PART II MATHEMATICS Let a, b, c be positive integers such that \frac{a\sqrt2+b}{b\sqrt2+c} is a rational number, then which of the following is always an integer?

Correct Answer is : a2+b2+c2a+c−b

KVPY SA Exam 2012 Question Paper

Question : 61 of 80

Marks: +1, -0

PART II MATHEMATICS

Let a, b, c be positive integers such that

{a√2+b}/{b√2+c}

is a rational number, then which of the following is always an integer?

Solution:

({a√2+b}/{b√2+c}) ({b√2-c}/{b√2-c})

={2ab-√2ac+√2b^2-bc}/{2b^2-c^2}

={b(2a-c)+√2(b^2-ac)}/{2b^2-c^2}

Is rational when

b^2 = ac

i,e. a, b, c are in GP
Here given option (A),(B), (C) does not satisfy the criteria
But option (D) always satisfy

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