KVPY SB SX Exam 2011 Question Paper

Question : 90 of 120

Marks: +1, -0

The arithmetic mean and the geometric mean of two distinct 2-digit numbers x and y are two integers one of which can be obtained by reversing the digits of the other (in base 10 representation). Then x + y equals

Validate

Solution:

{x+y}/2=10a+b,

√{xy}=10b+a

(a, b ∈ N)

xy = (10b + a)^2

(x - y)^2 = 4 (11a + 11b) (9a - 9b)

= 4 . 11 . (a + b) . 9(a - b)

⇒ a + b = 11, a – b = 1
a = 6, b = 5

((x - y)^2

is perfect square of an integer)
x + y = 130

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