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JEE Main202330 Jan 2023Evening ShiftMathematicsBinomial TheoremActual

Let x = 8 3 + 13 13 and y = 7 2 + 9 9 . If t denotes the greatest integer ≤ t , then

Options

  1. Ax + y is even
  2. Bx is odd but y is even
  3. Cx is even but y is odd
  4. Dx and y are both odd

Correct answer

A. x + y is even

Step-by-step solution

We know the binomial expansion of a + b n is, a + b n = C 0 n a n b 0 + C 1 n a n - 1 b 1 + C 2 n a n - 2 b 2 + . . . . . . + C n n a 0 b n Using the above expansion, x = 8 3 + 13 13 = C 0 13 8 3 13 + C 1 13 8 3 12 13 1 + … Let x ' = 8 3 - 13 13 = C 0 13 8 3 13 - C 1 13 8 3 12 13 1 + … So, x - x ' = 2 C 1 13 · 8 3 12 13 1 + C 3 13 8 3 10 · 13 3 … Therefore, x - x ' is an even integer, hence x is even Now, y = 7 2 + 9 9 = C 0 9 7 2 9 + C 1 9 7 2 8 9 1 + C 2 9 7 2 7 9 2 + . . . . N

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