If `s_(r )=x^(r )+y^(r )+z^(r )`, prove by considering the square of the determinant `|{:(1,1,1),(x,y,z),(x^(2),y^(2),z^(2)):}|` that `|{:(s_(0),s_(1),s_(2)),(s_(1),s_(2),s_(3)),(s_(2),s_(3),s_(4)):}|=(x-y)^(2)(y-z)^(2)(z-x)^(2)`

If Y=SX , Z=tX all the variables being differentiable functions of x and lower suffixes denote the derivative with respect to x and |{:(X,Y,Z),(X_(1),Y_(1),Z_(1)),(X_(2),Y_(2),Z_(2)):}| = |{:(S_(1),t_(1)),(S_(2),t_(2)):}| X^(n) , then n=

Solve for x, y and z, if {:((x+y,2),(1,0))={:((2,x-z),(2x-y,0)):}

The lines (x)/(1)=(y)/(2)=(z)/(3) and (x-1)/(-2)=(y-2)/(-4)=(3-z)/(6) are

If a^(x)=b^(y)=c^(z) and b^(2)=ac prove that (1)/(x)+(1)/(z)=(2)/(y)

Factorise : x(1+y^(2))(1+z^(2))+y(1+z^(2))(1+x^(2))+z(1+x^(2))(1+y^(2))+4xyz

If X, Y, Z are the angles of a DeltaXYZ then prove that cos^(2)""(X)/(2)+cos^(2)((Y+Z)/(2))=1

If x^(2)+y^(2)=z^(2) , then prove that 1/(log_(z-y)x) + 1/(log_(z+y)x) = 2 .

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

If x+y+z=x y z prove that (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)=(2x)/(1-x^2)(2y)/(1-y^2)(2z)/(1-z^2)dot

If S_(1), S_(2), S_(3) be respectively the sums of n, 2n and 3n terms of a G.P., prove that, S_(1)(S_(3) - S_(2)) = (S_(2) - S_(1))^(2) .