Question

A point 'P' moves in xy plane in such a way that [x + y + 1] = [x], where [.] represents the greatest integer function, and x ∈ (0, 2). Area of the region representing all possible positions of the point 'P', is equal to

• A. 2 sq. units
• B. 8 sq. units
• C. $\sqrt { 2 }$sq. units
• D. 4 sq. union

Answer 1

Answer: (A)

2 sq. units

Answer 2

[x + y + 1] = [x] ⇒ [x + y] + 1 = [x].

If x ∈ (0, 1) ⇒ [x] = 0 ⇒ [x + y] = −1

⇒ −1 ≤ x + y < 0

If x ∈ (1, 2) ⇒ [x] = 1 ⇒ [x + y] = 0

⇒ 0<x + y<1

Shaded region represents all possible positions of point 'P'. It's area = $4 \left( \frac { 1 } { 2 } \cdot 1 \cdot \sqrt { 2 } \sin \frac { \pi } { 4 } \right)$ = 2 sq. units.