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} – k^{3}
- (k + 2)
^{3} – (k +1)^{3}
- (k
^{2} + 1)
- None of these

^{3}– k^{3}^{3}– (k +1)^{3}^{2}+ 1)## The correct answer is: (k + 1)^{3} – k^{3}

### |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]

= (3k^{2} + 3k + 1).

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