Competitive Exams Mathematics Basic – You must know….. |
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In general we are using Decimal Numbering System. The base of Decimal Numbering system is 10. In decimal Numbering system total number of digits are 10 i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9. |
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Face Value of a digit in a numeral is its own value, at whenever place it may be.
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3456 Face of 4 is 4 in this numeral |
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Place value of digit is depends on the position of digit like
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123456 in this numeral Place or Local value of 6 is 6 x 1 = 6 (Unit Digit) |
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Place value of 5 is 50 = 5 x 10 (Ten’s Digit) |
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Place value of 4 is 400 = 4 x 100 (Hundred’s Digit) |
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Place value of 3 is 3000 = 3 x 1000 (Thousand’s Digit) |
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Place value of 2 is 20000 = 2 x 10000 (Ten Thousand’s Digit) |
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Place value of 1 is 100000 = 1 x 100000 (Lac’s Digit) |
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Natural Numbers: The group of numbers 1,2,3,4,5….. is called as Natural Numbers. |
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Whole Numbers: The group of 0,1,2,3,4,5…. is called as set of whole numbers. |
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The smallest Natural Number is 1 and smallest whole number is 0 |
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Even Number (rather than zero) which is exactly divisible by 2 is called an even number. |
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Odd Number: A number which when divided by 2 gives the remainder 1 is called an odd number. |
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The difference between any two consecutive even or odd numbers is 2 |
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Even number 4 & 6 = 6-4 = 2 |
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Odd number 5 & 7 = 7-5 = 2 |
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The sum of any two ODD numbers is an EVEN numbers |
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The product of any two ODD number is always ODD numbers |
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If any EVEN number divisible by ODD number then division is EVEN number |
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If any ODD number divisible by ODD number then division is ODD number |
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Prime Number : A number which is greater than 1 and is divisible only by 1 and the number it self, is called a prime number. |
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There are 25 prime numbers between 1 and 100. |
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They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 & 97. Out of 25 numbers only one number is EVEN ie 2 and all remaining numbers are ODD. |
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Composite Numbers: Number other than 1 which are not prime numbers are composite numbers. |
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1 is neither a PRIME number nor COMPOSITE number |
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Co-Prime Numbers – Two natural numbers a and b are said to be co-prime if their HCF is 1. OR Numbers which do not have common divisor other than 1 are also called co-primes eg (25,28), (12,35), (2,3), (4,5) (7,9), (8,11) are the examples of co-primes. |
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The sum of the numbers from 1 to 10 is 55. Out of them the sum of EVEN numbers is 30 and that of ODD number is 25. |
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Perfect Square never ends with 2,3,7,8 and odd number of 0, Ten’s digit along with 5 instead of 2 ( 15,35,55,75 end with these digits is not a perfect square)
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Square of number which is ends with 5 is always ends with 25. |
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Square of number which is ends with 0 is always ends with even number of zero. |
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10, 1000 are not square numbers but 100, 10000 are the square numbers. |
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Digital root of perfect square is always 1, 4, 7 and 9 (digit total of number)
e.g 625 = 6+2+5 = 13 = 1+3 = 4 , 1024 = 1+0+2+4 = 7
961 = 9+6+1 = 16 = 7 , 64 = 6+4 = 10 = 1 |
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Total number of digits between 1 to 100 = 192 |
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Roman Numbers |
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10 = X, 20 = XX, 30 = XXX, 40 = XL, 50 = L, 100 = C, |
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500 = D, 1000 = M |
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Between 1 to 100
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Digit ‘1’ repeats no. of times = 21 |
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Digit ‘0’ repeats no. of times = 11 |
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Any other digit 2,3,4,5 etc = 20 |
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Rule of BODMAS |
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Bracket, Division, Multiplication, Addition then Subtraction |
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| Law of Indices |
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am x an = am+n Example : 23 x 25 = 23+5 = 28 |
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am / an = am-n Example : 54 / 53 = 51 = 5 |
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(a x b)m = am x bm Example : (3 x 5)2 = 32 x 52 |
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(a / b)m = am / an Example: (2 / 5)2 = 22 / 52 |
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a-m = 1/am Example: 4-3 = 1/ 43 |
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a0 = 1 Example: 50 = 1 |
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(am)n = amxn Example: (52)3 = 52x3 = 56 |
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Square of the same number’s square root is the same number
Eg. (Ö3)2 = 3 |
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| Divisibility Test for 1 to 13 numbers |
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Divisor # 1
Any integer divisible by 1 |
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Divisor # 2
The last digit is even ie. 0, 2, 4, 6, or 8
Eg. 24/2, 50/2, 68/2 etc |
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Divisor # 3
The sum of the digits is divisible by 3. For large numbers digits may be
Summed iteratively.
Eg. 123/3 in this case sum of (1+2+3) = 6 is divisible by 3
471/3 in this case sum of (4+7+1) = 12 is divisible by 3 |
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Divisor # 4
Add the ones digit to twice the tens digit (All digits to the left of the tens digit can be ignored) OR last two digits divisible by 4
Eg. 624/4 in this case (2x2) + 4 = 8 and 8 is divisible by 4 or you can check just 24 it is also divisible by 4 |
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Divisor # 5
The last digit is 0 or 5
Eg. 50/5, 75/5, 430/5 etc... |
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Divisor # 6
Add the last digit to four times the sum of all remaining other digits.
Eg. 38712 / 6 in this case sum of (3+8+7+1) x 4 + 2 = 78 is divisible by 78/6=13 |
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Divisor # 7
Form the alternating sum of blocks of three digits from right to left.
1369851: 851 - 369 + 1 = 483 = 7 x 69
Subtract 2 times the last digit from the rest 483: 48 - (3 x 2) = 42 = 7 x 6
Or add 5 times the last digit to the rest 483: 48 + (3 x 5) = 63 = 7 x 9 |
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Divisor # 8
If the hundred digit is even examine the number formed by the last two digits 624: 24
If the hundred digit is odd examine the number obtained by the last two digit plus 4
352: 52 + 4 = 56
Add the last digit to twice the rest
56: (5 x 2) + 6 = 16 |
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Divisor # 9
The sum of the digits is divisible by 9. For large numbers digits may besummed iteratively.
Eg. 6174 / 9 in this case sum of (6+1+7+4) = 18 it is divisible by 9 |
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Divisor # 10
The last digit must be 0
Eg. 400/10, 12000/10, 360000/10 etc... |
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Divisor # 11
Alternate sum of digit should be “zero” or in times of 11 like 11, 22,33 etc. |
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Divisor # 12
Apply test of divisibility of 3 & 4. Both tests should be true. |
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Divisor # 13
Add four times of unit digit into the number which is form using tenth and hundredth digit.
Eg. 156 / 13 = 15 + (6 x 4) = 39 , and 39 is divisible by 13 so the number 156 is divisible by 13. |
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| Common Formulae |
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Addition of any natural from 1 to n, use following formulae |
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n (n+1) e.g. 1 to 10 natural numbers addition is 10 ( 10 +1) 110
---------- --------------- = ------ = 55
2 2 2 |
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Sum of first n odd number is = n2 |
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Sum of first n even number is = n(n+1) |
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n
Sum of Square of 1st n odd number is = ----- x (4n2-1)
3 |
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2n (n+1) (2n+1)
Sum of Square of 1st n even number is = --------------------
3 |
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Sum of natural numbers square from 1 to n2
1
---- n (n+1)(2n+1)
6 |
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Sum of natural numbers cubes from 1 to n3
1
---- n2 (n+1)2
4 |
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| Average |
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Average speed = 2xy / (x + y) |
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For Single Qty |
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Find the qty/weight of object = new average + (difference of both average x no. of qty) |
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For more than one qty |
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Find the qty/weight of object = new average + (difference of both average x no. of qty ¸ added qty) |
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Simple Interest |
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Principal = P, Rate of Interest = I, Period = T
( P x R x T )
SI = Simple Interest Formula = -------------------
100
( SI x 100 )
Principal = -----------------
R x T
( SI x 100 )
Rate % = -----------------
P x T
( SI x 100 )
Time = -----------------
P x R |
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| Compound Interest |
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Let Principal = P, Rate of Interest = R% per annum, Time = n yearsWhen interest is compounded Annually,
R
Amount = P( 1 + ------ )n
100
When interest is compounded half yearly,
R/2
Amount = P( 1 + ------ )2n
100When interest is compounded quarterly,
R/4
Amount = P( 1 + ------ )4n
100
2
When interest is compounded Annually but time is in fraction, say 3---
5
R/4
Amount = P( 1 + ------ )4n
100 |
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Profit and Loss |
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Profit = Selling Price – Cost Price
Loss = Cost Price – Selling Price
Selling Price = Cost Price + Profit
Cost Price = Selling price + Loss |
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| Direct and Invers |
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Direct Variation
When two interdependent quantities change in such way that their ratio remains constant, then it is an example of direct variation.
e.g The cost of pens varies directly as the number of pens. |
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Inverse Variation
When two interdependent quantities change in such way that their product remains constant, then it is an example of inverse variation.
If 6 persons finish a piece of work in 8 days, then 12 persons will finish the same work in 4 days, here the number of days varies inversely as the number of persons. |
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Problems on Train |
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Convert a km/hr to m/sec
5
a x ----- m/sec
18
Convert a m/sec to km/hr
18
a x ----- km/hr
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Relative speed & time:
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Suppose two trains or two bodies are moving in opposite direction at u m/s and v m/s, then their relative speed is = ( u + v ) m/s. |
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If two trains of length a and b meters are moving in the opposite directions at u m/s and v m/s, then time taken by the to cross each other = (a + b) / ( u - v ) |
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If two trains of length a and b meters are moving in the opposite directions at u m/s and v m/s, then time taken by the to cross each other = (a + b) / ( u + v ) |
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| Formulae |
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Perimeter
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Rectangle = 2(a + b) (where a is length and b is breadth) |
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Square = 4 x side |
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Triangle = a + b + c |
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Area
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Rectangle = a x b |
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Square = a2 |
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Triangle = ½ b x h |
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Parallelogram = Length of that side x Perpendicular |
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Circle = π r2 |
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Surface
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Surface of Area of Cuboids = 2(lb + bh + hl) |
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Surface of Area of Cube = 6a2 |
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Curved surface area of Cylinder = 2 π r h |
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Total surface area of Cylinder = 2 π r (r + h) |
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Curved surface area of a Cone = ½ l 2 π r = π r l |
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Total surface area of a Cone = π r l + π r2 = π r ( l + r) |
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Surface Area of a Sphere = 4 π r2 |
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Curved surface Area of a Hemisphere = 2 π r2 |
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Total surface area of a Hemisphere = 3 π r2 |
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Volume
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Volume of a Cuboid = length x breadth x height |
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Volume of Cube = a3 |
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Volume of Cylinder = π r2 h |
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Volume of Cone = 1/3 π r2 h |
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Volume of Sphere = 4/3 π r3 |
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Volume of hemisphere = 2/3 π r3 |
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Facts about Day and Time |
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Generally day on 1st January will be the same on 31st December, incase of leap year it will be the next day.
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1st Jan 2011 – Saturday |
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31st December 2011 - Saturday |
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Days on 1st, 2nd and 3rd are the same on 29th, 30th and 31st of every month.
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1st May 2011 – 29th May 2011 – Sunday |
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2nd May 2011 – 30th May 2011 – Monday |
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3rd May 2011 – 31st May 2011 - Tuesday |
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A day on 1st January will be repeated 53 times in the whole year. |
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A year number completely divisible by 4 and 400 is the leap year. |
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In a leap year 29 days are available in the month of February. |
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Days of the month February and March generally are same except in the Leap year. |
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In every year days are same on 14th November, 5th September, 15 August & 1st August. |
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Date & Year |
Festival/Anniversary |
Day |
1st August 2011 |
Tilak Jayanti |
Monday |
15th August 2011 |
Independence Day |
Monday |
5th September 2011 |
Teachers Day |
Monday |
14th November 2011 |
Child Day |
Monday |
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In every year days are same on 1st May, 2nd October & 25th December. |
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Date & Year |
Festival/Anniversary |
Day |
1st May 2011 |
Maharashtra Day |
Sunday |
2nd October 2011 |
Gandhi Jayanti |
Sunday |
25th December 2011 |
Cristmas |
Sunday |
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Counting of Odd days : |
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Odd days in Ordinary year = 365 days ( 365 / 7) = 52 weeks and 1 day
Odd days in Leap year = 2 days
100 years = 76 ordinary year + 24 years
= 76 x 1 + 24 x 2 = 124 odd days, 124 / 7 = 17 weeks + 5 days
So Odd days in 100 years = 5 days
Odd days in 200 years = ( 5 x 2 ) = 10 / 7 = 3 odd days
Odd days in 300 years = ( 5 x 3 ) = 15 / 7 = 1 odd days
Odd days in 400 years = ( 5 x 4 + 1) = 21 / 7 = 0 odd days
A century which is divisible by 400 = 0 odd days |
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| Odd days in month |
Ordinary Year |
Leap Year |
January
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3 |
3 |
| February |
0 |
1 |
| March |
3 |
3 |
| April |
2 |
2 |
| May |
3 |
3 |
| June |
2 |
2 |
| July |
3 |
3 |
| August |
3 |
3 |
| September |
2 |
2 |
| October |
3 |
3 |
| November |
2 |
2 |
| December |
3 |
3 |
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