เคคु เค्เคฏा เคเคธ เคธเคฎ्เคชूเคฐ्เคฃ เคฌ्เคฐเคน्เคฎाเคฃ्เคก เคा เคธ्เคตाเคฎी เคญी เคเค เคฎเคจुเคท्เคฏ เคนो เคธเคเคคा เคนै, เคถाเคฏเคฆ เคจเคนीं .... เคเคฌ เคฏเคนां เคชเคฐ เคเค เคช्เคฐเคถ्เคจ เคนै เคเค เคคा เคนै, เคि เค्เคฏा เคเคธ เคธเคฎ्เคชूเคฐ्เคฃ เคฌ्เคฐเคน्เคฎाเคฃ्เคก เคो เคญी เคिเคธी เคธ्เคตाเคฎी เคी เคเคตเคถ्เคฏเคเคคा เคนै? เคฏเคฆि เคिเคธी เคฆेเคถ เคो เคธ्เคตाเคฎी เคि เคเคตเคถ्เคฏเคเคคा เคนै, เคฏเคฆि เคชृเคฅ्เคตी เคो เคिเคธी เคธ्เคตाเคฎी เคी เคเคตเคถ्เคฏเคเคคा เคนै, เคคो เคฏเคीเคจเคจ เคเคธ เคฌ्เคฐเคน्เคฎांเคก เคो เคญी เคเค เคธ्เคตाเคฎी เคि เคเคตเคถ्เคฏเคเคคा เคนै। เคเคธ เคฌ्เคฐเคน्เคฎांเคก เคा เคธ्เคตाเคฎी เคो เคोเค เคญी เคนै, เคเคธเคे เคฒिเค เคฏे เคชूเคฐा เคฌ्เคฐเคน्เคฎांเคก เคเค เคฆेเคถ เคैเคธा เคนै, เคिเคธเคे เคญीเคคเคฐ เคนเคฎाเคฐे เคैเคธे เคเคธंเค्เคฏ เคीเคต เคฐเคน เคฐเคนे เคนै, เคเคธ เคฌ्เคฐเคน्เคฎांเคก เคฎें เคนเคฎ เคเคेเคฒे เคจเคนीं เคนै। เคฏเคฆि เคฏे เคชूเคฐा เคฌ्เคฐเคน्เคฎांเคก เคเค เคฆेเคถ เคนै, เคคो เคจिเคถ्เคिเคค เคนी เคเคธ เคฆेเคถ เคा เคญी เคเค เคจिเคฏเคฎ เคนोเคा เคोเค เคाเคจूเคจ เคนोเคा। เคฏเคฆि เคนเคฎ เคिเคธी เคฆेเคถ เคी เคฌाเคค เคเคฐें เคคो เคตเคนां เคाเคจूเคจ เคต्เคฏเคตเคธ्เคฅा เคฌเคจाเค เคฐเคเคจे เคे เคฒिเค เคธैเคจिเคों เคो เคคैเคจाเคค เคिเคฏा เคा…

### Metamorphosing Sociopolitical Matrix of India under rule of East India Company

Metamorphosing Sociopolitical Matrix of India under the Regime of East India Company till 1857

Under the colonial rule of the British Imperial Legislative Government and East India Company, the sociopolitical structure of India had undergone a massive change at several levels. East India Company was evolving as a crucial political strength in India by late eighteenth century after deposing prominent regional powers like Bengal, Bombay etc. The Company introduced repressive policies for expansion of territories as elaborated in the article Emergence of East India Company as an Imperialist Political Power in India.
Functioning as an administrative and political entity in India, EIC launched numerous political, social and education-related policies that considerably affected various sections of society like peasants, women, children, industrial sectors and handicrafters. The prime objective of this article is to shed light on the sociopolitical matrix of British India to understand the sta…

### Was East India Company supremely functioning as a Colonial Trading Group till 1857?

Was British East India Company supremely functioning as a Colonial Trading Group till 1857?

After acquiring the royal charter from the ruler of England in 1600, the British East India Company attained a monopoly on trade with East. The company eliminated competition in business; asserted control over Bengal after Battle of Plassey 1757; achieved Diwani rights ( i.e. revenue collection rights over Bengal, Bihar and Orisha) after Treaty of Allahabad 1765 and emerged as a supreme political power by the middle eighteenth century. But interestingly, the company experienced financial collapse by the second half of the eighteenth century because of nepotism and persistence of corruption in company officials. ( Such corrupt officials were often referred as nabobs- an anglicised form of the nawab.)
British Parliamentary Government investigated the inherent functioning of the company and introduced several acts to induce discipline in the company officials. Regulating Act/ Charter Act (1773):Thi…