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) more explanation on this area question

evaluate integral of (x ² +3x) with limits 1 to 3 using summation (limit of a sum)
answer: (62/3) explanation of integration by limit of sum question

evaluate integral of [ x*exp(x)] / (x+1) ² dx
answer: exp(x) / (x+1) + C explanation on integration by parts of exp(x)[f(x) + f '(x)] form

∫ { [log x] / x² } dx log x refers to lnx
answer: -(1+ln x ) /x +C explanation on integration by parts for ∫ { [log x] / x² } dx

∫ dx / [ (x-2) (1 + x²)]
answer:(1/5)ln(x-2) -(1/10)ln(1 + x²) -(2/5)arctan(x)+C explanation on integration by partial fraction

problem on matrices

answer: (x= -1) explanation of matrix problem


problem on determinant

explanation of determinant problem

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 explanation on solution of equations using cramer's rule

solve 3x+4y = 17 , -4x+3y =44 using matrix inversion
answer: x = -5; y =8 explanation on solving system of equations using matrix inversion

solve cos ² x (dy/dx) + y = tanx
answer : y = tanx - 1 + C exp(-tanx) explanation

ntegral of ln(sin x) from 0 to pi/2
answer: (pi/2)ln(1/2) for explanation: integral of ln(sin x) from 0 to pi/2

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

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

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

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

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) explanation


find the equation of the tangent at t = pi/3 on the curve x=2cost , y = 3sint
answer: 3x + 2ysqrt(3) =12 explanation of finding equation of tangent






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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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indian parliament passing bills without debates

It is sad that something like 8 bills are passed in around 20 minutes (on 23/12/2008 ?) in the indian parliament without much of a debate.(and another 4 on the previous day ! ?)
The number of sittings per year of parliament also seems to be decreasing.

integral of [sin(lnx) ]/ x ² using substitution

.∫ [sin(lnx) ]/ x ² dx

use the substitution lnx= u or x = e^u

then use the std. formula for integral of {e^(ax)} sinbx given here

or

use integration by parts and then manipulation as in this example

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If x+18y+c=0 is a normal to y = 5x ² -12x + 1 , find the value of c

let x+18y+c=0 be a normal at (h,k) on the curve y = 5x ² -12x + 1

implies k = 5h ² - 12h+ 1 -------------(1)

y = 5x ² -12x + 1
diff. w.r.t.x dy/dx = 10x-12

slope of normal at (h,k) = -1 / (dy/dx) = -1 / (10h-12)
also slope of the given line x+18y+c=0 is -1/18

therefore -1 / (10h-12) = -1/18

solving 10h-12 = 18 or h =3
using equation(1) k = 5*3 ² - 12*3+ 1 = 10

therefore h=3 , k=10

x+18y+c=0 is a normal at (3,10) and should be satisfied by (3,10)

3 + 18*10 +c =0

c= -183






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integration with adustment instead of long division/partial fractions

integral of x^4 / (x^4 - 1)

first we have to do long division since degree of numerator = degree of denominator
then we have to proceed to partial fraction
but in this method those have been replaced by adjusting the numerator in terms of the
denominator and then dividing term by term

integration formulae



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trigonometry problem

show that 1/(1-cosx) =(cscx+cotx)cscx

start from the LHS, introduce the conjugate
simplify the denominator using identity then divide term by term

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trigonometry formulae

for some other examples
prove that (cotA+cosecA-1)/(cotA-cosecA+1)=cosecA+cotA -----> (problem on trigonometry )

Prove that (tanxsinx) / (tanx+sinx) = (tanx-sinx) / (tanx sinx) -----> (problem on trigonometry )

find exact value of tan[22½ °] or tan (pi / 8 ) -----> (problem on trigonometry )

find exact value of tan15° without using calculator --------value of tan15° without using calculator

conditional identity : If A+B+C=π,
prove that: sinA+sinB+sinC = cot(A/2).cot(B/2) [sinA+sinB-sinC] -----> (problem on conditional identity (trigonometry) )


graphs of trigonometric functions-----> (graphs of trigonometric functions )

prove that
{ tan((π/4)+x) - tan((π/4)-x)} / { tan((π/4)+x) + tan((π/4)-x) } = sin2x ---------trigonometry problem

prove that (cos²x - sin²x) / (cos²x + sinxcosx) = 1 - tanx -------------trigonometry problem

show that
tan(x)/[1-cot(x)] + cot(x)/[1-tan(x)]= 1+sec(x)csc(x) , x≠ n π /4 ---------trigonometry


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solving a quadratic equation by graphing

solve (x^2)- 2x - 8 = 0 by graphing

graph y = x² -2x -8 using a calculator or a piece of paper
the graph is shown below
see where the graph cuts the x-axis
in this problem x = -2 and x=4 are the roots


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index of problems

Integration

Integration formulae

Online guide to certain topics on integration for isc, cbse , plus two etc
PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTIONPAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

PAGE 6 INTEGRATION USING PARTIAL FRACTIONS
PAGE 7 INTEGRATION OF 1 / [ACOSX +BSINX +C]

integral of 1 / [9 + 7 cos² x]

integral of x^3 / sqrt(9 - x^2) using substitution

integral of tan ² x secx using substitution

integral of ln(x) using integration by parts

evaluate ∫ [1 / {x lnx}]dx

integral of e^(2x) / sqrt [1 + e^(2x)]

integral of ∫ { lnx / [x^4]} dx

integral of [sin(lnx) ]/ x ² using substitution

integration using partial fraction

problem on integration using trigonometry formulae integral of sin(5x)sin(8x)

integral of arc(cosx) using integration by parts

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Differential calculus

derivative of sqrt(3x) from first principles

if sqrt(x) + sqrt(y) = 1 find y" second derivative with implicit differentiation
equation of tangent to y =x^4 at the point where x=2

equation of tangent at (e,e) on y = x lnx
equation of tangent at (0,4) on x(y^3)+y+7(x^4) = 4

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Laplace transform

laplace transform of sin(t) from definition using integration

Quadratic equation

solving a quadratic equation by graphing

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Trigonometry

trigonometric identities

if (A-B) = pi/4 show that (1+tanA)(1+tanB) = 2tanA
trigonometry problem : show that 1/(1-cosx) =(cscx+cotx)cscx
trigonometry problem :show that [( tanx -sinx ) / (2tanx) ] = sin² (x/2)
finding sin( arctan(2) )
exact vaue of sin(arctan(4/3)-arccos(12/13))
exact values of certain trigonometric ratio

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Logarithm

problem on changing the base of a log

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Complex

integral using cauchy's integral formula

real and imaginary parts of ln(x+iy)

show that e^(ix) = cosx +i sinx using power series

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problem on sequences AP,GP ...

problem on sequences If Sn is the sum of the cubes of the first n natural numbers, find the sum of 1 / sqrt(Sn) where n varies from 1 to 2000

find the sum of all even natural numbers between 2 and 200 (both inclusive ) excluding the multiples of 3

sum of n terms of a geometric progression 1 + 2 + 4 + 8 + 16 + ...

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vectorfinding a perpendicular vector using the concept of dot product

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problem on theory of equationssolving a fourth degree equation given one of its complex roots

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construction of truth tables logic
to construct truth tables for [~ (p ^ q ) ]v (~r)

proof of boundedness law a+1 = 1 and idempotent law a+a=a in a boolean algebra
boundedness law in a boolean algebra
idempotent law in a boolean algebra

abstract algebra

groups
If G is an abelian group ,
let f : G -->G be defined by f(x) = xֿ¹ show that f is an automorphism on G problem on automorphism on abelian group

If G is a group and f :G -->G is defined as f(x) = a ֿ¹ x a for x in G and a is a fixed element of G
then show that f is an isomorphism of G onto G problem on automorphism in a group

analytical geometry

centre and radius of a circle

correlation and regression

derivation of regression line of y on x

limits for the correlation coefficient

changing recurring decimal to fraction

derivation of mean and variance of binomial distribution

derivation of mode of the binomial distribution

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approximation of an integral using binomial series

mclaurin's power series for ln(1+x)

word problem A man has 10 paise and 25 paise coins in his purse. The man has a total of 60 coins which amount to Rs. 8.25.
How many 10 paise and 25 paise coins does the man have in his purse? solution

The ratio of the incomes of two persons is 9:7 and the ratio of their expenditures is 4:3 . If each of them saves Rs.200/- per month, find their monthly incomes. solution

X can do a piece of work in 2 hours, while Y can do the same piece of work in 4 hours.What is the time taken if X and Y do the work together? solution

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problem on changing the base of a log

show that log_a [/x] = log_(1/a) [x]

use the definition of log

refer note on properties of logarithms
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integral of tan ² x secx using substitution

integral of tan ² x secx using substitution

regroup as tanx (secxtanx)

change tanx to sqrt(sec ² x -1)

use the substitution u = secx ; du =secxtanxdx

see below for more stepssome other examples
evaluate integral of { e^(1 /x^2)} / {x^3} -------------(integration by substitution)

evaluate the integral of [e^(-x) +1] ^2 ------------- integration after expansion using identity

integral of e^{x^(1/3)}-------------- e^{x^(1/3)}

integral of x / (x+1)^2-------------integration by adjustment

integrate 1/ (x^7 - x) -----------integration by adjusting

integral of ( ( 4 - x^2 )^( 3/2) ) with limits 0 to 1--------------integration by substitution

integral of [cosx (1+sin²x)] with 0 to π /6 as limits-------------integration by substitution


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cbse model question paper for 2009

cbse has released the model sample question paper for Xth and XII th exams
there are answers/ marking scheme also as far as the the math question paper for 12th is concerned

the questions were available on their website(contains flash i guess) cbse wesite

or in particular http://cbse.nic.in/curric~1/sample2009.htm
or http://cbse.nic.in/curric~1/sample.htm


a collection of some questions for a quick revision in some topics for 12th students (CBSE, ISC etc) is available here

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verifying eulers theorem for homogeneous functions ?

if f(x,y) = x² cos(x/y) + 3xy , show that f satisfies



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variation of parameter method

solve xy" - 4y' = x^4 by method of variation of parameter
make the coefficient of y" unity by dividing by x

find CF in the form Ay1 + By2 (using trial and error or some other method)
find the wronskian
and then find the PI as shown below

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time and work problem

X can do a piece of work in 2 hours, while Y can do the same piece of work in 4 hours.What is the time taken if X and Y do the work together?

converting to rate (quantity of work per unit time)

X can do (1/2) of the work in one hour
Y can do (1/4)th of the work in one hour

X+Y combined can do (1/2)+(1/4)=(3/4)th of the work in one hour

therefore X and Y combined will need 1 / (3/4)
or (4/3) hours to complete the work

that is 1 hour and 20 minutes

try the following page for some formulae time and work

other word problems with some explanation


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partial fractions

put 1/ [x(1 + x ² )] into partial fractions

other examples
put into partial fraction 1 / {(x^3) + 1 } -----> (partial fractions problem)

put 1/[ (x)(x-2)(2x-1)] into partial fractions ----------partial fraction


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integral of ln(x) using integration by parts

find the integral of ln(x)

make it into ln(x )*1 {remember ILATE rule} and then start off integration by parts
some other examples
integral of (e^(ax))cosbx using integration by parts ----------> (integration by parts)

.∫ (x²) sin x dx using integration by parts ----------> (integration by parts)

integral of { sqrt{1+x^2} } / {x^2} using integration by parts ----------> (integration by parts)

∫ arc(tan4x) dx ----------> (integration by parts)

other examples on problems in math --------math problems

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integration by substitution

evaluate integral of e^(2x) / sqrt [1 + e^(2x)]

some other examples
integral of ( ( 4 - x^2 )^( 3/2) ) with limits 0 to 1--------------integration by substitution

integral of e^{x^(1/3)}-------------- e^{x^(1/3)}

integral of ( ( 4 - x^2 )^( 3/2) ) with limits 0 to 1--------------integration by substitution

more problems on integration -----------problems on integration

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obesity in children

I recently read an article on how the arteries and heart problems of some obese children as young as 10 year old was similar to that of 40 to 50 year old people.

Basically it seems to be the fault of parents, that a lot of small children end up getting fat. Some parents do not seem to know the importance of proper diet and good exercise.

I guess the children of today spend more time watching tv, and playing video games or on the computer, than playing outside. Also some parents (single working moms due to divorces etc) do not seem to have enough time to cook healthy meals for their children and instead just buy fast food /prepacked/processed /cooked food /junk food high in fat and carbohydrate content from chain stores which need not always be healthy.

Some of the government subsidies for corn farmers lead to some components like high fructose corn syrup (HFC) showing up in many places in place of ordinary cane sugar.

Children are easily misled by marketing gimmicks into consuming carbonated sweetened drinks which are fattening.

In the old days children would walk or ride cycles to school. But nowadays parents are afraid of traffic and drive their children to school or send them via auto's or buses.

Overweight children are more likely to have problem as overweight adults.
Some of the problems like heart problems, diabetes (type 2 ?), breathing difficulties, high blood pressure , depression etc may be due to obesity.

what do girls carry around in their handbags ?

The common items seem to be

hair brush
hair spray
perfume
nail file
cell / mobile phone
makeup items / kit
money
coins ,currency notes
credit cards
keys
tissues
pictures of spouse/kids
pens / pencils
tampon / pad
hand sanitizer
bills
receipts

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problem on L'Hospital's rule

evaluate limit of (1+2x)^cotx as x-->0

it is of the form 1^infinity which is indeterminate
hence take log and adjust into 0/0 form before starting
to apply L'Hospitals rule

for more details refer L'Hospital's rule and indeterminate forms







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evaluate .∫ [1 / {x lnx}]dx

evaluate ∫ [1 / {x lnx}]dx

put u = lnx find du, substitute into the given integral and then
evaluate





another example
evaluate integral of { e^(1 /x^2)} / {x^3} substitution

integration formulae



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derivation of mean and variance of binomial distribution

derive the mean and variance of the binomial distribution
now to find the variance, we rewrite x^2 as x(x-1) +x before we start out



for explanation of mean of poisson distribution try the link mean of poisson distribution

for explanation of mean and variance of discrete uniform distribution try the link mean of the discrete uniform distribution



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solving a second order differential equation

solve y"-y'+y = 1

rewrite the equation as (D²-D+1)y = 1
replace D with m to get the auxilary equation
write the C.F. and then proceed to the particular integral (PI)

german 480 billion euro bailout

While a lot of noise was being made about the american bailout, the German parliament has hastily approved a 480 billion euro (approx. $650 billion) rescue plan for the financial sector, even though there is not as much of a mess as in the USA., (but the economy sure seems to be slowing down to a crawl)
Most of the money is for lending guarantees for banks and some of the money may be spent on buying up risky assets.
A major difference seems to be that banks participating in the plan will have to obey government conditions on executive pay, directions on lending, more government oversight etc.

At the same time the german government does not seem inclined to help bailout the beleagured carmakers in the auto sector.


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