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

If sin^(-1) x + sin^(-1) y + sin^(-1) z = pi" , prove that " x sqrt(1-x^(2) ) + y sqrt(1 - y^(2)) + zsqrt( 1 - z^(2)) = 2 xyz .

The lines (x-1)/(3) = (y-1)/(-1) = (z+1)/0 and (x-4)/(2) = (y+0)/(0) = (z+1)/(3) are ......

Prove that the lines x=2 (y-1)/(3) = (z-2)/(1) and x = (y-1)/(1) = (z+1)/(3) are skew lines.

show that [((x+y)^2 , zx , zy),( zx, (z+y)^2 ,xy),(zy,xy,(z+x)^2)]=2xyz (x +y+z)^3

If Delta = abs{:(x+y+z^(2) , x^(2) + y+ z, x+y^(2) + z),(z^2, x^2, y^2),(x+y,y+z,x+z):} , (where ( x ne y ne z) x, y, z in R- {0} ) then Delta= ........

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

If a ,\ b ,\ c >0\ a n d\ x ,\ y ,\ z in R , then the determinant |\ \ (a^x+a^x)^2(a^x-a^(-x))^2 1(b^y+b^(-y))^2(b^y-b^(-y))^2 1(c^z+c^(-z))^2(c^z-c^(-z))^2 1| is equal to- a. a^x b^y c^x b. a^(-x)b^(-y)c^(-z)\ c. a^(2x)b^(2y)c^(2x) d. zero

The equation of the plane passing through the intersection of the planes 2x-5y = z = 3 and x+y+4z = 5 and parallel to the plane x+3y + 6z = 1 is x + 3y + 6z = k is..........