CBSE Class 12 Mathematics Linear Programming MCQs Set 06

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 \( XY \)-plane excluding the points on the line \( 2x + 3y = 6 \).
(d) entire \( XY \)-plane.
Answer: (b) half plane that neither contains the origin nor the points of the line \( 2x + 3y = 6 \).

Question. In 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 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 \( p, q > 0 \). Then, the 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 the feasible region in the graphical representation of a linear programming problem are \( (2, 72), (15, 20) \) and \( (40, 15) \). If \( Z = 18x + 9y \) be the objective function, then
(a) \( Z \) is maximum at \( (2, 72) \) and minimum at \( (15, 20) \).
(b) \( Z \) is maximum at \( (15, 20) \) and minimum at \( (40, 15) \).
(c) \( Z \) is maximum at \( (40, 15) \) and minimum at \( (15, 20) \).
(d) \( Z \) is maximum at \( (40, 15) \) and minimum at \( (2, 72) \).
Answer: (c) \( Z \) is maximum at \( (40, 15) \) and minimum at \( (15, 20) \)

Question. The number of corner points of the feasible region determined by the constraints \( x - y \ge 0, 2y \le x + 2 \) and \( x \ge 0, y \ge 0 \) is
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (a) 2

Question. The objective function \( Z = ax + by \) of LPP, if its maximum value 42 at \( (4, 6) \) and minimum value 19 at \( (3, 2) \). Which of the following is true?
(a) \( a = 9 \) and \( b = 1 \)
(b) \( a = 5 \) and \( b = 2 \)
(c) \( a = 3 \) and \( b = 5 \)
(d) \( a = 5 \) and \( b = 3 \)
Answer: (c) \( a = 3 \) and \( b = 5 \)

Question. The corner points of the feasible region of linear programming problem are \( (0, 4), (8, 0) \) and \( \left(\frac{20}{3}, \frac{4}{3}\right) \). If \( Z = 30x + 24y \) is the objective function, then (Maximum value of \( Z \) - Minimum value of \( Z \)) is equal to
(a) 144
(b) 96
(c) 120
(d) 136
Answer: (a) 144

Question. The feasible region of a linear programming problem is shown in the figure below.
Which of the following are the possible constraints?
(a) \( x + 2y \ge 4, x + y \le 3 \) and \( x \ge 0, y \ge 0 \)
(b) \( x + 2y \le 4, x + y \le 3 \) and \( x \ge 0, y \ge 0 \)
(c) \( x + 2y \ge 4, x + y \ge 3 \) and \( x \ge 0, y \ge 0 \)
(d) \( x + 2y \ge 4, x + y \ge 3 \) and \( x \le 0, y \le 0 \)
Answer: (c) \( x + 2y \ge 4, x + y \ge 3 \) and \( x \ge 0, y \ge 0 \)

Question. The solution set of the inequality \( 3x + 5y < 4 \) is
(a) an open half plane not containing the origin,
(b) an open half plane containing the origin,
(c) the whole \( XY \)-plane not containing the line \( 3x + 5y = 4 \).
(d) a closed half plane containing the origin.
Answer: (b) an open half plane containing the origin,

Question. The corner points of the shaded unbounded feasible region of an LPP are \( (0,4), (0.6,1.6) \) and \( (3,0) \) as shown in the figure. The minimum value of the objective function \( Z = 4x + 6y \) occurs at
(a) \( (0.6, 1.6) \) only.
(b) \( (3, 0) \) only.
(c) \( (0.6, 1.6) \) and \( (3, 0) \) only.
(d) at every point of the line segment joining the points \( (0.6, 1.6) \) and \( (3, 0) \).
Answer: (d) at every point of the line segment joining the points \( (0.6, 1.6) \) and \( (3, 0) \).

Question. A linear programming problem is as follows
Minimise \( Z = 30x + 50y \)
Subject to the constraints,
\( 3x + 5y \ge 15, 2x + 3y \le 18 \) and \( x \ge 0, y \ge 0 \).
In the feasible region, the minimum value of \( Z \) occurs at
(a) a unique point
(b) no point
(c) infinitely many points
(d) two points only
Answer: (c) infinitely many points

Question. In a linear programming problem, the constraints on the decision variables \( x \) and \( y \) are \( x - 3y \ge 0, y \ge 0, 0 \le x \le 3 \). The feasible region
(a) is not in the first quadrant,
(b) is bounded in the first quadrant,
(c) is unbounded in the first quadrant,
(d) Does not exist.
Answer: (b) is bounded in the first quadrant,

Question. In the given graph, the feasible region for a LPP is shaded. The objective function \( Z = 2x - 3y \) will be minimum at
(a) \( (4, 10) \)
(b) \( (6, 8) \)
(c) \( (0, 8) \)
(d) \( (6, 5) \)
Answer: (c) \( (0, 8) \)

Question. The value of objective function is maximum under linear constraints
(a) At the centre feasible region
(b) At \( (0,0) \)
(c) At any vertex of feasible region
(d) The vertex which is maximum distance from \( (0,0) \).
Answer: (c) At any vertex of feasible region

Question. The point which does not lie in the half plane \( 2x + 3y - 12 \le 0 \) is
(a) \( (1, 2) \)
(b) \( (2, 1) \)
(c) \( (2, 3) \)
(d) \( (-3, 2) \)
Answer: (c) \( (2, 3) \)

Assertion-Reason Based Questions

Assertion (A): The points \( (10, 50) \), \( (0, 60) \) or \( (20, 0) \) are feasible solutions.
Reason (R): Points within and on the boundary of the feasible region represent feasible solutions of the constraints.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The objective function, \( Z = -50x + 20y \) subject to the constraints, \( 2x - y \ge -5 \), \( 3x + y \ge 3 \), \( 2x - 3y \le 12 \) and \( x \ge 0, y \ge 0 \) in the feasible region has no minimum value.
Reason (R): If the open half plane determined by \( ax + by < m \), where \( m \) is the minimum value of \( Z \), has a point in common with feasible region, then \( Z \) has no minimum value.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The maximum value of \( Z = 11x + 7y \) subject to the constraints, \( 2x + y \le 6 \), \( x \le 2 \) and \( x \ge 0, y \ge 0 \) in the feasible region occurs at the corner point \( (0, 6) \).
Reason (R): If the feasible region of the given LPP is bounded, then the maximum and minimum value of the objective function occurs at corner points.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): If LPP attains its maximum value at two corner points of the feasible region, then it attains maximum value at infinitely many points.
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.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Let the feasible region of a LPP with objective function \( Z = ax + by \). \( M \) and \( m \) are maximum and minimum value of \( Z \).
Assertion (A): \( M \) is the maximum value of \( Z \), if the open half plane determined by \( ax + by > M \) has no point in common in the feasible region. Otherwise \( Z \) has no maximum value.
Reason (R): \( m \) is the minimum value of \( Z \), if the open half plane determined by \( ax + by < m \) has no point in common in the feasible region. Otherwise \( Z \) has no minimum value.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): Every point of the feasible region of a LPP is an optimal solution.
Reason (R): The optimal solution for a LPP exists only at one or more corner point(s) of the feasible region.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A): If a LPP 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 segment joining the two corner points.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The maximum value of \( Z = 3x + 4y \), subject to the constraints \( x \ge 0, y \ge 0 \) and \( x + y \le 1 \), is 4.
Reason (R): A feasible region may be bounded or unbounded.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): If the corner points of the feasible region of LPP are \( A(0, 3) \), \( B(3, 2) \) and \( C(0, 5) \), then the minimum value of \( Z = 11x + 7y \) is 21.
Reason (R): A feasible region is always bounded.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (c) A is correct; R is incorrect.

Question. Assertion (A): The corner points of the feasible region determined by the set of constraints are \( A(0, 5) \), \( B(3, 5) \), \( C(5, 0) \) and \( D(4, 1) \) and the objective function \( Z = ax + 2by \), where \( a, b > 0 \). The condition on \( a \) and \( b \) such that the maximum \( Z \) occurs at \( B \) and \( D \) is \( a - 8b = 0 \).
Reason (R): A feasible region is always unbounded.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (c) A is correct; R is incorrect.