revision of some math topics for class XII (12th) ISC , CBSE

find the area of the region {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) 
evaluate integral of (x ² +3x) with limits 1 to 3 using summation (limit of a sum)
answer: (62/3)
evaluate integral of [ x*exp(x)] / (x+1) ² dx
answer: exp(x) / (x+1) + C 

∫ { [log x] / x² } dx log x refers to lnx
answer: -(1+ln x ) /x +C 
∫ dx / [ (x-2) (1 + x²)]
answer:(1/5)ln(x-2) -(1/10)ln(1 + x²) -(2/5)arctan(x)+C

problem on matrices

answer: (x= -1)


problem on determinant



solve x+y = 2 ; 2y-z = 0 ; -x-y+z = -1 using cramer's rule (using determinants)
answer : x = 3/2 , y =1/2 , z=1 
solve 3x+4y = 17 , -4x+3y =44 using matrix inversion
answer: x = -5; y =8 

solve cos ² x (dy/dx) + y = tanx
answer : y = tanx - 1 + C exp(-tanx) 
ntegral of ln(sin x) from 0 to pi/2
answer: (pi/2)ln(1/2) for explanation: 

.∫ (x²) sin x dx
answer: -x² cosx +2xsinx+2cosx+C ;explanation:

if y= tanֿ¹ { √ [(1-x)/(1+x)] } find dy/dx
answer: -1 / sqrt (1-x²) ;explanation

show that the semivertical angle of a right circular cone
of maximum volume and given slant height is tan ֿ¹(√2 )


show that the height of a closed cylinder of given volume and minimum surface area
is equal to its diameter

There is a figure (norman window) in which a rectangle is surmounted by a semicircle with diameter along one side of the rectangle .If the perimeter is given find the radius of the semicircle if the area is to be maximum (maximum amount of light is to be admitted into the room)


find the equation of the tangent at t = pi/3 on the curve x=2cost , y = 3sint
answer: 3x + 2ysqrt(3) =






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some resources on weight loss

portion control
distinguish between hunger and emotional feelings like anxiety, boredom etc
chew your food slowly
drink sufficient quantities of water
increase physical activities
avoid stress
cut down on usage of butter
snack on fruits instead of candy
use steamed food instead of fried
cut down on cookies
cut down on carbonated beverages
use fresh fruits instead of canned fruits with syrup or juice
cut down on ice cream
avoid high fructose corn syrup (HFCS) this will get rid of most of the junk food
use more fish in your diet
use lean meat
try not to skip breakfast
cut down on chips
avoid fast food
use foods rich in fibre (avoid refined food)
use vegetables


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south africa win cricket series 2-0 against australia

south africa have won the cricket test series against australia 2-0 by winning the boxing day test match at MCG on 30/12/2008, by 9 wickets.

http://news.bbc.co.uk/sport2/shared/fds/hi/statistics/cricket/scorecards/2008/12/15095/html/scorecard.stm

http://content-ind.cricinfo.com/ausvrsa2008_09/engine/current/match/351682.html

This is australia's first home test series loss in nearly 16 years.

In this match australia seemed to be carrying some players who had injuries and some who were out of form.

The fall of aussie cricket team seems to have started at the beginning of the year when they lost the perth test against india (even though australia won that series 2-1) . Perth was the last place anyone expected india to beat the australians . In the CB one day series also , India easily defeated the aussies 2-0 in the best of three finals.



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beware of student education loans

In India, as education for medical and engineering studies are becoming more and more costly, particularly in private colleges, it is common for students to take out education loans.

Most of the students might be taking on these loans in the hope of landing jobs with high wages as soon as they graduate. But they have to remember that many things can happen in the duration of a 4 to 5 year course. If for some reason, the student has some arrear papers by the time he finishes the course, or if the job market for that particular branch of study is not very strong , the student will find it very hard to find a job to support himself and pay off the student loan (some loans might have a 1 year or so grace period after the course duration before repayment starts. ). Even an average student will only start at an entry level position in his job ladder and will take many years before he might have enough experience to get the high wages that he might have dreamt of as a student.

So before you take out huge loans, read and do research about the interest rates(fixed/floating etc) , repayment details, relative chance of jobs in fields related to current education, aptitude of the student in the chosen field of study etc

integral using cauchy's integral formula

evaluate integral of cosz / (2z-5) over |z-3|=2
pole is given by (2z-5)=0
z = 5/2 which lies inside the given circle
take the 2 common from the denominator to make it (z-a) form
take f(z)=cosz which is analytic inside and on the given contour




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disposable email

Disposable email address


Most internet users must have seen that some internet sites ask you for your email addresses for registering. After that you will get tons of unwanted spam mail in your inbox and you spend a lot of time to delete them. Spam filters can help a little bit.There is another approach. There are some sites which provide you with alternate disposable temporary email addresses. You can give these temporary addresses when registering. Some of these services will forward the mail to your proper email address. Some of the services have no password so that anyone who knows your id can read your spam mail at the site. Some of the services delete all mail after a certain number of messages/amount of time. Some of the addresses are valid only for a certain periond of ime. Some services require some registration procedure while some allow direct usage.Before using such services please read their terms of services

website counters

website counters

If you have a website you may like to see the number of people who visit your site, where your visitors come from, which pages are the most popular, which browsers and operating systems your visitors use and so on.

Sometimes your web-host themselves may provide you with some of these statistics. If you are using services like geocities they already provide you with some of these statistics.

Otherwise you will have to rely on counter solutions available on the internet or make your own counter.

Most of the counters available on the internet give you a piece of code which you copy and paste into your webpage. Most of the sites providing these counters will ask for your email address when you sign up. Some of the free counters may send small ads along with the counter.

Some examples of third party counters are

webstat.com (this counter is used extensively on my site They provide a free version for upto 20000 pageviews per month, and also a premium version/trial version )

statcounter.com(offers free edition for sites with upto 250000 pageloads per month and paid versions also if number of pageloads is higher)

Many more counters are available and can easily be located via search engines by searching for 'free page counters' or some similar search words.

Another approach to counters are php scripts.(first check if you can use php). There are many php scripts available on the net. Most of these require you to change your webpage from html extension to php extension. There may also be another small php file which contains the counter code to which you will link to from inside your webpage. Usually there may also be a text file to hold the count and optional image files to display the digits 0 to 9. (Also remember that the php code can be mangled by the auto correction features of webpage creating software like frontpage. Therefore it is better to enter the php code using a text editor after you have finished making your webpage). The php files are usually uploaded as ascii files.

some examples are

php without sql

php with sql

More counters based on php are available on the internet and can be located by searching for 'php counters' search engines.

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