If $$a, b, c$$ are non-zero real numbers and if the system of equations $$(a-1) x-y-z=0, -x+(b-1) y-z=0,-x-y+(c-1) z=0$$ has a non-trivial solution, then $$a b+b c+c a$$ equals to

If $$f(x)=x^2-2 x+1$$ and $$f \circ g(x)=x^2+2 x+1$$, then $$g(x)$$ is equal to

If $$z_1$$ and $$z_2$$ be nth root of unity which subtend a right angled at the origin. Then, $$n$$ must be of the form

A tower $$T_1$$ of the height $$60 \mathrm{~m}$$ is located exactly opposite to a tower $$T_2$$ of height $$80 \mathrm{~m}$$ on a straight road. From the top of $$T_1$$, if the angle of depression of the foot of $$T_2$$ is twice the angle of elevation of the top of $$T_2$$, then the width (in $$\mathrm{m}$$) of the road between the feet of the towers $$T_1$$ and $$T_2$$ is