Q45 / 90 · KVPY 2011 (SB/SX) · Mathematics

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-

  • A
    82
  • B
    116
  • Correct answer
    C
    130
  • D
    148

Solution

$\frac{x+y}{2}=10 a+b, \sqrt{x y}=10 b+a \quad(a, b \in N)$ $x y=(10 b+a)^{2}$ $(x-y)^{2}=4(11 a+11 b)(9 a-9 b)$ $=4 \cdot 11 \cdot(a+b) \cdot 9(a-b)$ $\Rightarrow a+b=11, a-b=1$ $a=6, b=5$ $\left((x-y)^{2}\right.$ is perfect square of an integer) $x+y=130$

Hence, the correct answer is option: 'C'.

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