Given x = cy+bz, y = az + cx, z = b x + ay where, x,y,z are not all zero prove that `a^2+b^2 +c^2 + 2 a b c = 1.`

If x,y,z are in G.P. and a^x= b^y=c^z , prove that log_b a. log_b c=1 .

Given x=cy+bz,y=az+cx and z=bx+ay, then prove a^(2) +b^(2) +c^(2) +2abc =1.

If a^x=b^y=c^z and a, b, c are in G.P. then x,y,z are in :

If a , b , c are non-zero real numbers and if the system of equations (a-1)x=y+z , (b-1)y=z+x , (c-1)z=x+y has a non-trivial solution, then prove that a b+b c+c a=a b c

Statement -1 : If a transversal cuts the sides OL, OM and diagonal ON of a parallelogram at A, B, C respectively, then (OL)/(OA) + (OM)/(OB) =(ON)/(OC) Statement -2 : Three points with position vectors veca , vec b , vec c are collinear iff there exist scalars x, y, z not all zero such that x vec a + y vec b +z vec c = vec 0, " where " x +y + z=0.

If x, y, z are not all zero & if ax + by + cz=0, bx+ cy + az=0 & cx + ay + bz = 0 , then prove that x: y : z = 1 : 1 : 1 OR 1 :omega:omega^2 OR 1:omega^2:omega , where omega is one ofthe complex cube root of unity.

If the lines (a-b-c) x + 2ay + 2a = 0 , 2bx + ( b- c - a) y + 2b = 0 and (2c+1) x + 2cy + c - a - b = 0 are concurrent , then prove that either a+b+ c = 0 or (a+b+c)^(2) + 2a = 0

If the planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a line, then the value of a^2+b^2+c^2+2abc is

If cos^(-1) x + cos^(-1) y + cos^(-1) z = pi , prove that x^(2) + y^(2) + z^(2) + 2xyz = 1

Factorise the following expressions : a x^2 y + b x y^2 + c x y z