Wednesday, December 31, 2008

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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index of math problems

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