CBSE Class 12 Mathematics Linear Programming MCQs Set 03

Question. In an LPP, if the objective function \( z = ax + by \) has the same maximum value on two corner points of the feasible region, then the number of points at which \( Z_{max} \) occurs is : 
(a) 0
(b) 2
(c) finite
(d) infinite
Answer: (d) infinite

Question. The graph of the inequality \( 2x + 3y > 6 \) is : 
(a) half plane that contains the origin
(b) half plane that neither contains the origin nor the points of the line \( 2x + 3y = 6 \)
(c) whole XOY–plane excluding the points on the line \( 2x + 3y = 6 \)
(d) entire XOY plane.
Answer: (b) half plane that neither contains the origin nor the points of the line \( 2x + 3y = 6 \)

Question. Which of the following types of problems cannot be solved by linear programming methods?
(a) Transportation problem
(b) Manufacturing problems
(c) Traffic signal control
(d) Diet problems
Answer: (c) Traffic signal control

Question. The optimal value of the objective function is attained at the points:
(a) Corner points of the feasible region
(b) Any point of the feasible region
(c) on x-axis
(d) on y-axis
Answer: (a) Corner points of the feasible region

Question. An optimisation problem may involve finding :
(a) maximum profit
(b) minimum cost
(c) minimum use of resources
(d) All of the options
Answer: (d) All of the options

Question. The condition \( x \ge 0, y \ge 0 \) are called :
(a) restrictions only
(b) negative restrictions
(c) non-negative restrictions
(d) None of the options
Answer: (c) non-negative restrictions

Question. In which of the following problems (s), linear programming can be used :
(a) manufacturing problems
(b) diet problems
(c) transportation problems
(d) All of the options
Answer: (d) All of the options

Question. Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let \( F = 4x + 6y \) be the objective function. The minimum value of F occurs at :
(a) (0, 2) only
(b) (3, 0) only
(c) the mid-point on the line segment joining the points (0, 2) and (3, 0) only
(d) any point on the line segment joining the points (0, 2) and (3, 0)
Answer: (d) any point on the line segment joining the points (0, 2) and (3, 0)

Question. The corner points of the feasible region determined by the following system of linear inequalities \( 2x + y \le 10, x + 3y \le 15, x, y \ge 0 \) are (0, 0), (5, 0), (3, 4) and (0, 5). Let \( Z = px + qy \), where \( p, q > 0 \). Condition on p and q, so that the maximum of Z occurs at both (3, 4) and (0, 5) is :
(a) \( p = q \)
(b) \( p = 2q \)
(c) \( p = 3q \)
(d) \( q = 3p \)
Answer: (d) \( q = 3p \)

Question. The variable x and y in a linear programming problem are called :
(a) decision variables
(b) linear variables
(c) optimal variables
(d) None of the options
Answer: (a) decision variables

Question. The linear inequalities or equations or restrictions on the variables of a linear programming problem are called :
(a) linear relations
(b) constraints
(c) functions
(d) objective functions
Answer: (b) constraints

Question. The objective function of an LPP is :
(a) a constraint
(b) a function to be optimised
(c) a relation between the variables
(d) None of the options
Answer: (b) a function to be optimised

Question. Which of the term is not used in a linear programming problem ?
(a) Optimal solution
(b) Feasible solution
(c) Concave region
(d) Objective function
Answer: (c) Concave region

Question. Which of the following sets are not convex ?
(a) \( (x, y) : 2x + 5y < 7 \)
(b) \( (x, y) : x^2 + y^2 \le y \)
(c) \( (x|x|) = 5 \)
(d) \( (x, y) : 3x^2 + 2y^2 \le 6 \)
Answer: (c) \( (x|x|) = 5 \)

Question. The optimal value of the objective function is attained at the point is :
(a) given by intersection of inequations with axes only
(b) given by intersection of inequations with X-axis only
(c) given by corner points of the feasible region
(d) None of the options
Answer: (c) given by corner points of the feasible region

Question. Refer to Question 18. (Maximum value of Z + Minimum value of Z) is :
(a) 13
(b) 1
(c) – 13
(d) – 17
Answer: (d) – 17

Question. Feasible region in the set of points which satisfy :
(a) The objective functions
(b) Some the given functions
(c) All of the given constraints
(d) None of the options
Answer: (c) All of the given constraints

Question. The region of feasible solution in LPP grapline method is called.
(a) Infeasible region
(b) unbounded region
(c) Infinite region
(d) feasible region
Answer: (d) feasible region

Question. In equation \( 3x - y \ge 3 \) and \( 4x - 4y > 4 \) :
(a) Have solution for positive x and y
(b) Have no solution for positive x and y
(c) Have solution for all x
(d) Have solution for all y
Answer: (a) Have solution for positive x and y

Question. The corner point of the feasible region determined by the system of linear constraints are (0, 0), (0, 30), (20, 40), (60, 20), (50, 0). The objective function is \( Z = 4x + 3y \). Compare the quantity in Column A and Column B :
Col. A: Max Z
Col. B: 340
(a) Quantity in column A is greater
(b) Quantity in column B is greater
(c) Two quantities are equal
(d) Relationship cannot be determined and the basis of information supplied
Answer: (b) Quantity in column B is greater

Question. In a LPP, the objective function is always :
(a) cubic
(b) quadratic
(c) linear
(d) constant
Answer: (c) linear

Question. Maximise, \( Z = - x + 2y \), subject to the constraints \( x \ge 3, x + y \ge 5, x + 2y \ge 6, y \ge 0 \) :
(a) Max, Z = 12 at (2, 6)
(b) Z has no max. value
(c) Max., Z = 10 at (2, 6)
(d) Max., Z = 14 at (2, 6)
Answer: (b) Z has no max. value

Question. A linear programming problem is one that is concerned with :
(a) finding the upper limits of a linear function of several variables
(b) finding the lower limit of a linear function of several variables
(c) finding the limiting values of a linear function of several variables
(d) finding the optimal value (max or min) of a linear function of several variables
Answer: (d) finding the optimal value (max or min) of a linear function of several variables

Question. Objective function is expressed in terms of the ............................. .
(a) Numbers
(b) Symbols
(c) Decision variables
(d) None of the options
Answer: (c) Decision variables

Question. In a transportation problem, with 4 supply point and 5 demand points how many number of constraints required in its formulation :
(a) 20
(b) 1
(c) 0
(d) 9
Answer: (d) 9

Question. The position of points O(0, 0) and P(2, – 2) in the region of graph of inequation \( 2x - 3y < 5 \), will be :
(a) O inside and P outside
(b) O and P both inside
(c) O and P both outside
(d) O outside and P inside
Answer: (a) O inside and P outside

Question. Let \( Z = ax + by \) is a linear objective function variables x and y are called ................... variables.
(a) Independent
(b) Continuous
(c) Decision
(d) Dependent
Answer: (c) Decision

Question. Infeasibility means that the number of solutions to the linear programming models that satisfies all constraints is :
(a) At least
(b) An infinite number
(c) Zero
(d) At least 2
Answer: (c) Zero

Assertion and Reason Questions

Choose the correct option :
(a) Both (A) and (B) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : The region represented by the set \( \{ (x, y) : 4 \le x^2 + y^2 \le 9 \} \) is a convex set.
Reason (R) : The set \( \{ (x, y) : 4 \le x^2 + y^2 \le 9 \} \) represents the region between two concentric circles of radii 2 and 3.
Answer: (d) A is false but R is true.

Question. Assertion (A) : If a L.P.P. admits two optimal solutions then it has infinitely many optimal solutions.
Reason (R) : If the value of the objective function of a LPP is same at two corners then it is same at every point on the line joining two corner points.
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Case Study 1

Corner points of the feasible region for an LPP are (0, 3), (5, 0), (3, 2), (2, 3). Let \( z = 4x - 6y \) be the objective function.

Question. The minimum value of z occurs at :
(a) (6, 8)
(b) (5, 0)
(c) (0, 3)
(d) (0, 8)
Answer: (d) (0, 8)

Question. Maximum value of z occurs at :
(a) (5, 0)
(b) (0, 8)
(c) (0, 3)
(d) (6, 8)
Answer: (a) (5, 0)

Question. \( \text{Max } z - \text{min } z = \)
(a) 58
(b) 68
(c) 78
(d) 88
Answer: (b) 68

Question. The feasible solution of LPP belongs to :
(a) First and second quadrant
(b) First and third quadrant
(c) Only second quadrant
(d) Only first quadrant
Answer: (d) Only first quadrant

Case Study 2

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 equation or inequalities.

Question. The optimal value of the objective function is attained at the points :
(a) x-axis
(b) on y-axis
(c) which are corner points of the feasible region
(d) none of these
Answer: (c) which are corner points of the feasible region

Question. The graph of the inequality \( 3x + 4y \le 12 \) is
(a) Half plane that contains the origin
(b) Half plane that neither contains the origin nor the points one the line \( 3x + 4y = 12 \)
(c) Whole XOY plane excluding the points on the line \( 3x + 4y = 12 \)
(d) None of these
Answer: (a) Half plane that contains the origin

Question. 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 :
(a) \( p = q \)
(b) \( p = 2q \)
(c) \( q = 2p \)
(d) \( q = 3p \)
Answer: (d) \( q = 3p \)

Question. The corner points of 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: Max Z
Column B: 325
(a) The quantity in column A is greater
(b) The quantity in column B is greater
(c) The two quantities are equal
(d) None of these
Answer: (b) The quantity in column B is greater