Prove that the points `O(0,0,0), A(2.0,0), B(1,sqrt3,0) and C(1,1/sqrt3,(2sqrt2)/sqrt3)` are the vertices of a regular tetrahedron.,

If A(1,1), B(sqrt3+1,2) and C(sqrt3,sqrt3+2) are three vertices of a square, then the diagonal through B is

Prove that the tetrahedron with vertices at the points O(0,0,0),\ A(0,1,1), B(1,0,1)a n d\ C(1,1,0) is a regular one.

The orthocentre of a trianlge whose vertices are (0,0) , (sqrt3,0) and (0,sqrt6) is

lim_(xrarr0) (sqrt(x+1)+sqrt(x+4)-3)/(sqrt(x+2)-sqrt2)

Statement 1: Distance of point D( 1,0,-1) from the plane of points A( 1,-2,0) , B ( 3, 1,2) and C( -1,1,-1) is 8/sqrt229 Statement 2: volume of tetrahedron formed by the points A,B, C and D is sqrt229/ 2

If A(0, 0), B(1, 0) and C(1/2,sqrt(3)/2) then the centre of the circle for which the lines AB,BC, CA are tangents is

Prove that the lines sqrt(3)x+y=0,\ sqrt(3)y+x=0,\ sqrt(3)x+y=1\ a n d\ sqrt(3)y+x=1 form a rhombus.

Prove that tan [7(1)/(2)]^0 = (sqrt(3) - sqrt(2)) (sqrt(2) - 1) .

If in a "Delta"A B C ,\ A=(0,0),\ B=(3,3sqrt(3)), C-=(-3sqrt(3),3) then the vector of magnitude 2sqrt(2) units directed along A O ,\ w h e r e\ O is the circumcentre of A B C is a. (1-sqrt(3)) hat i+(1+sqrt(3)) hat j b. (1+sqrt(3)) hat i+(1-sqrt(3)) hat j c. (1+sqrt(3)) hat i+(sqrt(3)-1) hat j d. none of these

The perimeter of the triangle formed by the points (0,\ 0),\ (1,\ 0) and (0,\ 1) is 1+-sqrt(2) (b) sqrt(2)+1 (c) 3 (d) 2+sqrt(2)