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Clocks Concepts :

The dial of the clock is circular in shape and was divided into 60 equal minute spaces

60 minute spaces traces an angle of 3600. Therefore, 1minute space traverses an angle of 60 In 1 hour, Minute hand traverses 60 minute space or 3600 ,Hour hand traverses 5 minute space or 300 The hands of the clock are perpendicular in 15 minute spaces apart

The hands of the clock are in straight line and opposite to each other in 30 minute spaces apart.

The hands of the clock are in straight line when they coincide or opposite to each other.

The hands of the clock are perpendicular to each other for 22 times in 12 hours and for 44 times in a day.

The hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.

The hands of the clock coincides with each other for 11 times in 12 hours and 22 times per day

The hands of the clock are 44 times in a straight line per day

The minute hand gain 55 minutes over hour hand per hour.

Hence x minute space to be gained by minute hand over hour hand can be calculated as x.(60/55) or x.(12/11)

Ex : At what time between 2'O clock and 3'O clock the hands of the clock are opposite to each other.

1. 34( 6/11 ) past 2'Oclock 2. 43( 7/11 ) past 2'Oclock

3. 56( 8/11 ) past 2'Oclock 4. 64(9/11past 2'Oclock

Sol At 2'O clock the minute hand will be at 12 as shown below

The minutes hand to coincide with the hour hand it should trace at first 10 minute spaces

And then the hands of the clocks to be opposite to each other minute hand should trace 30 minute spaces i.e. totally it should gain 10+30=40 minute spaces to be opposite to that of hour hand

We know that,

Minute hand gains 55 minute spaces over hour hand in 1 hour

Therefore, Minute hand gain 40 minute spaces over hour hand in 40 × (60/55) = 43(7/11)

Hence the hand of the clock will minutes be opposite to each at 43( 7/11 ) past 2'Oclock

Therefore, Correct option is 2'

When clock is too fast, too slow

If a clock or watch indicates 6 hr 10 min when the correct time is 6, it is said that the clock is 10 min too fast If it indicates 6. 40 when the correct time is 7, it is said to be 20 min too slow.

Now let us have an example based on this concept

Ex. My watch, which gains uniformly, is 2 min, & show at noon on Sunday, and is 4 min 48 seconds fast at 2 p.m on the following Sunday when was it correct ?

Sol: From Sunday noon to the following Sunday at 2 p.m there are 7 days 2 hours or 170 hours. The watch gains 2+4 4/5 min in 170 hrs.

Therefore, the watch gains 2 min in 2 *170 hrs i.e., 50 hours

6 4/5 Now 50 hours = 2 days 2 hrs. Therefore, 2 days 2 hours from Sunday noon = 2 p.m on Tuesday.

Calendars Concept :

The time in which the earth travels round the sun is a solar year and is equal to 365 days 5 hrs. 48 minutes and 47 1/2 seconds

Year is 365.2422 days approximately.

The common year consists of 365 days.

The difference between a common year and a solar year is therefore 0.2422 of a day and we consider it by adding a whole day to every fourth year.

Consequently in every 4th year there are 366 days.

The years which have the extra day are called leap years. The day is inserted at the end of February, The difference between 4 common years and 4 solar years is 0.969 of a day.

If therefore, we add a whole day to every 4th year, we add too much by 0.0312 of a day. To take account of this, we omit the extra day three times every 400 years,

The thing is to ensure that each season may fall at the same time of the year in all years. In course of time, without these corrections, we should have winter in July and summer in January also .

With the very small variation, the present divisions of the year are those given in B.C 46 by Julius Caesar . The omission of the extra day three times in 400 years is called the Gregorian Correction. This correction was adopted at once in 1582 in Roman Catholic Countries. but not in England until, 1752. The Gregorian mode of reckoning is called the New Style, the former, the Old Style.

The New Style has not yet been adopted in Russia , so that they are now 13 days behind us as an Example What we call Oct. 26th they call 13th Oct . They have Christmas day on 7th of January and we have on 25th December every year.

In an ordinary year there are 365 days i.e., 52 weeks + 1 day

Therefore an ordinary years contains 1 odd day.

A leap year contains 2 odd days.

100 year = 76 ordinary years + 24 leap years.

= 76 odd days + 48 odd days

= 124 odd days = 17 weeks + 5 days. (in the consideration of weeks)

Therefore, 100 years contain 5 odd days.

200 years contains 3 odd days.

300 years contain 1 odd days

Since there are 5 odd days in 100 years, there will be 20 days in 400 years. But every 4th century is a leap year.

Therefore, 400 years contain 21 days. Here 400 years contain no odd days.

As First January 1 AD was Monday. we must count days from Sunday

i.e. Sunday for 0 odd days , Monday for 1 odd day , Tuesday for 2 odd days and so on.

Last day of a century cannot be either Tuesday. Thursday or Saturday.

The first day of a century must either be Monday. Tuesday, Thursday or Saturday.

Now let us observe the Examples

Ex How many times does the 29th days of the month occur in 400 consecutive years

1) 97 times 2) 4400 times 3) 4497 times 4) none

Sol: In 400 consecutive years there are 97 leap years. Hence in 400 consecutive years, February has the 29th day 97 times, and the remaining 11 months have the 29th day 400 x 11 or 4400 times.

Therefore, 29th day of the month occurs (4400 + 97) or 4497 times

Ex Given that on 10th November 1981 is Tuesday, what was the day on 10th November 1581

1) Monday 2) Thursday 3) Sunday 4) Tuesday

Sol: After every 400 years, the same day comes.

Thus if 10th November1981 was Tuesday, before 400 years i.e on 10th November 1581, it has to be Tuesday.