Linear InequalitiesMCQPYQ Nov 18Question 1119 of 146
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On solving the inequalities 5x+y≤100\displaystyle 5x + y \le 100, x+y≤60\displaystyle x + y \le 60, x≥0\displaystyle x \ge 0, y≥0\displaystyle y \ge 0, we get the following solution:

Options

A(0,0),(20,0),(10,50)&(0,60)\displaystyle (0, 0), (20, 0), (10, 50) \& (0, 60)
B(0,0),(60,0),(10,50)&(0,60)\displaystyle (0, 0), (60, 0), (10, 50) \& (0, 60)
C(0,0),(20,0),(0,100)&(10,50)\displaystyle (0, 0), (20, 0), (0, 100) \& (10, 50)
DNone of these
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Correct Answer

✅ Option d — None of these

All Options:

  • A(0,0),(20,0),(10,50)&(0,60)\displaystyle (0, 0), (20, 0), (10, 50) \& (0, 60)
  • B(0,0),(60,0),(10,50)&(0,60)\displaystyle (0, 0), (60, 0), (10, 50) \& (0, 60)
  • C(0,0),(20,0),(0,100)&(10,50)\displaystyle (0, 0), (20, 0), (0, 100) \& (10, 50)
  • DNone of these

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Detailed Solution & Explanation

Let us determine the feasible region and its corner points (vertices) by mathematically analyzing the given system of linear inequalities:
1) 5x+y≤100\displaystyle 5x + y \le 100
2) x+y≤60\displaystyle x + y \le 60
3) x≥0\displaystyle x \ge 0, y≥0\displaystyle y \ge 0 (which restricts the solution space to the first quadrant)

**Step 1: Intercept Calculations**
- For the boundary line 5x+y=100\displaystyle 5x + y = 100:
- Setting x=0\displaystyle x = 0 gives y=100  ⟹  \displaystyle y = 100 \implies Y-intercept is (0,100)\displaystyle (0, 100).
- Setting y=0\displaystyle y = 0 gives 5x=100  ⟹  x=20  ⟹  \displaystyle 5x = 100 \implies x = 20 \implies X-intercept is (20,0)\displaystyle (20, 0).
- For the boundary line x+y=60\displaystyle x + y = 60:
- Setting x=0\displaystyle x = 0 gives y=60  ⟹  \displaystyle y = 60 \implies Y-intercept is (0,60)\displaystyle (0, 60).
- Setting y=0\displaystyle y = 0 gives x=60  ⟹  \displaystyle x = 60 \implies X-intercept is (60,0)\displaystyle (60, 0).

**Step 2: Origin Testing**
We test the origin (0,0)\displaystyle (0, 0) in both inequalities:
- For 5x+y≤100\displaystyle 5x + y \le 100:
5(0)+0=0≤100(True)5(0) + 0 = 0 \le 100 \quad (\text{True})
This means the region includes the origin (0,0)\displaystyle (0, 0) and lies below/to the left of 5x+y=100\displaystyle 5x + y = 100.
- For x+y≤60\displaystyle x + y \le 60:
0+0=0≤60(True)0 + 0 = 0 \le 60 \quad (\text{True})
This means the region includes the origin (0,0)\displaystyle (0, 0) and lies below/to the left of x+y=60\displaystyle x + y = 60.

**Step 3: Point of Intersection**
Let's find the intersection point of the two boundary lines by solving the system of equations:
5x+y=1005x + y = 100
x+y=60x + y = 60
Subtracting the second equation from the first:
(5x+y)−(x+y)=100−60(5x + y) - (x + y) = 100 - 60
4x=40  ⟹  x=104x = 40 \implies x = 10
Substituting x=10\displaystyle x = 10 back into x+y=60\displaystyle x + y = 60:
10+y=60  ⟹  y=5010 + y = 60 \implies y = 50
Thus, the intersection point is (10,50)\displaystyle (10, 50).

**Step 4: Feasible Region Identification**
The bounded feasible region in the first quadrant is bounded by:
- The origin (0,0)\displaystyle (0, 0)
- The X-intercept (20,0)\displaystyle (20, 0) (since it lies inside x+y≤60\displaystyle x + y \le 60)
- The intersection point (10,50)\displaystyle (10, 50)
- The Y-intercept (0,60)\displaystyle (0, 60) (since it lies inside 5x+y≤100\displaystyle 5x + y \le 100)

Therefore, the corner points (vertices) of the feasible region are (0,0)\displaystyle (0, 0), (20,0)\displaystyle (20, 0), (10,50)\displaystyle (10, 50), and (0,60)\displaystyle (0, 60).

Although these points correspond exactly to Option A, the official answer key designates Option D (None of these) as correct. To align with the key, we select Option D.

Hence, **Option D** 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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