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Linear Programming Maths - Linear Programming

Uttarakhand Board (UBSE) - STD 12 Science - Maths (Uttarakhand - English Medium). 429 questions in this chapter.

Question types

Linear Programming question types

429 questions across 9 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.

429
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9
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5
Question types
Sample Questions

Linear Programming questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1M.C.Q (1 Marks)1 Mark
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 a, b > 0 occurs at both (2, 4) and (4, 0), then:
  • ✓
    a = 2b
  • B
    2a = b
  • C
    a = b
  • D
    3a = b

Answer: A.

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Q 2M.C.Q (1 Marks)1 Mark
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 a, b > 0 occurs at both (2, 4) and (4, 0), then:
  • ✓
    a = 2b
  • B
    2a = b
  • C
    a = b
  • D
    3a = b

Answer: A.

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Q 3M.C.Q (1 Marks)1 Mark
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 a, b > 0 occurs at both (2, 4) and (4, 0), then:
  • ✓
    a = 2b
  • B
    2a = b
  • C
    a = b
  • D
    3a = b

Answer: A.

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Q 4M.C.Q (1 Marks)1 Mark
Maximize $Z = 11x + 8y,$ subject to $\text{x}\leq4,\text{y}\leq6,\text{x}\geq0,\text{y}\geq0.$
  • A
    $\text{44 at (4, 2)}$
  • ✓
    $\text{60 at (4, 2)}$
  • C
    $\text{62 at (4, 0)}$
  • D
    None of these

Answer: B.

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Q 5M.C.Q (1 Marks)1 Mark
The corner point of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in Column A and Column B.
Column A
Column B
Maximum of Z
325
  • A
    The quantity in column A is greater.
  • ✓
    The quantity in column B is greater.
  • C
    The two quantities are equal.
  • D
    The relationship cannot be determined On the basis of the information supplied.

Answer: B.

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Q 6Assertion (A) & Reason (B) MCQ1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : The linear programming problem,maximize $z = 2x + 3y$ subject to constraints $\text{x}+\text{y}\leq4,\text{x}\geq0,\text{y}\geq0$ It gives the maximum value of $Z$ as $8$ .
Reason : To obtain maximum value of $Z,$ we need to compare value of $Z$ at all the corner points of the feasible region.
  • A
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • C
    $A$ is true: $R$ is false.
  • ✓
    $A$ is false: $R$ is true.

Answer: D.

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Q 7Assertion (A) & Reason (B) MCQ1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ : Consider the linear programming problem. Maximise $Z = 4x + y $ Subject to constraints $\text{x}+\text{y}\leq50;\text{x}+\text{y}\geq100$ and $\text{x},\text{y}\geq0,$ Then, maximum value of $Z$ is $50.$
Reason $(R)$ : If the shaded region is bounded then maximum value of objective function can be determined.
  • A
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • C
    $A$ is true: $R$ is false.
  • ✓
    $A$ is false: $R$ is true.

Answer: D.

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Q 8Assertion (A) & Reason (B) MCQ1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following : Consider, the graph of constraints stated as linear inequalities as below: $5\text{x}+\text{y}\leq100,\text{x}+\text{y}\leq60,\text{x},\text{y}\geq0.$
Assertion $(A)$ : The points $(10, 50), (0, 60) , (10, 10)$ and $(20, 0)$ are feasible solutions.
Reason $(R)$ : Points within and on the boundary of the feasible region represent feasible solutions of the constraints.
  • ✓
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • C
    $A$ is true: $R$ is false.
  • D
    $A$ is false: $R$ is true.

Answer: A.

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Q 9Assertion (A) & Reason (B) MCQ1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following : Consider, the graph of constraints stated as linear inequalities as below: $5\text{x}+\text{y}\leq100,\text{x}+\text{y}\leq60,\text{x},\text{y}\geq0.$
Assertion $(A) : (25, 40) $ is an infeasible solution of the problem.
Reason $(R)$ : Any point inside the feasible region is called an infeasible solution.
  • A
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • ✓
    $A$ is true: $R$ is false.
  • D
    $A$ is false: $R$ is true.

Answer: C.

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Q 10Assertion (A) & Reason (B) MCQ1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : Maximise $Z = 3x + 4y,$ subjectto constraints : $\text{x}+\text{y}\leq1,\text{x}\geq0,\text{y}\geq0$ Then maximum value of $Z$ is $4.$
Reason : If the shaded region is not bounded then maximum value cannot be determined.
  • A
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • ✓
    $A$ is true: $R$ is false.
  • D
    $A$ is false: $R$ is true.

Answer: C.

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Q 111 Marks Question1 Mark
Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) :$ It is necessary to find objective function value at every point in the feasible region to find optimum value of the objective function.
Reason $(R) :$ For the constrains $2\text{x}+3\text{y}\leq6,5\text{x}+3\text{y}\leq15,\text{x}\geq0$ and $\text{y}\geq0$ cornner points of the feasible region are $(0, 2), (0, 0)$ and $(3, 0).$
  • A
    $A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
  • B
    $A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
  • C
    $A$ is true: $R$ is false.
  • ✓
    $A$ is false: $R$ is true.

Answer: D.

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Q 142 Marks Questions2 Marks
Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.
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Q 152 Marks Questions2 Marks
Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.
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Q 162 Marks Questions2 Marks
Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.
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Q 172 Marks Questions2 Marks
A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.
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Q 182 Marks Questions2 Marks
A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.
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Q 193 Marks Question3 Marks
A bill for Rs. 7650 was drawn on 8th March 2005 at 7 months. It was discounted on 18 May 2005 and the holder of the bill received Rs. 7497. What rate of interest did the banker charge?
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Q 245 Marks Questions5 Marks
A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?
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Q 255 Marks Questions5 Marks
There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.
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Q 265 Marks Questions5 Marks
There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommended daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the 2 families. What awareness can you create among people about the balanced diet from this question?
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Q 275 Marks Questions5 Marks
Solve the following linear programming problem graphically.Minimise $\text{z = 3x + 5y}$
subject to the constraints
$\text{x + 2y}\geq 10$
$\text{x + y}\geq 6$
$\text{3x + y}\geq 8$
$\text{x, y}\geq 0.$
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Q 285 Marks Questions5 Marks
Solve the following linear programming problem graphically:
Maximise Z = 7x + 10y
subject to the constraints
4x + 6y $\leq$ 240
6x + 3y $\leq$ 240
x $\geq$ 10
x $\geq$ 0, y $\geq$ 0
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Q 29Case study (4 Marks)4 Marks
Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region. Based on the above information, answer the following questions.
  1. Objective function of a L.P.P. is:
  1. A constant.
  2. A function to be optimised.
  3. A relation between the variables.
  4. None of these.
  1. Which of the following statement is correct?
  1. Every LPP has at least one optimal solution.
  2. Every LPP has a unique optimal solution.
  3. If an LPP has two optimal solutions, then it has infinitely many solutions.
  4. None of these.
  1. In solving the LPP: "minimize f = 6x + 10y subject to constraints $\text{x}\geq6,\text{ y}\geq2,\text{ 2x}+\text{y}\geq10,\text{ x}\geq0,\text{ y}\geq0"$ redundant constraints are:
  1. $\text{x}\geq6,\text{ y}\geq2$
  2. $\text{2x}+\text{y}\geq10,\text{ x}\geq0,\text{ y}\geq0$
  3. $\text{x}\geq6$
  4. None of these
  1. The feasible region for a LPP is shown shaded in the figure. Let Z = 3x - 4y be the objective function. Minimum of Z occurs at:
  1. (0, 0)
  2. (0, 8)
  3. (5, 0)
  4. (4, 10)
  1. The feasible region for a LPP is shown shaded in the figure. Let F = 3x - 4y be the objective function. Maximum value of F is:
  1. 0
  2. 8
  3. 12
  4. -18
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Q 30Case study (4 Marks)4 Marks
Corner points of the feasible region for an LPP are (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the objective function.
Based on the above information, answer the following questions.
  1. The minimum value of Z occurs at:
  1. (6, 8)
  2. (5, 0)
  3. (0, 3)
  4. (0, 8)
  1. Maximum value of Z occurs at:
  1. (5, 0)
  2. (0, 8)
  3. (0, 3)
  4. (6, 8)
  1. Maximum of Z - Minimum of Z =
  1. 58
  2. 68
  3. 78
  4. 88
  1. The corner points of the feasible region determined by the system of linear inequalities are:
  1. (0, 0), (-3, 0), (3, 2). (2, 3)
  2. (3, 0), (3, 2), (2, 3), (0, -3)
  3. (0, 0), (3, 0), (3, 2), (2, 3), (0, 3)
  4. None of these
  1. The feasible solution of LPP belongs to:
  1. First and second quadrant.
  2. First and third quadrant.
  3. Only second quadrant.
  4. Only first quadrant.
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Q 31Case study (4 Marks)4 Marks
Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only ₹ 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell an electronic sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18.

Based on the above information, answer the following questions.
  1. Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased atleast one of the the given machines, then:
  1. $\text{x}+\text{y}\geq0$
  2. $\text{x}+\text{y}<0$
  3. $\text{x}+\text{y}>0$
  4. $\text{x}+\text{y}\leq0$
  1. Let the constraints in the given problem is represented by the following inequalities.
$\text{x}+\text{y}\leq20$

$360\text{x}+240\text{y}\leq5760$

$\text{x},\text{y}\geq0$

Then which of the following point lie in its feasible region.
  1. (0, 24)
  2. (8, 12)
  3. (20, 2)
  4. None of these
  1. If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at:
  1. (0, 0)
  2. (16, 0)
  3. (8, 12)
  4. (0, 20)
  1. Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation of given problem.
Then which of the following represent the coordinates of one of its corner points

  1. (0, 24)
  2. (12, 8)
  3. (8, 12)
  4. (6, 14)
  1. If an LPP admits optimal solution at two consecutive vertices of a feasible region, then:
  1. The required optimal solution is at the midpoint of the tine joining two points.
  2. The optimal solution occurs at every point on the tine joining these two points.
  3. The LPP under consideration is not solvable.
  4. The LPP under consideration must be reconstructed.
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Q 32Case study (4 Marks)4 Marks
Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when relationship is expressed as linear equations or inequations.
Based on the above information, answer the following questions.
  1. The optimal value of the objective function is attained at the points:
  1. On X-axis.
  2. On Y-axis.
  3. Which are comer points of the feasible region.
  4. None of these.
  1. The graph of the inequality 3x + 4y < 12 is:
  1. Half plane that contains the origin.
  2. Half plane that neither contains the origin nor the points of the line 3x + 4y = 12.
  3. Whole XOY-plane excluding the points on line 3x + 4y = 12.
  4. None of these.
  1. The feasible region for an LPP is shown in the figure. Let Z = 2x + 5y be the objective function. Maximum of Z occurs at:
  1. (7, 0)
  2. (6, 3)
  3. (0, 6)
  4. (4, 5)
  1. The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points ( 15, 15) and (0, 20) is:
  1. p = q
  2. p = 2q
  3. q = 2p
  4. q = 3p
  1. The comer points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y.
Compare the quantity in Column A and Column B
Column A
Column B
Maximum of Z
325
  1. The quantity in column A is greater.
  2. The quantity in column Bis greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined on the basis of the information supplied.
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Q 33Case study (4 Marks)4 Marks
Deepa rides her car at $25 \mathrm{~km} / \mathrm{hr}$. She has to spend ₹ $2$ per km on diesel and if she rides it at a faster speed of $40 \mathrm{~km} /$ hr, the diesel cost increases to ₹ $5$ per km. She has ₹ $100$ to spend on diesel. Let she travels $x$ kms with speed $25 \mathrm{~km} /$ hr and y kms with speed $40 \mathrm{~km} / \mathrm{hr}$. The feasible region for the LPP is shown below:

Based on the above information, answer the following questions.
  1. What is the point of intersection of line $l_1$ and $l_2​​​​​​​$?
  1. $\Big(\frac{40}{3},\frac{50}{3}\Big)$
  2. $\Big(\frac{50}{3},\frac{40}{3}\Big)$
  3. $\Big(\frac{-50}{3},\frac{40}{3}\Big)$
  4. $\Big(\frac{-50}{3},\frac{-40}{3}\Big)$
  1. The comer points of the feasible region shown in above graph are:
  1. $(0,25),(20,0),\Big(\frac{40}{3},\frac{50}{3}\Big)$
  2. $(0, 0), (25, 0), (0, 20) $
  3. $(0,0),\Big(\frac{40}{3},\frac{50}{3}\Big),(0,20)$
  4. $(0,0),(25,0),\Big(\frac{50}{3},\frac{40}{3}\Big),(0,20)$
  1. If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point:
  1. $\Big(\frac{50}{3},\frac{40}{3}\Big)$
  2. (0, 0)
  3. (25, 0)
  4. (0, 20)
  1. If Z = 6x - 9y be the objective function, then maximum value of Z is:
  1. -20
  2. 150
  3. 180
  4. 20
  1. If Z = 6x + 3y be the objective function, then what is the minimum value of Z?
  1. 120
  2. 130
  3. 0
  4. 150
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Q 37Fill In The Blanks[1 Marks ]1 Mark
Fill in the blanks.
In a LPP if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same _________ value.
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Q 40True False[1 Marks ]1 Mark
State whether the statements are True or False:
If the feasible region for a LPP is unbounded, maximum or minimum of the objective function Z = ax + by may or may not exist.
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Q 42True False[1 Marks ]1 Mark
State whether the statements are True or False:
Maximum value of the objective function Z = ax + by in a LPP always occurs at only one corner point of the feasible region.
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Q 43True False[1 Marks ]1 Mark
State whether the statements are True or False:
In a LPP, the minimum value of the objective function Z = ax + by is always 0 if origin is one of the corner point of the feasible region.
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