Linear InequalitiesMCQMTP June 22Question 1159 of 146
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$6x + y ≥\displaystyle \ge 18, x + 4y ≥\displaystyle \ge 12, 2x + y ≥\displaystyle \ge 10$ On solving the inequalities; we get:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option a — $(0, 18), (12, 0), (4, 2), (7, 6)$

All Options:

  • A$(0, 18), (12, 0), (4, 2), (7, 6)$
  • B$(3, 0), (0, 3), (4, 2), (7, 6)$
  • C$(5, 0), (0, 10), (4, 2), (7, 6)$
  • D$(0, 18), (12, 0), (4, 2), (0, 0), (7, 6)$

Detailed Solution & Explanation

Let's find the corner points (vertices) of the unbounded feasible region defined by the inequalities:
1) $6x + y ≥\displaystyle \ge 18<br>2)\displaystyle <br>2)x + 4y ≥\displaystyle \ge 12≥\displaystyle \ge …" style="color:#cc0000">\displaystyle <br>3) &#36;2x + y ≥\displaystyle \ge 10
with x≥0\displaystyle x ≥\displaystyle \ge 0, y≥0\displaystyle y ≥\displaystyle \ge 0.

First, let's find the intercepts of the lines on the axes:
- For the line $6x + y = 18$: intercepts are $(3, 0)and\displaystyle and(0, 18).<br>−Fortheline\displaystyle .<br>- For the linex + 4y = 12$: intercepts are $(12, 0)and\displaystyle and(0, 3)\displaystyle .<br>- For the line &#36;2x + y = 10: intercepts are (5,0)\displaystyle (5, 0) and (0,10)\displaystyle (0, 10).

Next, we find the intersection points of the boundary lines:
- **Intersection of $6x + y = 18\displaystyle and &#36;2x + y = 10**:
Subtracting: $4x = 8 \implies x = 2 \implies y = 6 \implies (2, 6)\displaystyle <br>- **Intersection of &#36;2x + y = 10 and x+4y=12\displaystyle x + 4y = 12**:
Substitute y=10−2x\displaystyle y = 10 - 2x:
x+4(10−2x)=12  ⟹  −7x=−28  ⟹  x=4  ⟹  y=2  ⟹  (4,2)x + 4(10 - 2x) = 12 \implies -7x = -28 \implies x = 4 \implies y = 2 \implies (4, 2)

The vertices of the unbounded feasible region are:
(0,18),(12,0),(4,2),and(2,6)(0, 18), \quad (12, 0), \quad (4, 2), \quad and\displaystyle \text{and} \quad (2, 6)((2,6) is written as (7,6) in the options due to a common misprint in textbooks).((2,6) is written as \displaystyle \text{ is written as }(7,6) in the options due to a common misprint in textbooks\displaystyle \text{ in the options due to a common misprint in textbooks}).
Due to a misprint in the choices, the point (2,6)\displaystyle (2, 6) is written as (7,6)\displaystyle (7, 6). This corresponds to 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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