Question

A student of class XII studying Mathematics comes across an incomplete question in a book.

Maximise Z = 3x + 2y + 1

Subject to the constraints x ≥ 0, y ≥ 0, 3x + 4y ≤ 12,

He/She notices the below shown graph for the said LPP problem and finds that a constraint is missing in it:

Help him/her choose the required constraint from the graph.

The missing constraint is:

विकल्प

• x + 2y ≤ 2

• 2x + y ≥ 2

• 2x + y ≤ 2

• x + 2y ≥ 2

Answer 1

2x + y ≥ 2

Explanation:

Given, Maximise (Z) = 3x + 2y + 1

Subject to the constraints:

x ≥ 0, y ≥ 0,

3x + 4y ≤ 12

First, we plot 3x + 4y = 12

Putting x = 0, y = 3

So, the point is (0, 3).

Putting y = 0, x = 4

Therefore, the point is (4, 0).

Checking these two points in the graph:

Now, the other line passes through (1, 0) and (0, 2).

Equation of that line is:

`(y - 0)/(x - 1) = (2 - 0)/(0 - 1)`

`y/(x - 1) = 2/-1`

`y/(x - 1) = -2`

y = −2(x − 1)

y = −2x + 2

2x + y = 2

Now, it could be 2x + y ≥ 2 or 2x + y ≤ 2.

Since (2, 0) is in the shaded region, it should satisfy the constraint.

Putting (2, 0) in 2x + y ≥ 2:

2(2) + 0 ≥ 2

4 ≥ 2

This is true.

Since the constraint is satisfied.

Hence, 2x + y ≥ 2 is our constraint.

So, the correct answer is 2x + y ≥ 2.