This is a CSIR-NET June 2016 Part A question of Life Sciences, Physical Sciences.
Question discussed: Two iron spheres of radii 12 cm and 1 cm are melted and fused. Two new spheres are made without any loss of iron. Their possible radii could be:
(1) 9 and 4 cm (2) 9 and 10 cm (3) 8 and 5 cm (4) 2 and 11 cm
History: 1729 is the Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Srinivasa Ramanujan. In Hardy’s words: “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” The two different ways are: 1729 = 1cube + 12cube = 9cube + 10cube
P.S. If you know about the Ramanujan Number, the step of finding the values of x and y using trial and error can be skipped as these are the only two different ways the number 1729 can be expressed as the sum of two cubes.
Discussed by – Christy Varghese