### quicker maths

ALGEBRA :
1. 1.      Sum of first n natural numbers = n(n+1)/2
2. 2.      Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6
3. 3.      Sum of the cubes of first n natural numbers = [n(n+1)/2]^2
4. 4.      Sum of first n natural odd numbers = n^2
5. 5.      Average = (Sum of items)/Number of items

Arithmetic Progression (A.P.):
An A.P. is of the form a, a+d, a+2d, a+3d, ...
where a is called the 'first term' and d is called the 'common difference'
1. 1.      nth term of an A.P. tn = a + (n-1)d
2. 2.      Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or Sn = n/2(first term + last term)

Geometrical Progression (G.P.):
A G.P. is of the form a, ar, ar2, ar3, ...
where a is called the 'first term' and r is called the 'common ratio'.
1. 1.      nth term of a G.P. tn = arn-1
2. 2.      Sum of the first n terms in a G.P. Sn = a|1-rn|/|1-r|

Permutations and Combinations :
1. nPr = n!/(n-r)!
2. nPn = n!
3. nP1 = n

1. nCr = n!/(r! (n-r)!)
2. nC1 = n
3. nC0 = 1 = nCn
4. nCr = nCn-r
5. nCr = nPr/r!

Number of diagonals in a geometric figure of n sides = nC2-n

Tests of Divisibility :
1. A number is divisible by 2 if it is an even number.
2. A number is divisible by 3 if the sum of the digits is divisible by 3.
3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
4. A number is divisible by 5 if the units digit is either 5 or 0.
5. A number is divisible by 6 if the number is divisible by both 2 and 3.
6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
7. A number is divisible by 9 if the sum of the digits is divisible by 9.
8. A number is divisible by 10 if the units digit is 0.
9. A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places, is divisible by 11.

H.C.F and L.C.M :
H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common Divisor (G.C.D) and Greatest Common Measure (G.C.M).
The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.
The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
Two numbers are said to be co-prime if their H.C.F. is 1.
H.C.F. of fractions = H.C.F. of numerators/L.C.M of denominators
L.C.M. of fractions = G.C.D. of numerators/H.C.F of denominators

Product of two numbers = Product of their H.C.F. and L.C.M.

PERCENTAGES :
1. If A is R% more than B, then B is less than A by R / (100+R) * 100
2. If A is R% less than B, then B is more than A by R / (100-R) * 100
3. If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100
4. If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is : R/(100-R)*100

PROFIT & LOSS :
1. Gain = Selling Price(S.P.) - Cost Price(C.P)
2. Loss = C.P. - S.P.
3. Gain % = Gain * 100 / C.P.
4. Loss % = Loss * 100 / C.P.
5. S.P. = (100+Gain%)/100*C.P.
6. S.P. = (100-Loss%)/100*C.P.

RATIO & PROPORTIONS:
1. The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent.
2. The equality of two different ratios is called proportion.
3. If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.
4. In a : b = c : d, then we have  a* d = b * c.
5. If a/b = c/d then ( a + b ) / ( a – b  ) = ( d + c ) / ( d – c ).

TIME & WORK :
1. If A can do a piece of work in n days, then A's 1 day's work = 1/n
2. If A and B work together for n days, then (A+B)'s 1 days's work = 1/n
3. If A is twice as good workman as B, then ratio of work done by A and B = 2:1

PIPES & CISTERNS :
1. If a pipe can fill a tank in x hours, then part of tank filled in one hour = 1/x
2. If a pipe can empty a full tank in y hours, then part emptied in one hour = 1/y
3. If a pipe can fill a tank in x hours, and another pipe can empty the full tank in y hours, then on opening both the pipes,

the net part filled in 1 hour = (1/x-1/y)  if y>x
the net part emptied in 1 hour = (1/y-1/x) if x>y

TIME & DISTANCE :
1. Distance = Speed * Time
2. 1 km/hr = 5/18 m/sec
3. 1 m/sec = 18/5 km/hr
4. Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is 2xy/(x+y) kmph.

PROBLEMS ON TRAINS :
1. Time taken by a train x metres long in passing a signal post or a pole or a standing man is equal to the time taken by the train to cover x metres.
2. Time taken by a train x metres long in passing a stationary object of length y metres is equal to the time taken by the train to cover x+y metres.
1. Suppose two trains are moving in the same direction at u kmph and v kmph such that u>v, then their relative speed = u-v kmph.
2. If two trains of length x km and y km are moving in the same direction at u kmph and v kmph, where u>v, then time taken by the faster train to cross the slower train = (x+y)/(u-v) hours.
3. Suppose two trains are moving in opposite directions at u kmph and v kmph. Then, their relative speed = (u+v) kmph.
4. If two trains of length x km and y km are moving in the opposite directions at u kmph and v kmph, then time taken by the trains to cross each other = (x+y)/(u+v)hours.
5. If two trains start at the same time from two points A and B towards each other and after crossing they take a and b hours in reaching B and A respectively, then A's speed : B's speed = (√b : √

SIMPLE & COMPOUND INTERESTS :
Let P be the principal, R be the interest rate percent per annum, and N be the time period.
1. Simple Interest = (P*N*R)/100
2. Compound Interest = P(1 + R/100)N – P
3. Amount = Principal + Interest

LOGORITHMS :
If am = x , then m = logax.
Properties :
1. log xx = 1
2. log x1 = 0
3. log a(xy) = log ax + log ay
4. log a(x/y) = log ax - log ay
5. log ax = 1/log xa
6. log a(xp) = p(log ax)
7. log ax = log bx/log ba
Note : Logarithms for base 1 does not exist.

AREA & PERIMETER :
Shape                          Area                            Perimeter
Square                         (side)2                          4(side)

1. Area of a triangle = 1/2*Base*Height or
2. Area of a triangle = √ (s(s-(s-b)(s-c)) where a,b,c are the lengths of the sides and s = (a+b+c)/2
3. Area of a parallelogram = Base * Height
4. Area of a rhombus = 1/2(Product of diagonals)
5. Area of a trapezium = 1/2(Sum of parallel sides)(distance between the parallel sides)
6. Area of a quadrilateral = 1/2(diagonal)(Sum of sides)
7. Area of a regular hexagon = 6(√3/4)(side)2
8. Area of a ring = ∏(R2-r2) where R and r are the outer and inner radii of the ring.

VOLUME & SURFACE AREA :
Cube :
Let a be the length of each edge. Then,
1. Volume of the cube = a3 cubic units
2. Surface Area = 6a2 square units
3. Diagonal = √ 3 a units
Cuboid :
Let l be the length, b be the breadth and h be the height of a cuboid. Then
1. Volume = lbh cu units
2. Surface Area = 2(lb+bh+lh) sq units
3. Diagonal = √ (l2+b2+h2)
Cylinder :
Let radius of the base be r and height of the cylinder be h. Then,
1. Volume = ∏r2h cu units
2. Curved Surface Area = 2∏rh sq units
3. Total Surface Area = 2∏rh + 2∏r2 sq units
Cone :
Let r be the radius of base,  h be the height, and l be the slant height of the cone. Then,
1. l2 = h2 + r2
2. Volume = 1/3(∏r2h) cu units
3. Curved Surface Area = ∏rl sq units
4. Total Surface Area = ∏rl + ∏r2 sq units
Sphere :
Let r be the radius of the sphere. Then,
1. Volume = (4/3)∏r3 cu units
2. Surface Area = 4∏r2 sq units
Hemi-sphere :
Let r be the radius of the hemi-sphere. Then,
1. Volume = (2/3)∏r3 cu units
2. Curved Surface Area = 2∏r2 sq units
3. Total Surface Area = 3∏r2 sq units
Prism :
Volume = (Area of base)(Height)

### UNIT 13 FEATURES OF 73rd AND 74th CONSTITUTIONAL AMENDMENT

Structure

13.0 Learning Outcome

13.1 Introduction

13.2 Initiatives towards Constitutional Status to Local Governance

13.2.1 Features of 73rd Constitutional Amendment

13.2.2 Features of 74th Constitutional Amendment

13.2.3 Decentralised Planning in Context of 73rd and 74th Constitutional Amendment Act

13.3 Initiatives after Economic Reforms

13.4 Functioning of PRIs in Various States after 73rd Amendment

13.5 Functioning of Local Governance after 73rd and 74th Constitutional Amendment: Observations

13.6 Conclusion

13.7 Key Concepts

13.9 Activities

13.0 LEARNING OUTCOME

After studying this Unit you should be able to:

• Identify the background of revitalisation of local governance;

• Understand the features of 73rd and 74th constitutional amendment;

• Discuss the initiatives after economic reforms; and

• Outlines the functioning of local governance in various states after the amendment.

13.1 INTRODUCTION

The revitalization of Pancha…

### Q. What is the meaning of the terms like ‘Pardon’, ‘Reprieve’, ‘Respite’, ‘Remission’ and ‘Commutation’ with respect to the power of the President to grant pardon to convicted persons?

Ans. In terms of their scope and effect, these terms have specific connotations. The effect of Pardon is to abolish punishment and to absolve the convict of all charges. If Pardon is granted, it is assured as if the convict has not committed any crime. The convict will not face any disabilities due to the allegations and charges made against him. ‘Remission’ means reducing the punishment without changing the nature of punishment. For example, the imprisonment for 20 years may be reduced to the imprisonment for 10 years. ‘Commutation’ means reducing the punishment by changing the nature of punishment. For example, punishment to death may be changed to life imprisonment. ‘Respite’ means reducing or changing the nature of punishment in view of the specific facts and circumstances of the convict. For example, the punishment to death awarded to a pregnant woman, may be changed to simple life imprisonment. Respite means delay in execution of punishment especially that of death, in order to …

### UNIT 1 CONCEPT, EVOLUTION AND SIGNIFICANCE OF DEMOCRATIC DECENTRALISATION

Structure

1.0 Learning outcome

1.1 Introduction

1.2 Concept of Democratic Decentralisation

1.3 Evolution of Democratic Decentralisation

1.4 Significance of Democratic Decentralisation

1.5 Democratic Decentralisation in India

1.6 Conclusion

1.7 Key concepts

1.9 Activities

1.0 LEARNING OUTCOME

After studying this unit, you should be able to:

• Understand the concept of Democratic Decentralization;

• Know the evolution and significance of Democratic Decentralization; and

• Describe the Democratic Decentralization pattern in India.

1.1 INTRODUCTION

The dawn of 21st century is marked by decentralized governance both as a strategy and philosophy of brining about reforms and changes in democracies. These changes led to such virtues of transparency, responsiveness and accountability and ensures good governance. Today decentralization and democracy are the most significant themes in the development discourse. In the present contex…