(a) A balloon is observed simultaneously from three points $$A, B$$ and $$C$$ on a straight road directly beneath it. The angular elevation at $$B$$ is twice that at $$A$$ and the angular elevation at $$C$$ is thrice that at $$A$$. If the distance between $$A$$ and $$B$$ is a and the distance between $$B$$ and $$C$$ is $$b$$, find the height of the balloon in terms of $$a$$ and $$b$$.

(b) Find the area of the smaller part of a disc of radius $$10$$ cm, cut off by a chord $$AB$$ which subtends an angle of at the circumference.

If the cube roots of unity are $$1,\,\omega ,\,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0$$ are

If $$\alpha + \beta + \gamma = 2\pi ,$$ then

If x + iy = $$\sqrt {{{a + ib} \over {c + id}}} $$, prove that $${({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {d^2}}}$$.