Question
The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x| 
k, |y| k, |x – y| k ; is-
- (k + 1)3 – k3
- (k + 2)3 – (k +1)3
- (k2 + 1)
- None of these
The correct answer is: (k + 1)3 – k3
|x| k 
–k x k ….(1)
& |y| k –k y k ….(2)
& |x – y| k |y – x| k ….(3)
– k y – x k x – k y x + k
Number of points having integral coordinates
= (2k + 1)2 – 2[k + (k – 1) + …. + 2 + 1]
= (3k2 + 3k + 1).
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