Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2x^2 – 1$ as shown in the following diagram:
[Diagram shows a region R between two parabolas]
Two distinct lines are drawn such that each of these lines partitions the region R into at least two parts. If ‘n’ is the total number of regions generated by these lines, then :
[amp_mcq option1=”‘n’ can be 4 but not 3″ option2=”‘n’ can be 4 but not 5″ option3=”‘n’ can be 5 but not 6″ option4=”‘n’ can be 6″ correct=”option2″]
Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R).
Let’s consider the number of regions created by 2 lines inside a bounded region R:
1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R.
2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R.
3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions.
The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves.
Thus, ‘n’ can be 3 or 4.
Evaluating the options:
A) ‘n’ can be 4 but not 3 (False, n can be 3)
B) ‘n’ can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6)
C) ‘n’ can be 5 but not 6 (False, n cannot be 5)
D) ‘n’ can be 6 (False, n cannot be 6)