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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