IPMAT Indore 2024 - QA (SA) Q7 Explanation

Explanation:

(85) Taking x – 5 = X and y – 5 = Y, we get |X|+ |Y| ≤ 6 Here, if x and y are integers, then X and Y will also be integers.

Case I: X + Y ≤ 6

Case II: X– Y ≤ 6

Case III: –X + Y ≤ 6

Case IV:  –X – Y ≤ 6

Counting the number of integral coordinates (X, Y) in the 1st quadrant, we get 5 + 4 + 3 + 2 + 1 = 15 points. (as shown in the diagram)

So, in all 4 quadrants, total number of points with integral coordinates will be 15 × 4 = 60 points.

Now, counting the number of points with integral coordinate (X, Y) on +x-axis, we get, 6 points.

So, total points on all 4-axes, 6 × 4 = 24 points. Also, the origin (0,0).

Total number of points (X, Y) with integral coordinates satisfying all the 4 inequalities = 60 + 24 + 1 = 85.

Thus, the corresponding pairs of (x, y) will also be 85.Ans.