## CBSE Class 12 Maths Chapter 12 Linear Programming Important Questions

**Question 1. A Linear programming problem is as follows: Minimize Z = 2x + y**

**Subject to the constraints:**

- x ≥ 3, x ≤ 9, y ≥ 0,
- x-y ≥ 0, x + y ≤ 14

**The feasible region has:**

- 5 corner points including (0, 0) and (9, 5)
- 5 corner points including (7, 7) and (3, 3)
- 5 corner points including (14, 0) and (9, 0)
- 5 corner points including (3, 6) and (9, 5)

**Solution: **2. 5 corner points including (7, 7) and (3, 3)

Given; Z = 2x + y subject to the constraints

x ≥ 3, x ≤ 9, y ≥ 0, x – y ≥ 0, x + y ≤ 14

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Hence, the feasible region has 5 corner points including (7, 7) and (3, 3)

**Question 2. A Linear Programming Problem is as follows: Maximise/Minimise objective function Z = 2x – y + 5**

**Subject to the constraints:**

- 3x + 4y ≤ 60,
- x + 3y ≤ 30,
- x ≥ 0, y ≥ 0

**If the corner points of the feasible region are A(0, 10), B(12, 6), C(20, 0) and 0(0,0); then which of the following is true?**

**The maximum value of Z is 40****The minimum value of Z is -5****The difference between the maximum and minimum values of Z is 35****At two corner points, the value of Z is equal**

**Solution:** 2. The minimum value of Z is -5.

**Question 3. The corner points of the feasible region determined by a set of constraints (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0), and S(4, 1) and the objective function is Z = ax + 2by; where a, b > 0. The condition on ‘a’ and ‘b’ such that the maximum of Z occurs at Q and S is:**

- a-5b=0
- a-3b = 0
- a – 2b = 0
- a – 8b = 0

**Solution:** 4. a – 8b = 0

Given points arc P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and Z = ax + 2by

Since the maximum of Z occurs at Q and S;

⇒ 3a + 10b = 4a + 2b

⇒ 8b = a ⇒ a – 8b = 0

**Question 4. For an L.P.P.. the objective function is Z = 4x + 3y and the feasible region is determined by a set of constraints (linear inequations) as shown in the graph.**

**Which one of the following statements is true?**

- The maximum value of Z is at R
- The maximum value of Z is at Q
- The value of Z at R is less than the value of P
- The value of Z at Q is less than the value of R

**Solution:** 2. Maximum value of Z is at Q

Z = 4x + 3y

A maximum value of Z is at Q.

**Question 5. Solve the following problem graphically: Maximize Z = 3x + 9y**

**Subject to the constraints:**

- x + 3y ≤ 60,
- x + y ≥ 10,
- x ≤ y,
- x ≥ 0, y ≥ 0

**Solution:**

First of all, let us graph the feasible region of the system of linear inequalities given above. The feasible region ABCD is shown.

Note that the region is bounded. The coordinates of the comer points A, B, C, and D are (0, 10), (5, 5), (15, 15), and (0, 20) respectively.

The maximum value of Z is 180. which occurs at every point of the line segment joining the points C and D.