Question

Match the LIST-I with LIST-II

	LIST-I Inequality		LIST-II Solution Space
A.	\(2x+3y \geq 6\)	I.	half plane contains the origin, also points on the line \(2x+3y=6\)
B.	\(2x+3y \leq 6\)	II.	half plane contains the origin but not points on the line \(2x+3y=6\)
C.	\(2x+3y > 6\)	III.	half plane not contains the origin but contains points on the line \(2x+3y=6\)
D.	\(2x+3y < 6\)	IV.	half plane neither contains the origin nor contains points on the line \(2x+3y=6\)

Choose the correct answer from the options given below:

• A. A-II, B-III, C-I, D-IV
• B. A-III, B-II, C-I, D-IV
• C. A-III, B-I, C-IV, D-II
• D. A-II, B-IV, C-I, D-III

Hint:

Test the origin in each inequality. A strict sign leaves out the line, a non-strict sign keeps it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A linear inequality in two variables shades one side (a half plane) of its boundary line. If the sign is \(\geq\) or \(\leq\), the boundary line is part of the solution. If the sign is \(>\) or \(<\), the line is left out.
To decide which side is shaded, test the origin \((0,0)\) in the inequality.

Step 2: Key Formula or Approach:
The boundary line is \(2x+3y=6\). At the origin, the left side is \(2(0)+3(0)=0\). So we only need to check whether \(0 \geq 6\), \(0 \leq 6\), \(0 > 6\) or \(0 < 6\) is true.

Step 3: Check A: \(2x+3y \geq 6\).
At the origin, \(0 \geq 6\) is false. So the origin is not in the half plane. The sign includes equality, so points on the line are included. This matches III.

Step 4: Check B: \(2x+3y \leq 6\).
At the origin, \(0 \leq 6\) is true. So the origin is in the half plane. The line is included because of the equality sign. This matches I.

Step 5: Check C: \(2x+3y > 6\).
At the origin, \(0 > 6\) is false, so the origin is not in the half plane. The sign is strict, so the line is excluded. This matches IV.

Step 6: Check D: \(2x+3y < 6\).
At the origin, \(0 < 6\) is true, so the origin is in the half plane. The sign is strict, so the line is excluded. This matches II.

Step 7: Compare with the options.
We got A-III, B-I, C-IV, D-II. Option 1 has A-II, so it is wrong. Option 2 has B-II, so it is wrong. Option 4 has A-II, so it is wrong. Only option 3 matches.

Final Answer:
The correct matching is A-III, B-I, C-IV, D-II, which is option 3. \[ \boxed{\text{Option 3: A-III, B-I, C-IV, D-II}} \]