If `S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n` terms then prove that `(x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)]`.

If sqrt(1-x^6)+sqrt(1-y^6)=a(x^3-y^3), then prove that (dy)/(dx)=(x^2)/(y^2)sqrt((1-y^6)/(1-x^6))

The HCF of x^(2) - y^(2), x^(3) - y^(3), …... x^(n) - y^(n) , where n in N is ………. .

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(x-y)(y-z)(z-x)

Given (x_(1) + iy_(1)) (x_(2) + iy_(2))…(x_(n) + iy_(n)) = a + ib, show that (x_(1)^(2)+y_(1)^(2))(x_(2)^(2) + y_(2)^(2))(x_(3)^(2) + y_(3)^(2))…(x_(n)^(2) + y_(n)^(2)) = a^(2) + b^(2)

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(1)=2 .

If y=1+x/(1!)+(x^2)/(2!)+(x^3)/(3!)++(x^n)/(n !), show that (dy)/(dx)-y+(x^n)/(n !)=0.

(a) If y=cos(msin^(-1)x) prove that (1-x^(2))y_(3)-3xy_(2)+(m^(2)-1)y_(1)=0 (b) Find y'' if x^(4)+y^(4)=16 .

2x+3y=2,x-(y)/(2)=(1)/(2) .