isc syllabus change

Some change has happened in the syllabus for the isc 2010 12th exam for maths, compared to syllabus for last year. You can view the latest syllabus at cisce.org or in particular cisce.org/syllabus.jsp

The question papers of past years of the karnataka board +2 pre university course puc may be available at
pue.kar.nic.in or pue.kar.nic.in/questionbank.htm









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collection of problems in integration, differentiation, areas, volumes, trigonometry, matrices, limits

Calculus
problems on integration ------------------------ integration

problems on application of integration like
area, volume, arc length etc ------------------------integration application

problems on differential equations ------differential equation

problems on differentiation------------------------ differentiation

problems on limits ------------------limits

problems on matrices --------------matrices

problems on trigonometry ------trigonometry


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orthogonal trajectory of y(1+x ² ) = Cx

find the orthogonal trajectory of y(1+x ² ) = Cx
first form the differential equation of the given family
replace (dy/dx) with -1 / (dy/dx) to get the differential equation of the orthogonal trajectory
and solve the resulting equation

another example:orthogonal trajectory of y = (k/x)
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integral of (5x^3 + 8x^2 + 36x + 36) / (x^4 + 9x^2)

integral of (5x^3 + 8x^2 + 36x + 36) / (x^4 + 9x^2)
first do the partial fraction decomposition
refer partial fraction decomposition
then split the (x+4) /( x² +9) into two terms
the first one can be manipulated into u' / u form


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limits for the correlation coefficient

find limits for the correlation coefficient

show that -1 <= r <=1 or |r| <=1

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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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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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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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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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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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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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