Paper 2 CS · UNIT 01 · MOCK TEST

Mock 11 – Optimisation (LPP, Transportation, Assignment)

Unit 1: Discrete Structures and Optimisation · 27 questions · about 32 minutes. Press End Mock any time to see your score and the correct answers.

All 14 topic-wise mocks in this unit

0 / 27 answered 00:00

Q1. Consider a linear program {max cᵀx | Ax ≤ b}. Which of the following statements is true?

Athe dual program is {min bᵀy | Aᵀy ≤ c, y ≥ 0} Bthe dual program is {min bᵀy | Aᵀy ≥ c, y ≥ 0} Cthe dual program is {min bᵀy | Aᵀy = c, y ≥ 0} Dthe dual program is {max bᵀY | AᵀY = c}

Q2. A linear programming problem (LPP) is as follows: Min z = 30x − 18y, subject to the constraints 3x + 4y ≤ 60, 5x − 3y ≥ 20 and x, y ≥ 0. In this feasible region, the solution(s) of the LPP is/are
A. (4, 0)
B. (2, 0)
C. (7, 5)
D. (0, 15)
E. (8, 5)
Choose the correct answer from the options given below:

AA and C only BB only CE only DD only

Directions (Q91–95): Food X contains 6 units of vitamin D per gram and 7 units of vitamin E per gram and costs Rs 12 per gram. Food Y contains 8 units of vitamin D per gram and 12 units of vitamin E per gram and costs Rs 20 per gram. The daily minimum requirements of vitamin D and E are 100 units and 120 units respectively. Let x be the quantity (in grams) of food X and y the quantity (in grams) of food Y.

Q3. Which of the following are the constraints when formulating the LPP?

A6x + 7y ≤ 100, 8x + 12y ≤ 120, x, y ≥ 0 B6x + 8y ≤ 100, 7x + 12y ≤ 120, x, y ≥ 0 C6x + 7y ≥ 100, 8x + 12y ≥ 120, x, y ≥ 0 D6x + 8y ≥ 100, 7x + 12y ≥ 120, x, y ≥ 0

Q4. The dual of the formulated LPP is:

AMax Z = 100u + 120v s.t. 6u + 7v ≤ 12, 8u + 12v ≤ 20, u, v ≥ 0 BMax Z = 12u + 20v s.t. 6u + 7v ≤ 100, 8u + 12v ≤ 120, u, v ≥ 0 CMax Z = 100u + 120v s.t. 6u + 7v ≤ 12, 8u + 7v ≤ 20, u, v unrestricted DMax Z = 100u + 120v s.t. 6u + 7v ≥ 12, 8u + 12v ≥ 20, u, v ≥ 0

Directions (Q91–95): A firm can produce two types of cloth, A and B. Three kinds of wool are required: red, green and blue. One unit length of type A cloth needs 2 yards of red and 3 yards of green wool. One unit length of type B cloth needs 3 yards of red wool, 2 yards of green wool and 2 yards of blue wool. The firm has a stock of only 8 yards of red wool, 10 yards of green wool and 15 yards of blue wool. The income from one unit length of type A and type B cloth is ₹30 and ₹50 respectively. Let x and y be the unit lengths of type A and type B cloth produced.

Q5. When the LPP is formulated from the passage, which of the following is the objective function of its dual problem?

AMin P = 5u + 8v + 15w BMin P = 8u + 10v + 15w CMin P = 15u + 5v + 8w DMax P = 10u + 8v + 15w

Q6. Which of the following is the appropriate objective function for the given problem?

A2x + 3y B3x + 2y + 2z C30x + 50y D3x + 2y

Q7. How many unit lengths of cloth A are used when income is maximised (as per the keyed answer)?

A0.8 unit B2.4 unit C2.8 unit D3.0 unit

Q8. If x unit lengths of type A and y unit lengths of type B cloth are produced, the constraints of the LPP are:

Ay ≤ 7.5, 2x + 3y ≤ 8, 3y + 2x ≤ 10 By ≤ 7.5, 2x + 3y ≤ 8, 3x + 2y ≤ 10 C2x + 3y ≤ 8, 3x + 2y + 2z < 15 D2x + 3y ≤ 10, 3x + 2y ≤ 10, 2y ≤ 15

Q9. Consider the statements:
I. There does not exist a polynomial-time algorithm to solve integer linear programming problems.
II. The main focus of PERT is 'minimising time'.
Choose the correct option about statements I and II.

AI is true; II is true BI is true; II is false CI is false; II is true DI is false; II is false

Q10. Consider the primal problem:
Maximize z = 5x1 + 12x2 + 4x3
subject to x1 + 2x2 + x3 = 10
2x1 − x2 + 3x3 = 8
x1, x2, x3 ≥ 0
Its dual problem is
Minimize w = 10y1 + 8y2
subject to y1 + 2y2 ≥ 5
2y1 − y2 ≥ 12
y1 + 3y2 ≥ 4
Which of the following is correct?

Ay1 ≥ 0, y2 unrestricted By1 ≥ 0, y2 ≥ 0 Cy1 is unrestricted, y2 ≥ 0 Dy1 is unrestricted, y2 restricted

Q11.

An LPP: Minimize z = 30x − 18y subject to 3x + 4y ≤ 60, 5x − 3y ≥ 20, x, y ≥ 0. In this feasible region, the solution(s) of the LPP is/are: A. (4, 0) B. (2, 0) C. (7, 5) D. (0, 15) E. (8, 5)

AA and C only BB only CE only DD only

Directions (Q91–95): A firm produces two types of cloth, A and B, using red, green and blue wool. One unit length of type A needs 2 yards of red and 3 yards of green wool. One unit length of type B needs 3 yards of red, 2 yards of green and 2 yards of blue wool. The firm has 8 yards of red, 10 yards of green and 15 yards of blue wool. Income is ₹30 per unit length of type A and ₹50 per unit length of type B.

Q12.

When we formulate the LPP from the above-given passage. Which of the following is an objective function for its dual problem?

AMin P = 5u + 8v + 15w BMin P = 8u + 10v + 15w CMin P = 15u + 5v + 8w DMax P = 10u + 8v + 15w

Q13.

How many unit length used for cloth A when income is maximized ?

A0.8 unit B2.4 unit C2.8 unit D3.0 unit

Q14.

If x unit length of type A cloths and y unit length of type B cloths, then contrains formulate for the LPP, is :

Ay ≤ 7.5, 2x + 3y ≤ 8, 3y + 2x ≤ 10 By ≤ 7.5, 2x + 3y ≤ 8, 3x + 2y ≤ 10 C2x + 3y ≤ 8, 3x + 2y + 2z < 15 D2x + 3y ≤ 10, 3x + 2y ≤ 10, 2y ≤ 15

Directions (Q91–95): Food X contains 6 units of vitamin D and 7 units of vitamin E per gram and costs ₹12 per gram. Food Y contains 8 units of vitamin D and 12 units of vitamin E per gram and costs ₹20 per gram. The daily minimum requirements of vitamins D and E are 100 units and 120 units. Let x and y be the grams of food X and Y.

Q15.

The cost function of total food is:

AZ=6 x+7 y BZ=8 x+12 y CZ=12 x+20 y DZ=20 x+12 y

Q16.

Which of the following constraints when formulating the LPP?

A6 x+7 y<= 100,8 x+12 y <= 120, x, y >= 0 B6 x+8 y <=100,7 x+12 y <=120, x, y>= 0 C6 x+7 y >= 100,8 x+12 y >=120, x, y >= 0 D6 x+8 y >= 100,7 x+12 y >=120, x, y >= 0

Q17.

The dual of the formulated LPP is :

AMax Z=100 u+120 v s.t. 6 u+7 v <=12 8 u+12 v <= 20 u, v >= 0 BMax Z=12 u+20 v s.t. 6 u+7 v <=100 8 u+12 v <= 120 u, v >= 0 CMax Z=100 u+120 v 6 u+7 u <=12 8 u+7 v <= 20 u, v are unrestricted DMax Z=100 u+120 u s.t. 6 u+7 v >= 12 8 u+12 v >=20 u, v >= 0

Q18.

Consider the primal problem: maximize z = 5x1 + 12x2 + 4x3 subject to x1 + 2x2 + x3 = 10, 2x1 − x2 + 3x3 = 8, x1, x2, x3 ≥ 0. Its dual is: minimize w = 10y1 + 8y2 subject to y1 + 2y2 ≥ 5, 2y1 − y2 ≥ 12, y1 + 3y2 ≥ 4. Which of the following is correct?

Ay1 ≥ 0, y2 unrestricted By1 ≥ 0, y2 ≥ 0 Cy1 unrestricted, y2 ≥ 0 Dy1 unrestricted, y2 restricted

Q19.

Maximize Z = 6x + 5y subject to 2x − 3y ≤ 5, x + 3y ≤ 11, 4x + y ≤ 15, x, y ≥ 0. The optimal value is

A15 B25 C31.72 D41.44

Q20.

Maximize z = 2x₁ + 3x₂ subject to 2x₁ + x₂ ≤ 4, x₁ + 2x₂ ≤ 5, x₁, x₂ ≥ 0. The optimum value is

A23 B9.5 C13 D8

Q21.

A basic feasible solution of an transportation problem is said to be non- degenerate, if basic feasible solution contains exactly------------- number of individual allocation in-------------- positions.

Am + n + 1, independent Bm + n − 1, independent Cm + n − 1, appropriate Dm − n + 1, independent

Q22.

Maximize z = x₁ + x₂ subject to x₁ + 2x₂ ≤ 2000, x₁ + x₂ ≤ 1500, x₂ ≤ 600, x₁, x₂ ≥ 0. The solution is

Ax₁ = 750, x₂ = 750, z = 1500 Bx₁ = 500, x₂ = 1000, z = 1500 Cx₁ = 1000, x₂ = 500, z = 1500 Dx₁ = 900, x₂ = 600, z = 1500

Q23.

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 optimum solution, then the other problem has no feasible solution at all. Which of the statements is (are) correct?

Aonly (a) and (b) Bonly (a) and (c) Conly (b) and (c) D(a). (b) and (c)

Q24.

For the LPP: maximize Z = 2x₁ − x₂ + 2x₃ subject to 2x₁ + x₂ ≤ 10, x₁ + 2x₂ − 2x₃ ≤ 20, x₁ + 2x₃ ≤ 5, x₁, x₂, x₃ ≥ 0, what is the solution after the first simplex iteration?

Ax₁ = 5/2, x₂ = 0, x₃ = 0, Z = 5 Bx₁ = 0, x₂ = 0, x₃ = 5/2, Z = 5 Cx₁ = 0, x₂ = 5/2, x₃ = 0, Z = −5 Dx₁ = 0, x₂ = 0, x₃ = 10, Z = 20

Q25.

In PERT/CPM, the merge event represents ___ of two or more events.

Acompletion Bbeginning Csplitting Djoining

Q26.

Use the dual simplex method to solve: Maximize z = −2x₁ − 3x₂ subject to x₁ + x₂ ≥ 2, 2x₁ + x₂ ≤ 10, x₁ + x₂ ≤ 8, x₁, x₂ ≥ 0.

Ax₁ = 2, x₂ = 0 and z = −4 Bx₁ = 2, x₂ = 6 and z = −22 Cx₁ = 0, x₂ = 2 and z = −6 Dx₁ = 6, x₂ = 2 and z = −18

Q27.

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

ASolution: x₁ = 100, x₂ = 0, x₃ = 0 BUnbounded solution CNo solution DSolution: x₁ = 50, x₂ = 70, x₃ = 60

Next: Mock 12 – Mixed Practice (Part 1) →