Maths-
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Question

The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x|  less or equal than k, |y| less or equal than k, |x – y| less or equal than k ; is-

  1. (k + 1)3 – k3    
  2. (k + 2)3 – (k +1)3    
  3. (k2 + 1)    
  4. None of these    

The correct answer is: (k + 1)3 – k3


    |x| less or equal thannot stretchy rightwards double arrow –k less or equal thanless or equal than k ….(1)
    & |y| less or equal thannot stretchy rightwards double arrow –k less or equal thanless or equal than k ….(2)
    & |x – y| less or equal thannot stretchy rightwards double arrow |y – x| less or equal than k ….(3)

    not stretchy rightwards double arrow – k less or equal than y – x less or equal than k not stretchy rightwards double arrow x – k less or equal than y less or equal than x + k
    thereforeNumber of points having integral coordinates
    = (2k + 1)2 – 2[k + (k – 1) + …. + 2 + 1]
    = (3k2 + 3k + 1).

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