1. If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that
(a) Right hand side of the primal constraint can be altered without affecting the optimum solution
(b) Changing the right hand side of the primal constraint will disturb the optimum solution
(c) The objective function is unbounded
(d) The problem is degenerate
(1 Mark, 1996)

Ans: a
2. The supply at three sources is 50, 40 and 60 units respectively whilst the demand at the four destinations is 20, 30, 10 and 50 units. In solving this transportation problem
(a) A dummy source of capacity 40 units is needed
(b) A dummy destination of capacity 40 units is needed
(c) No solution exists as the problem is infeasible
(d) None solution exists as the problem is degenerate
(2 Mark, 2002)

Ans: b
3. A manufacturer produces two types of products, 1 and 2, at production levels of x_{1} and x_{2}. The profit is given is 2x_{1} + 5x_{2}. The production constraints are
x_{1} + 3x_{2} \le 40
3x_{1} + x_{2} \le 24
x_{1} + x_{2} \le 10
x_{1} > 0
x_{2} > 0
The maximum profit which can meet the constraints is
(a) 29
(b) 38
(c) 44
(d) 75
(2 Mark, 2003)

Ans: a
Explanation:
4. A company produces two types of toys: P and Q. Production time of Q is twice that of P and the company has a maximum of 2000 time units per day. The supply of raw material is just sufficient to produce 1500 toys (of any type) per day. Toy type Q requires an electric switch which is available @ 600 pieces per day only. The company makes a profit of Rs.3 and Rs.5 on type P and Q respectively. For maximization of profits, the daily production quantities of P and Q toys should respectively be
(a) 1000, 500
(b) 500, 1000
(c) 800, 600
(d) 1000, 1000
(2 Mark, 2004)

Ans: a
Explanation:
5. A component can be produced by any of the four processes I, II, III and IV.
Process I has a fixed cost of Rs.20 and variable cost of Rs.3 per piece. Process II has a fixed cost of Rs.50 and variable cost of Re.1 per piece. Process III has a fixed cost of Rs.40 and variable cost of Rs.2 per piece. Process IV has a fixed cost of Rs.10 and variable cost of Rs.4 per piece. If the company wishes to produce 100 pieces of the component, from economic point of view it should choose.
(a) Process I
(b) Process II
(c) Process III
(d) Process IV
(2 Mark, 2005)

Ans: b
6. A company has two factories S1, S2 and two warehouses D1, D2. The supplies from S1 and S2 are 50 and 40 units respectively. Warehouse D1 requires a minimum of 20 units and a maximum of 40 units. Warehouse D2 requires a minimum of 20 units and, over and above, it can take as much as can be supplied. A balanced transportation problem is to be formulated for the above situation. The number of supply points, the number of demand points, and the total supply (or total demand) in the balanced transportation problem respectively are
(a) 2, 4, 90
(b) 2, 4, 110
(c) 3, 4, 90
(d) 3, 4, 110
(2 Mark, 2005)

Ans: c
Statement for Linked Questions 7 & 8:
Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize X_{1} + X_{2}. The corner points of the feasible region are (0,0), (0,2), (2,0) and (4/3, 4/3).
7. If an additional constraint X_{1} + X_{2} \le 5 is added, the optimal solution is
(a) (5/3, 5/3)
(b) (4/3, 4/3)
(c) (5/2, 5/2)
(d) (5, 0)
(2 Mark, 2005)

Ans: b
8. Let Y_{1} and Y_{2} be the decision variables of the dual and v_{1} and v_{2} be the slack variables of the dual of the given linear programming problem. The optimum dual variables are
(a) Y_{1} and Y_{2}
(b) Y_{1} and v_{1}
(c) Y_{1} and v_{2}
(d) v_{1} and v_{2}
(2 Mark, 2005)

Ans: d
9. A firm is required to procure three items (P, Q and R). The prices quoted for these items (in Rs.) by suppliers S1, S2 and S3 are given in table. The management policy requires that each item has to be supplied by only one supplier and one supplier supply only one item. The minimum total cost (in Rs.) of procurement to the firm is:
(a) 350
(b) 360
(c) 385
(d) 395
(2 Mark, 2006)

Ans: c
Common Data for Questions 10 and 11:
Consider the Linear Programme (LP)
Max 4x + 6y
Subject to
3x + 2y ≤ 6
2x +3y ≤ 6
x, y \ge 0
10. After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0).
After some simplex iterations, the following table is obtained
From this, one can conclude that
(a) The LP has a unique optimal solution
(b) The LP has an optimal solution that is not unique
(c) The LP is infeasible
(d) The LP is unbounded
(2 Mark, 2008)

Ans: b
11. The dual for the LP in Q.10 is
(a) Min 6u + 6v
subject to 3u + 2v ≥ 4
2u + 3v ≥ 6
u, v ≥ 0
(b) Max 6u + 6v
subject to 3u + 2v ≤ 4
2u + 3v ≥ 6
u, v ≥ 0
(c) Max 4u + 6v
subject to 3u + 2v ≥ 6
2u + 3v ≥ 6
u, v ≥ 0
(d) Min 4u + 6v
subject to 3u + 2v ≤ 6
2u + 3v ≥ 6
u, v ≥ 0
(2 Mark, 2008)

Ans: a
12. For the standard transportation linear programme with m sources and n destinations and total supply equaling total demand, an optimal solution (lowest cost) with the smallest number of nonzero x_{ij} values (amounts from source i to destination j) is desired. The best upper bound for this number is
(a) mn
(b) 2(m + n)
(c) m + n
(d) m + n – 1
(2 Mark, 2008)

Ans: d
13. Consider the following Linear Programming Problem (LPP):
Maximize z = 3x_{1} + 2x_{2}
Subject to
x_{1} \le 4
x_{2} \le 6
3x_{1}+ 2x_{2} \le 18
x_{1} \ge 0, x_{2} \ge 0
(a) The LPP has a unique optimal solution
(b) The LPP is infeasible
(c) The LPP is unbounded
(d) The LPP has multiple optimal solutions
(2 Mark, 2009)

Ans: d
14. Simplex method of solving linear programming problem uses
(a) All the points in the feasible region
(b) Only the corner points of the feasible region
(c) Intermediate points within the infeasible region
(d) Only the interior points in the feasible region.
(1 Mark, 2010)

Ans: b
Common Data Questions 15 & 16:
One unit of product P_{1} requires 3 kg of resource R_{1} and 1kg of resource R_{2}. One unit of product P_{2} requires 2kg of resource R_{1} and 2kg of resource R_{2} . The profits per unit by selling product P_{1} and P_{2} are Rs.2000 and Rs.3000 respectively. The manufacturer has 90kg of resource R_{1} and 100kg of resource R_{2}.
15. The unit worth of resource R_{2 }i.e., dual price of resource R_{2} in Rs. Per kg is
(a) 0
(b) 1350
(c) 1500
(d) 2000
(2 Mark, 2011)

Ans: a
16. The manufacturer can make a maximum profit of Rs.
(a) 60000
(b) 135000
(c) 150000
(d) 200000
(2 Mark, 2011)

Ans: b
17. A linear programming problem is shown below.
Maximize 3x + 7y
Subject to 3x + 7y ≤ 10
4x + 6y ≤ 8
x, y ≥ 0
It has
(a) An unbounded objective function
(b) Exactly one optimal solution.
(c) Exactly two optimal solutions
(d) Infinitely many optimal solutions.
(2 Mark, 2013)

Ans: b
18. If there are m sources and n destinations in a transportation matrix, the total number of basic variables in a basic feasible solution is
(a) m + n
(b) m + n + 1
(c) m + n – 1
(d) m
(1 Mark, 2014[2])

Ans: c
19. Consider an objective function Z(x_{1}, x_{2}) = 3x_{1} + 9x_{2} and the constraints
x_{1}+ x_{2} \le 8
x_{1}+ 2x_{2} \le 4
x_{1} \ge0, x_{2} \ge0
The maximum value of the objective function is _______.
(2 Mark, 2014[3])

Ans: 18
20. The total number of decision variables in the objective function of an assignment problem of size n × n (n jobs and n machines) is
(a) n^{2}
(b) 2n
(c) 2n−1
(d) n
(1 Mark, 2014[4])

Ans: a
21. For the linear programming problem:
Maximize Z = 3x_{1} + 2x_{2}
Subject to −2x_{1} + 3x_{2} ≤ 9
x_{1 }− 5x_{2} ≥ −20
x_{1}, x_{2} ≥ 0
The above problem has
(a) Unbounded solution
(b) Infeasible solution
(c) Alternative optimum solution
(d) Degenerate solution
(2 Mark, 2015[3])

Ans: a
22. Maximize Z = 15X_{1} + 20X_{2}
subject to:
12X_{1} + 4X_{2} ≥ 36
12X_{1} − 6X_{2} ≤ 24
X_{1}, X_{2} ≥ 0
The above linear programming problem has
(a) Infeasible solution
(b) Unbounded solution
(c) Alternative optimum solutions
(d) Degenerate solution
(2 Mark, 2016[1])

Ans: b
23. Two models, P and Q of a product earn profits of Rs. 100 and Rs. 80 per piece, respectively. Production times for P and Q are 5 hours and 3 hours, respectively, while the total production time available is 150 hours. For a total batch size of 40, to maximize profit, the member of units of P to be produced is ____.
(2 Mark, 2017[1])

Ans: 15
24. A product made in two factories, P and Q, is transported to two destinations, R and S. The per unit costs of transportation (in Rupees) from factories to destinations are as per the following matrix:
Factory P produces 7 units and factory Q produces 9 units of the product. Each destination requires 8 units. If the northwest corner method provides the total transportation cost X(in Rupees) and the optimized (the minimum) total transportation cost is Y (in rupees), then (X Y), in Rupees, is
(a) 0
(b) 15
(c) 35
(d) 105
(2 Mark, 2017[2])

Ans: The answer is not in the options. So, Grace mark is awarded to all the students.
25. Maximize Z = 5x_{1} + 3x_{2}
Subject to:
x_{1} + 2x_{2} ≤ 10
x_{1 }− x_{2} ≤ 8
x_{1}, x_{2} ≥ 0
In the starting simplex tableau, x_{1} and x_{2} are nonbasic variables and the value of Z in the next simplex tableau is _____.
(2 Mark, 2017[2])

Ans: 40
26. The minimum value of 3x + 5y
such that:
3x + 5y ≤ 15
4x + 9y ≤ 8
13x + 2y ≤ 2
x ≥ 0, y ≥ 0
is _______.
(2 Mark, 2018[1])

Ans: 0
27. The problem of maximizing z = x_{1} – x_{2} subject to constraints x_{1} + x_{2 }≤ 10, x_{1} \ge 0, x_{2} \ge 0 and x_{2 }≤ 5 has
(a) No solution
(b) One solution
(c) Two solutions
(d) More than two solutions
(2 Mark, 2018[2])

Ans: b
RAJU Kumar GEC
September 23, 2019 @ 11:13 am
Ur encouragement regarding gate exam for mechanical student is very helpful for us but sir I want to give one suggestion “please upload theoretical concept of each subject related mechanical engineering “
BIGHNESH KUMAR SAHU
October 10, 2019 @ 3:38 pm
Hi Raju,
I will try my best to include the theoretical things into this site.
Thanks…..