KVPY SA Exam 2016 Question Paper

Question : 12 of 80

Marks: +1, -0

Consider a cuboid all of whose edges are integers and whose base is a square. Suppose the sum of all its edges is numerically equal to the sum of the areas of all its six faces. Then the sum of all its edges is

Validate

Solution:

Let sides are a, a, h
So, 4a + 4h + 4a =

2(a^2 + ah + ah)

2a + 2h + 2a

= a^2 + 2ah

⇒ a^2- 4a = 2h (1 - a)

⇒ (a^2- 1) + 1 - 4 (a - 1) - 4

= 2h (1- a)

⇒ (a - 1) (a + 1) -4 (a - 1) - 3

= 2h (1 - a)

⇒ 2h = 3/{a-1} + 4 - (a + 1)

So a = 2 & h = 2 are the only integral solution (a & h are positive integers)

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