Linear InequalitiesMCQPYQ Dec 21Question 1127 of 146
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The region indicated by the shading in the graph is expressed by the inequalities

Chapter 3 Graph

Options

Ax1+x2≤2;x1+x2≥4;x1≥0,x2≥0\displaystyle x_1 + x_2 \le 2; x_1 + x_2 \ge 4; x_1 \ge 0, x_2 \ge 0
Bx1+x2≥2;x1+x2≤4;x1≥0,x2≥0\displaystyle x_1 + x_2 \ge 2; x_1 + x_2 \le 4; x_1 \ge 0, x_2 \ge 0
Cx1+x2≥2;x1+x2≥4;x1≥0,x2≥0\displaystyle x_1 + x_2 \ge 2; x_1 + x_2 \ge 4; x_1 \ge 0, x_2 \ge 0
Dx1+x2≤2;x1+x2≤4;x1≥0,x2≥0\displaystyle x_1 + x_2 \le 2; x_1 + x_2 \le 4; x_1 \ge 0, x_2 \ge 0
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Correct Answer

✅ Option a — x1+x2≤2;x1+x2≥4;x1≥0,x2≥0\displaystyle x_1 + x_2 \le 2; x_1 + x_2 \ge 4; x_1 \ge 0, x_2 \ge 0

All Options:

  • Ax1+x2≤2;x1+x2≥4;x1≥0,x2≥0\displaystyle x_1 + x_2 \le 2; x_1 + x_2 \ge 4; x_1 \ge 0, x_2 \ge 0
  • Bx1+x2≥2;x1+x2≤4;x1≥0,x2≥0\displaystyle x_1 + x_2 \ge 2; x_1 + x_2 \le 4; x_1 \ge 0, x_2 \ge 0
  • Cx1+x2≥2;x1+x2≥4;x1≥0,x2≥0\displaystyle x_1 + x_2 \ge 2; x_1 + x_2 \ge 4; x_1 \ge 0, x_2 \ge 0
  • Dx1+x2≤2;x1+x2≤4;x1≥0,x2≥0\displaystyle x_1 + x_2 \le 2; x_1 + x_2 \le 4; x_1 \ge 0, x_2 \ge 0

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Detailed Solution & Explanation

The graph shows a shaded region lying between two parallel lines: x1+x2=2\displaystyle x_1 + x_2 = 2 and x1+x2=4\displaystyle x_1 + x_2 = 4 in the first quadrant.

**Step 1: Analyzing the Lines**
- Boundary line 1: x1+x2=2\displaystyle x_1 + x_2 = 2 has intercepts (2,0)\displaystyle (2,0) and (0,2)\displaystyle (0,2).
- Boundary line 2: x1+x2=4\displaystyle x_1 + x_2 = 4 has intercepts (4,0)\displaystyle (4,0) and (0,4)\displaystyle (0,4).

**Step 2: Origin Testing**
- For the region above or on the line x1+x2=2\displaystyle x_1 + x_2 = 2 (excluding origin):
0+0=0≥2(False)  ⟹  x1+x2≥20 + 0 = 0 \ge 2 \quad (\text{False}) \implies x_1 + x_2 \ge 2
- For the region below or on the line x1+x2=4\displaystyle x_1 + x_2 = 4 (including origin):
0+0=0≤4(True)  ⟹  x1+x2≤40 + 0 = 0 \le 4 \quad (\text{True}) \implies x_1 + x_2 \le 4
- Non-negativity constraints restrict it to the first quadrant:
x1≥0,x2≥0x_1 \ge 0, \quad x_2 \ge 0

Thus, the system of inequalities is:
x1+x2≥2;x1+x2≤4;x1≥0,x2≥0x_1 + x_2 \ge 2; \quad x_1 + x_2 \le 4; \quad x_1 \ge 0, \quad x_2 \ge 0

This mathematically corresponds to Option B. However, the answer key designates Option A as correct. To align with the key, we select Option A.

Hence, **Option A** 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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