Verify that ` x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]`

Find the vaoue of each of the folllowing polynomials for the indicated value of variables: (i) p(x) = 4x ^(2) - 3x + 7 at x =1 (ii) q (y) = 2y ^(3) - 4y + sqrt11 at y =1 (iii) r (t) = 4t ^(4) + 3t ^(3) -t ^(2) + 6 at t =p, t in R (iv) s (z) = z ^(3) -1 at z =1 (v) p (x) = 3x ^(2) + 5x -7 at x =1 (vii) q (z) = 5z ^(3) - 4z + sqrt2 at z =2

using properties of determinant prove that {:[( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(2)),( z,z^(2) , 1+pz^(2)) ]:} =( 1+pxyz ) ( x-y) ( y-z ) (z-x) , where p is any scalar .

x+y+z=0 Show that x^(3)+y^(2)+z^(3)=3xyz

If x_(1) = 3y_(1) + 2y_(2) -y_(3), " " y_(1)=z_(1) - z_(2) + z_(3) x_(2) = -y_(1) + 4y_(2) + 5y_(3), y_(2)= z_(2) + 3z_(3) x_( 3)= y_(1) -y_(2) + 3y_(3)," " y_(3) = 2z_(1) + z_(2) express x_(1), x_(2), x_(3) in terms of z_(1) ,z_(2),z_(3) .

Solve the system of equations (2)/( x) + ( 3)/( y) +(10 )/( z) = 4 (4)/( x) -(6)/( y) +(5)/(z) =1 (6)/(x) +(9)/( y) -(20 )/( z) = 2

If (x(y+z-x))/log x = (y(z+x-y))/log y = (z(x+y-z))/log z ," prove that "x^(y) y^(x) = z^(y) y^(z) = x^(z) z^(x) .

Show that |(x,x^(2),yz),(y,y^(2),zx),(z,z^(2),xy)|=(x-y)(y-z)(z-x)(xy+yz+zx)

If x^(18)= y^(21) = z^(28) , then 3, 3log_(y)x, 3log_(z)y, 7 log_(x)z are in :

The shortest distance between the lines L_(1) : (x+1)/3 = (y+2)/1 =(z+1)/2 L_(2) = (x-2)/1 = (y+2)/2 = (z-3)/3 is

If cos ^(-1) x+cos ^(-1) y+cos ^(-1) z=3 pi then x y+y z+z x=