Find the sum of n terms of the series .

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA =12 cm, QC = QD = 3 cm, then find PC + PD.

If m^th term of an A.P. is 1/n and n^th term is 1/m, then find the sum of its first mn terms.

If the equation (1 + m²)x² + 2mcx + c² - a² = 0 has equal roots then show that c² = a²(1 +m²).

The (3/4)^th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.