Linear Programming

Previous Years' CBSE Board Questions

12.2 Linear Programming Problem and its Mathematical Formulation

MCO

1 Which of the following points satisfies both the inequations <math>2x + y \le 10</math> and <math>x + 2y \ge 8</math>?

1. (-2, 4) (b) (3, 2) (c) (-5, 6) (d) (4, 2) (2023) U

2 The solution set of the inequation <math>3x + 5y < 7</math> is

1. whole xy-plane except the points lying on the line <math>3x + 5y = 7</math>.
2. whole xy-plane along with the points lying on the line <math>3x + 5y = 7</math>.
3. open half plane containing the origin except the points of line <math>3x + 5y = 7</math>.
4. open half plane not containing the origin. (2023)

YA (4, 10)

1. If the corner points of the feasible region of an LPP are (0, 3), (3, 2) and (0, 5), then the minimum value of <math>Z = 11x + 7y</math> is
2. 21 (b) 33 (c) 14 (d) 35 (Term I, 2021-22) EV

(0, 8) (6, 8)

(6, 5)

1. The number of solutions of the system of inequations <math>x + 2y \le 3</math>, <math>3x + 4y \ge 12</math>, <math>x \ge 0</math>, <math>y \ge 1</math> is
2. 0 (b) 2 (c) finite (d) infinite (Term I, 2021-22) [ U ]

(0, 0) (5, 0)

1. The maximum value of <math>Z = 3x + 4y</math> subject to the constraints <math>x \ge 0</math>, <math>y \ge 0</math> and <math>x + y \le 1</math> is
2. 7 (b) 4 (c) 3 (d) 10 (Term I, 2021-22) Ev

(b) (0, 8)

(d) (4, 10)

(NCERT Exemplar, 2020) (Ap)

1. The feasible region of an LPP is given in the following figure

(0.104)

(0, 38)

0 (52,0)\(76,0)

Then, the constraints of the LPP are <math>x \ge 0</math>, <math>y \ge 0</math> and

1. <math>2x + y \le 52</math> and <math>x + 2y \le 76</math>
2. <math>2x + y \le 104</math> and <math>x + 2y \le 76</math>
3. <math>x + 2y \le 104</math> and <math>2x + y \le 76</math>
4. <math>x + 2y \le 104</math> and <math>2x + y \le 38</math> (Term I, 2021-22) (Ap

SAII (3 marks)

1. If the minimum value of an objective function <math>Z = ax + by</math> occurs at two points (3, 4) and (4, 3) then
2. <math>a+b=0</math> (b) a = b
3. 3a = b (d) <math>a = 3b</math> (Term I, 2021-22) III

1. For the following LPP, maximise <math>Z = 3x + 4y</math> subject to constraints <math>x - y \ge -1</math>, <math>x \le 3</math>, <math>x \ge 0</math>, <math>y \ge 0</math>

the maximum value is

1. 0 (b) 4 (c) 25 (d) 30 (Term I, 2021-22) Ap

The corner points of the feasible region determined

by the system of linear inequalities are (0, 0), (4, 0),

(2, 4) and (0, 5). If the maximum value of z = ax + by,

where <math>a, b > 0</math> occurs at both (2, 4) and (4, 0), then

1. <math>a = 2b</math> (b) <math>2a = b</math>
2. a = b (d) <math>3a = b</math> (2020)

In an LPP, if the objective function <math>z = ax + by</math> has the

same maximum value on two corner points of the

feasible region, then the number of points at which

Z_max occurs is

1. -0 (b) 2 (c) finite (d) infinite (2020) | U

The feasible region for an LPP is shown below:

Let <math>z = 3x - 4y</math> be the objective function. Minimum

of z occurs at

1. (0,0)
2. (5,0)

The graph of the inequality <math>2x + 3y > 6</math> is

(a) half plane that contains the origin

(b) half plane that neither contains the origin nor

the points of the line <math>2x + 3y = 6</math>.

1. (d) entire XOY-plane. (2020) 🕕

The objective function of an LPP is (d)

1. a constant
2. (c) an inequality

Solve the following linear programming problem

graphically: Maximise <math>z = -3x - 5y</math>

Subject to the constraints

<math>-2x + y \le 4</math> <math>x+y \ge 3</math> <math>x - 2y \le 2</math> <math>x \ge 0, y \ge 0</math>. (2023) [[V]

<math>A \perp</math> (4 marks)

1. Solve the following linear programming problem graphically: Maximize <math>z = 3x + 9y</math> Subject to constraints <math>x + 3y \le 60</math> <math>x+y \ge 10</math> <math>x \leq y</math> <math>x, y \ge 0</math> (2021) Ev

The corner points of the feasible region determined by the system of linear inequations are as shown below:

4 C(3, 4) 3 D(0, 2) B(5, 2)

1 (4,0) 0

Answer each of the following:

1. Let <math>z = 13x - 15y</math> be the objective function. Find the maximum and minimum values of z and also the corresponding points at which the maximum and minimum values occur.
2. Let z = kx + y be the objective function. Find k, if the value of z at A is same as the value of z at B.

(2021)

1. Solve the following LPP graphically: Minimize <math>z = 5x + 7y</math> Subject to the constraints <math>2x + y \ge 8, x + 2y \ge 10, x, y \ge 0</math>
2. Solve the following LPP graphically: Minimise <math>Z = 5x + 10y</math> Subject to constraints <math>x + 2y \le 120</math>, <math>x + y \ge 60</math>, <math>x - 2y \ge 0</math> and <math>x, y \ge 0</math> (NCERT Exemplar, Delhi 2017) [FV]
3. Maximise <math>Z = x + 2y</math> Subject to the constraints: <math>x + 2y \ge 100, 2x - y < 0, 2x + y \le 200, x, y \ge 0</math> Solve the above LPP graphically. (NCERT, AI 2017) EV

(5 / 6 marks)

1. Solve the following linear programming problem graphically. Maximize: <math>P = 70x + 40y</math> Subject to : <math>3x + 2y \le 9</math>, <math>3x + y \le 9</math>, <math>x \ge 0</math>, <math>y \ge 0</math>. (2023) Ev
2. Solve the following linear programming problem graphically. Minimize: <math>Z = 60x + 80y</math> Subject to constraints: <math>3x + 4y \ge 8</math> <math>5x + 2y \ge 11</math> <math>x, y \ge 0</math> (2023) Cr
3. Find graphically, the maximum value of <math>z = 2x + 5y</math>, subject to constraints given below: <math>2x + 4y \le 8</math>, <math>3x + y \le 6</math>, <math>x + y \le 4</math>; <math>x \ge 0</math>, <math>y \ge 0</math> (Delhi 2015) (Ev)
4. Maximise <math>z = 8x + 9y</math> subject to the constraints given below: <math>2x + 3y \le 6</math>, <math>3x - 2y \le 6</math>, <math>y \le 1</math>; <math>x, y \ge 0</math> (Foreign 2015) Ey

(2020) Ev

CBSE Sample Questions

12.2 Linear Programming Problem and its Mathematical Formulation

MCQ

1. The solution set of the inequality <math>3x + 5y < 4</math> is

A(0.4)

1. an open half-plane not containing the origin.

3

1. an open half-plane containing the origin.

1. the whole XY-plane not containing the line <math>3x + 5y = 4</math>.

2 B(0.6, 1.6) 1 C(3,0)

(2022-23) Ev -2 -1 0 2 5

1. a closed half plane containing the origin.
2. The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the

objective function <math>Z = 4x + 6y</math> occurs at