If `((x^(-1)y^2)/(x^3y^(-2)))^1.\ ((x^6\ y^(-3))/(x^2\ y^3))^(1/2)=x^a y^b ,` prove that `a+b=-1` , where `x\ a n d\ y` are different positive primes.

6/(x+y)=7/(x-y)+3 , 1/(2(x+y))=1/(3(x-y))

For any a ,\ b ,\ x ,\ y >0 , prove that : 2/3tan^(-1)((3a b^2-a^3)/(b^3-3a^2b))+2/3tan^(-1)((3x y^2-x^3)/(y^3-3x^2y))=tan^(-1)(2alphabeta)/(alpha^2-beta^2) , where alpha=-a x+b y beta=b x+a ydot

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)

If x!=y!=z and |x x^2 1+x^3 y y^2 1+y^3 z z^2 1+z^3|=0 , then prove that x y z=-1 .

Prove that the lines 2x-3y+1=0,\ x+y=3,\ 2x-3y=2\ a n d\ x+y=4 form a parallelogram.

If (x+y)^3-\ (x-y)^3-6y\ (x^2-y^2)=\ k y^3, then k= (a)1 (b) 2 (c) 4 (d) 8

Show that: ((a^(x+1))/(a^(y+1)))^(x+y)\ ((a^(y+2))/(a^(z+2)))^(y+z)\ ((a^(z+3))/(a^(x+3)))^(z+x)=1

Factorize : (1) (x+y)(2x+3y)-(x+y)(x+1) (2) (x+y)(2a+b)-(3x-2y)(2a+b)

If |(x, x^2, x^3 +1), (y, y^2, y^3+1), (z, z^2, z^3+1)| = 0 and x ,y and z are not equal to any other, prove that, xyz = -1

If y=(log)_e(x/a)^x ,then x^3y_2= (a) (xy_1-y)^2 (b) ((y-x y_1)/(y_1))^2 (c) none of these