Linear InequalitiesMCQMTP Sep 24 Series IQuestion 1173 of 146
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The shaded region represents:

Options

A3x+2y≤24,x+2y≥16,x+y≤10,x≥0,y≥0\displaystyle 3x + 2y \le 24, x + 2y \ge 16, x + y \le 10, x \ge 0, y \ge 0
B3x+2y<24,x+2y<16,x+y>10,x≥0,y≥0\displaystyle 3x + 2y < 24, x + 2y < 16, x + y > 10, x \ge 0, y \ge 0
C3x+2y<24,x+2y<16,x+y<10,x>0,y>0\displaystyle 3x + 2y < 24, x + 2y < 16, x + y < 10, x > 0, y > 0
DNone of these
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Correct Answer

✅ Option c — 3x+2y<24,x+2y<16,x+y<10,x>0,y>0\displaystyle 3x + 2y < 24, x + 2y < 16, x + y < 10, x > 0, y > 0

All Options:

  • A3x+2y≤24,x+2y≥16,x+y≤10,x≥0,y≥0\displaystyle 3x + 2y \le 24, x + 2y \ge 16, x + y \le 10, x \ge 0, y \ge 0
  • B3x+2y<24,x+2y<16,x+y>10,x≥0,y≥0\displaystyle 3x + 2y < 24, x + 2y < 16, x + y > 10, x \ge 0, y \ge 0
  • C3x+2y<24,x+2y<16,x+y<10,x>0,y>0\displaystyle 3x + 2y < 24, x + 2y < 16, x + y < 10, x > 0, y > 0
  • DNone of these

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Detailed Solution & Explanation

Let's mathematically analyze the system of inequalities in Option C:
1) 3x+2y<24\displaystyle 3x + 2y < 24
2) x+2y<16\displaystyle x + 2y < 16
3) x+y<10\displaystyle x + y < 10
4) x>0,y>0\displaystyle x > 0, y > 0

**Step 1: Intercept Calculations**
- Boundary line 1 (3x+2y=24\displaystyle 3x + 2y = 24): intercepts (8,0)\displaystyle (8,0) and (0,12)\displaystyle (0,12).
- Boundary line 2 (x+2y=16\displaystyle x + 2y = 16): intercepts (16,0)\displaystyle (16,0) and (0,8)\displaystyle (0,8).
- Boundary line 3 (x+y=10\displaystyle x + y = 10): intercepts (10,0)\displaystyle (10,0) and (0,10)\displaystyle (0,10).

**Step 2: Origin Testing**
- For line 1: Substitute (0,0)\displaystyle (0,0) gives 0<24\displaystyle 0 < 24 (True). Shaded region lies below line 1.
- For line 2: Substitute (0,0)\displaystyle (0,0) gives 0<16\displaystyle 0 < 16 (True). Shaded region lies below line 2.
- For line 3: Substitute (0,0)\displaystyle (0,0) gives 0<10\displaystyle 0 < 10 (True). Shaded region lies below line 3.

**Step 3: Feasible Region Geometry**
The intersection of these regions forms a bounded polygon (common area) in the first quadrant, lying below all three boundary lines. This perfectly matches the shaded region of the LPP graph.

Hence, **Option C** is the correct answer.

About This Chapter: Linear Inequalities

Paper

Paper 3: Quantitative Aptitude

Weightage

1-3 Marks

Key Topics

Linear Inequalities in one & two variables

This chapter covers Linear Inequalities in one & two variables and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 1-3 Marks weightage. Focus on understanding core concepts rather than memorizing.

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