NTA UGC NET Linear Programming Problems Question with Solutions


Q.1➡ | UGC NET DEC 2023
Comprehension:
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively.
Suppose x is quantity (in gram) of food X, y is quantity (in gram) of food Y. Answering the following question based on the above paragraph given.

The cost function of total food is :
i ➥ Z = 6x + 7y
ii ➥ Z = 8x + 12y
iii ➥ Z = 12x + 20y
iv ➥ Z = 20x + 12y
Best Explanation:
Answer: (III)
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Q.2➡ | UGC NET DEC 2023
Comprehension:
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively.
Suppose x is quantity (in gram) of food X, y is quantity (in gram) of food Y. Answering the following question based on the above paragraph given.

The dual of the formulated LPP is:
i ➥ Max Z = 100u + 120v
s.t.
6u + 7v <= 12
8u + 12v <= 20
u, v srcset== 0″>
ii ➥ Max Z = 12u + 20u
s.t.
6u + 7v <= 100
8u + 12v <= 120
u, v srcset== 0″>
iii ➥ Max Z = 100u + 120v
s.t.
6u + 7u <= 12
8u + 7v <= 20
u, v are unrestricted
iv ➥ Max Z = 100u + 120u
s.t.
6u + 7v srcset== 12 8u + 12v >= 20 u, v >= 0″>
Best Explanation:
Answer: (I)
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Q.3➡ | UGC NET DEC 2023
Comprehension:
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively.
Suppose x is quantity (in gram) of food X, y is quantity (in gram) of food Y. Answering the following question based on the above paragraph given.

Which of the following constrains when formulating the LPP ?
i ➥ 6x + 7y <= 100, 8x + 12y <= 120, x, y >= 0
ii ➥ 6x + 8y <= 100, 7x + 12y <= 120, x, y >= 0
iii ➥ 6x + 7y >= 100, 8x + 12y >= 120, x, y >= 0
iv ➥ 6x + 8y >= 100, 7x + 12y >= 120, x, y >= 0
Best Explanation:
Answer: (IV)
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Q.4➡ | UGC NET DEC 2023
Comprehension:
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively.
Suppose x is quantity (in gram) of food X, y is quantity (in gram) of food Y. Answering the following question based on the above paragraph given.

Which of the following are quantities (in grams) of food X and Y respectively when the cost of food is minimum:
i ➥
ii ➥
iii ➥
iv ➥ 0 and 10
Best Explanation:
Answer: (II)
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Q.5➡ | UGC NET DEC 2023
Comprehension:
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively.
Suppose x is quantity (in gram) of food X, y is quantity (in gram) of food Y. Answering the following question based on the above paragraph given.

The minimum cost of food is :
i ➥ 205
ii ➥ 250
iii ➥ 330
iv ➥ 200
Best Explanation:
Answer: (I)
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Q.6➡ | UGC NET JUNE 2023
Match List I with List II
Match List I with List II
List I	List II
A. PERT	I.  Used for nonrepetitive jobs
B. Optimistic time	II. Used for repetitive jobs
C. CPM	III.  The shortest possible time to complete the   
       activity if all goes well.
D. Pessimistic time	IV. The longest time that an activity takes

Choose the correct answer from the options given below:
i ➥ A-III B-IV C-II D-I
ii ➥ A-IV B-I C-I D-III
iii ➥ A-I B-III C-II D-III
iv ➥ A-I B-III C-II D-IV
Best Explanation:
Answer: IV
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Q.7➡ | UGC NET JUNE 2023
How many integral solutions are there to x + y + z + w = 29 ,where x >= 1, y >= 2, z >= 3 and w >= 0 ?
i ➥ 2400
ii ➥ 2600
iii ➥ 2800
iv ➥ 3000
Best Explanation:
Answer: II
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Q.8➡ | UGC NET JUNE 2023
How will you free the memory allocated by the following program?

include stdio.h
include stdlib.h
define MAXROW 3
define MAXCOL 4
int main()
{
int **p, i, j;
p = (int **) malloc(MAXROW * sizeof(int*));
return 0;
}
i ➥ memfree (int p);
ii ➥ dealloc (p);
iii ➥ malloc (p,0);
iv ➥ free (p);
Best Explanation:
Answer: IV
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Q.9➡ | UGC NET JUNE 2023
There are M points on one straight line AB and n points on another straight line AC none of them being A. How many triangles can be formed with these points as vertices?
i ➥ mn(m + n – 2)
ii ➥ 1/2 * mn(m + n – 2)
iii ➥1/2 * mn(m + n – 1)
iv ➥mn(m + n – 1)
Best Explanation:
Answer: II
Explanation:

The task is to find the number of triangles that can be formed using the given points on two straight lines AB and AC, where there are M points on line AB and n points on line AC (none of them being A).
To form a triangle, we need to choose three points—one from each line. Let’s go through the reasoning for the correct answer:
Option ii. 1/2 * mn(m + n – 2) :

1/2: This factor accounts for the fact that choosing points A, B, and C is the same as choosing points A, C, and B. It eliminates overcounting.

mn: This part represents the number of ways to choose one point from line AB (M choices) and one point from line AC (n choices).

(m+n−2): This part represents the number of ways to choose the third point, considering that the chosen points cannot be A. There are M + n points in total, and we subtract 2 (A and one of the chosen points) to ensure that the third point is not A and is distinct from the chosen points on AB and AC.

Putting it all together,

1/2*mn(m+n−2) gives the correct count of non-degenerate triangles formed by choosing three points from the given lines.
This expression accounts for the necessary combinations and avoids overcounting, making it a suitable formula for calculating the number of triangles. Therefore, option ii is the correct answer: 1/2*mn(m+n−2)
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Q.10➡ | NTA UGC NET November 2020
i ➥ 23
ii ➥ 9.5
iii ➥ 13
iv ➥ 8

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Answer: IV
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Q.11➡ | NTA UGC NET December 2019
i ➥ x1=750, x2=750, z=1500
ii ➥ x1=500, x2=100, z=1500
iii ➥ x1=1000, x2=500, z=1500
iv ➥ x1=900, x2=600, z=1500

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Answer: III
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Q.12➡ | NTA UGC NET December 2019
Consider the following statements with respect to duality in LPP:
(a) The final simplex table giving optimal solution of the primal also contains optimal solution of its dual in itself.
(b) If either the primal or the dual problem has a finite optimal solution, then the other problem also has a finite optimal solution.
(c) If either problem has an unbounded optimal solution, then the other problem has no feasible solution at all. Which of the statements is (are) correct?
i ➥ only (a) and (b)
ii ➥ only (a) and (c)
iii ➥ only (b) and (c)
iv ➥ (a), (b) and (c)

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Answer: IV
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Q.13➡ | NTA UGC NET December 2019
A basic feasible solution of an mXn Transportation-Problem is said to be non-degenerate, if basic feasible solution contains exactly____number of individual allocation in ___positions
i ➥ m+n+1, independent
ii ➥ m+n-1, independent
iii ➥ m+n-1, appropriate
iv ➥ m-n+1, independent

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Answer: II
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Q.14➡ | NTA UGC NET JUNE 2019
i ➥ x1=5/2, x2=0, x3=0, Z=5
ii ➥ x1=0, x2=0, x3=5/2, Z=5
iii ➥ x1=0, x2=5/2, x3=0, Z= -5/2
iv ➥ x1=0, x2=0, x3=10, Z=20

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Answer: II
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Q.15➡ | NTA UGC NET December 2018
i ➥ x1 =2, x2 =0 and z = -4
ii ➥ x1 =2, x2 =6 and z = -22
iii ➥ x1 =0, x2 =2 and z = -6
iv ➥ x1 =6, x2 =2 and z = -18

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Answer: I
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Q.16➡ | NTA UGC NET JUNE 2018
Which of the following statements is false about convex minimization problem ?
i ➥ If a local minimum exists, then it is a global minimum
ii ➥ The set of all global minima is convex set
iii ➥ The set of all global minima is concave set
iv ➥ For each strictly convex function, if the function has a minimum, then the minimum is unique

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Answer: III
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Q.17➡ | NTA UGC NET JUNE 2018
The following LPP
Maximize z=100x1+2x2 +5x3
Subject to 14x1+x2 −6x3 +3x4 =7
32x1 +x2 −12x3 ≤10
3x1 −x2 −x3 ≤0
x1 , x2 , x3 , x4 ≥ 0
has
i ➥ x1 =100, x2 =0, x3 =0
ii ➥ Unbounded solution
iii ➥ No solution
iv ➥ x1 =50, x2 =70, x3 =60

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Answer: II
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Q.18➡ | NTA UGC NET November 2017 Paper -3
Which of the following is a valid reason for causing degeneracy in a transportation problem ?
Here m is no. of rows and n is no. of columns in transportation table
i ➥ When the number of allocations is m+n−1.
ii ➥ When two or more occupied cells become unoccupied simultaneously.
iii ➥ When the number of allocations is less than m+n−1.
iv ➥ When a loop cannot be drawn without using unoccupied cells, except the starting cell of the loop.

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Answer: III
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Q.19➡ | NTA UGC NET November 2017 Paper -3
Consider the following LPP :
Max Z = 15x1 + 10x2
Subject to the constraints
4x1 + 6x2 ≤ 360
3x1 + 0x2 ≤ 180
0x1 + 5x2 ≤ 200
x1, x2 / 0
The solution of the LPP using Graphical solution technique is :
i ➥ x1=60, x2=0 and Z=900
ii ➥ x1=60, x2=20 and Z=1100
iii ➥ x1=60, x2=30 and Z=1200
iv ➥ x1=50, x2=40 and Z=1150

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Answer: II
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Q.20➡ | NTA UGC NET November 2017 Paper -3
Consider the following LPP :
Min Z = 2x1 + x2 + 3x3
Subject to :
x1 − 2x2 + x3 ≥ 4
2x1 + x2 + x3 ≤ 8
x1 − x3 ≥ 0
x1, x2, x3 ≥ 0
The solution of this LPP using Dual Simplex Method is :
i ➥ x1=0, x2=0, x3=3 and Z=9
ii ➥ x1=0, x2=6, x3=0 and Z=6
iii ➥ x1=4, x2=0, x3=0 and Z=8
iv ➥ x1=2, x2=0, x3=2 and Z=10

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Answer: I (Marks to all )
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Q.21➡ | NTA UGC NET January 2017 Paper -3
Which of the following statements is true?
i ➥ The sentence S is a logical consequence of S1,…., Sn if and only if S1 ∧ S2 ∧…. ∧ Sn → S is satisfiable.
ii ➥ The sentence S is a logical consequence of S1,…., Sn if and only if S1 ∧ S2 ∧…. ∧ Sn → S is valid.
iii ➥ The sentence S is a logical consequence of S1,…., Sn if and only if S1 ∧ S2 ∧…. ∧ Sn → S is consistent.
iv ➥ The sentence S is a logical consequence of S1,…., Sn if and only if S1 ∧ S2 ∧…. ∧ Sn → S is inconsistent.

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Answer: II
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Q.22➡ | NTA UGC NET January 2017 Paper -3
A t-error correcting q-nary linear code must satisfy:

Where M is the number of code words and X is
i ➥ qn
ii ➥ qt
iii ➥ q-n
iv ➥ q-t

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Answer: I
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Q.23➡ | NTA UGC NET January 2017 Paper -3
With respect to a loop in the transportation table, which one of the following is not correct?
i ➥ Every loop has an odd no. of cells and at least 5.
ii ➥ Closed loops may or may not be square in shape.
iii ➥ All the cells in the loop that have a plus or minus sign, except the starting cell, must be occupied cells.
iv ➥ Every loop has an even no. of cells and at least four

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Answer: I
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Q.24➡ | NTA UGC NET January 2017 Paper -3
At which of the following stage(s), the degeneracy do not occur in transportation problem ? (m, n represents number of sources and destinations respectively)
(1) While the values of dual variables ui and vj cannot be computed.
(2) While obtaining an initial solution, we may have less than m + n – 1 allocations.
(3) At any stage while moving towards optimal solution, when two or more occupied cells with the same minimum allocation become unoccupied simultaneously.
(4) At a stage when the no. of +ve allocation is exactly m + n – 1.
i ➥ (a), (b) and (c)
ii ➥ (a), (c) and (d)
iii ➥ (a) and (d)
iv ➥ (a), (b), (c) and (d)

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Answer: III
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Q.25➡ | NTA UGC NET January 2017 Paper -3
Consider the following LPP :

The standard form of this LPP shall be :
i ➥
ii ➥
iii ➥
iv ➥

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Answer: I
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Q.26➡ | NTA UGC NET August 2016 Paper -3
Which of the following is used for the boundary representation of an image object?
i ➥ Quad Tree
ii ➥ Projections
iii ➥ Run length coding
iv ➥ Chain codes

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Answer: IV
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Q.27➡ | NTA UGC NET August 2016 Paper -3
The region of feasible solution of a linear programming problem has a _ property in geometry, provided the feasible solution of the problem exists.
i ➥ concavity
ii ➥ convexity
iii ➥ quadratic
iv ➥ polyhedron

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Answer: II
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Q.28➡ | NTA UGC NET August 2016 Paper -3
Consider the following statements:
(1) Revised simplex method requires lesser computations than the simplex method.
(2) Revised simplex method automatically generates the inverse of the current basis matrix.
(3) Less number of entries are needed in each table of revised simplex method than usual simplex method.
Which of these statements are correct?
i ➥ (1) and (2) only
ii ➥ (1) and (3) only
iii ➥ (2) and (3) only
iv ➥ (1), (2) and (3)

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Answer: IV
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Q.29➡ | NTA UGC NET August 2016 Paper -3
The following transportation problem: is

The above solution of a given transportation problem is:
i ➥ infeasible solution
ii ➥ optimum solution
iii ➥ non-optimum solution
iv ➥ unbounded solution

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Answer: II
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Q.30➡ | NTA UGC NET December 2015 Paper -3
A basic feasible solution of a linear programming problem is said to be __ if at least one of the basic variables is zero.
i ➥ Degenerate
ii ➥ non-degenerate
iii ➥ Infeasible
iv ➥ Unbounded

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Answer: I
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Q.31➡ | NTA UGC NET December 2015 Paper -3
Consider the following conditions:
(a)The solution must be feasible, i.e. it must satisfy all the supply and demand constraints.
(b)The number of positive allocations must be equal to m+n-1, where m is the number of rows and n is the number of columns.
(c)All the positive allocations must be in independent positions.
The initial solution of a transportation problem is said to be non-degenerate basic feasible solution if it satisfies:
i ➥ (a) and (b) only
ii ➥ (a) and (c) only
iii ➥ (b) and (c) only
iv ➥ (a), (b) and (c)

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Answer: IV
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Q.32➡ | NTA UGC NET December 2015 Paper -3
Consider the following transportation problem:

The transportation cost in the initial basic feasible solution of the above transportation problem using Vogel’s Approximation method is:
i ➥ 1450
ii ➥ 1465
iii ➥ 1480
iv ➥ 1520

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Answer: II
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Q.33➡ | NTA UGC NET June 2015 Paper -3
In the Hungarian method for solving assignment problem, an optimal assignment requires that the maximum number of lines that can be drawn through squares with zero opportunity cost is equal to the number of:
i ➥ rows or columns
ii ➥ rows + columns
iii ➥ rows + columns – 1
iv ➥ rows + columns + 1

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Answer: I
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Q.34➡ | NTA UGC NET June 2015 Paper -3
Given the following statements with respect to linear programming problem:
S1 : The dual of the dual linear programming problem is again the primal problem
S2 : If either the primal or the dual problem has an unbounded objective function value, the other problem has no feasible solution.
S3 : If either the primal or dual problem has a finite optimal solution, the other one also possesses the same, and the optimal value of the objective functions of the two problems are equal.
Which of the following is true?
i ➥ S1 and S2
ii ➥ S1 and S3
iii ➥ S2 and S3
iv ➥ S1 , S2 and S3

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Answer: IV
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Q.35➡ | NTA UGC NET June 2015 Paper -3
Consider the following transportation problem:

The initial basic feasible solution of the above transportation problem using Vogel’s Approximation Method (VAM) is given below:

The solution of the above problem.
i ➥ is degenerate solution
ii ➥ is optimum solution
iii ➥ needs to improve
iv ➥ is infeasible solution

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Answer: II
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