CBSE Class 12 Mathematics Linear Programming MCQs Set 04

Question. The solution set of the inequation \( 2x + y > 5 \), is
(a) Half plane that contains the origin
(b) Open half plane not containing the origin
(c) Whole \( xy \)-plane except the points lying on the line \( 2x + y = 5 \)
(d) None of the options
Answer: (b) Open half plane not containing the origin

Question. Inequation \( y - x \le 0 \) represents
(a) The half plane that contains the positive x-axis
(b) Closed half plane above the line \( y = x \) which contains positive y-axis
(c) Half plane that contains the negative x-axis
(d) None of the options
Answer: (a) The half plane that contains the positive x-axis

Question. If a point \( (h, k) \) satisfies an inequation \( ax + by \ge 4 \), then the half plane represented by the inequation is
(a) The half plane containing the point \( (h, k) \) but excluding the points on \( ax + by = 4 \)
(b) The half plane containing the point \( (h, k) \) and the points on \( ax + by = 4 \)
(c) Whole xy-plane
(d) None of the options
Answer: (b) The half plane containing the point \( (h, k) \) and the points on \( ax + by = 4 \)

Question. If the constraints in a linear programming problem are changed
(a) The problem is to be re-evaluated
(b) Solution is not defined
(c) The objective function has to be modified
(d) The change in constraints is ignored.
Answer: (a) The problem is to be re-evaluated

Question. The optimal value of the objective function is attained at the points
(a) Given by intersection of inequations with the axes only
(b) Given by intersection of inequation 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. Let \( X_1 \) and \( X_2 \) are optimal solutions of a LPP, then
(a) \( X = \lambda X_1 + (1 - \lambda) X_2, \lambda \in R \) is also an optimal solution
(b) \( X = \lambda X_1 + (1 - \lambda) X_2, 0 \le \lambda \le 1 \) gives an optimal solution
(c) \( X = \lambda X_1 + (1 + \lambda) X_2, 0 \le \lambda \le 1 \) gives an optimal solution
(d) \( X = \lambda X_1 + (1 + \lambda) X_2, \lambda \in R \) gives an optimal solution
Answer: (b) \( X = \lambda X_1 + (1 - \lambda) X_2, 0 \le \lambda \le 1 \) gives an optimal solution

Question. The position of points \( O(0, 0) \) and \( P(2, -2) \) in the region of graph of inequations \( 2x - 3y < 5 \), 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. The solution set of constraints \( x + 2y \ge 11, 3x + 4y \le 30, 2x + 5y \le 30, x \ge 0, y \ge 0 \) includes the point
(a) \( (2, 3) \)
(b) \( (1, 1) \)
(c) \( (3, 4) \)
(d) \( (4, 3) \)
Answer: (c) \( (3, 4) \)

Question. The solution set of linear constraints \( x - 2y \ge 0, 2x - y \le -2 \) and \( x, y \ge 0 \), is
(a) \( \left(-\frac{4}{3}, -\frac{2}{3}\right) \)
(b) \( (1, 1) \)
(c) \( \left(0, \frac{2}{3}\right) \)
(d) \( (0, 2) \)
Answer: (a) \( \left(-\frac{4}{3}, -\frac{2}{3}\right) \)

Question. For the constraints of a L.P. problem given by \( x_1 + 2x_2 \le 2000, x_1 + x_2 \le 1500, x_2 \le 600 \) and \( x_1, x_2 \ge 0 \), which one of the following points does not lie in the positive bounded region
(a) \( (1000, 0) \)
(b) \( (0, 500) \)
(c) \( (2, 0) \)
(d) \( (2000, 0) \)
Answer: (d) \( (2000, 0) \)

Question. The graph of \( x \le 2 \) and \( y \ge 2 \) will be situated in the
(a) First and second quadrant
(b) Second and third quadrant
(c) First and third quadrant
(d) Third and fourth quadrant
Answer: (a) First and second quadrant

Question. The true statements for the graph of inequations \( 3x + 2y \le 6 \) and \( 6x + 4y \ge 20 \), is
(a) Both graphs are disjoint
(b) Both do not contain origin
(c) Both contain point \( (1, 1) \)
(d) None of the options
Answer: (a) Both graphs are disjoint

Question. In which quadrant, the bounded region for inequations \( x + y \le 1 \) and \( x - y \le 1 \) is situated
(a) I, II
(b) I, III
(c) II, III
(d) All the four quadrants
Answer: (d) All the four quadrants

Question. The region represented by the inequation system \( x, y \ge 0, y \le 6, x + y \le 3 \), is
(a) Unbounded in first quadrant
(b) Unbounded in first and second quadrants
(c) Bounded in first quadrant
(d) None of the options
Answer: (c) Bounded in first quadrant

Question. If the number of available constraints is 3 and the number of parameters to be optimized is 4, then
(a) The objective function can be optimized
(b) The constraints are short in number
(c) The solution is problem oriented
(d) None of the options
Answer: (b) The constraints are short in number

Question. The intermediate solutions of constraints must be checked by substituting them back into
(a) Object function
(b) Constraint equations
(c) Not required
(d) None of the options
Answer: (b) Constraint equations

Question. A basic solution is called non-degenerate, if
(a) All these basic variables are zero
(b) None of the basic variables is zero
(c) At least one of the basic variable is zero
(d) None of the options
Answer: (b) None of the basic variables is zero

Question. Objective function of a L.P.P. is
(a) A constraint
(b) A function to be optimized
(c) A relation between the variables
(d) None of the options
Answer: (b) A function to be optimized

Question. "The maximum or the minimum of the objective function occurs only at the corner points of the feasible region". This theorem is known as Fundamental Theorem of
(a) Algebra
(b) Arithmetic
(c) Calculus
(d) Extreme points
Answer: (d) Extreme points

Question. Which of the terms is not used in a linear programming problem
(a) Slack variable
(b) Objective function
(c) Concave region
(d) Feasible region
Answer: (c) Concave region

Question. Which of the following is not true for linear programming problems
(a) A slack variable is a variable added to the left hand side of a less than or equal to constraint to convert it into an equality
(b) A surplus variable is a variable subtracted from the left hand side of a greater than or equal to constraint to convert it into an equality
(c) A basic solution which is also in the feasible region is called a basic feasible solution
(d) A column in the simplex tableau that contains all of the variables in the solution is called pivot or key column
Answer: (d) A column in the simplex tableau that contains all of the variables in the solution is called pivot or key column

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

Question. Which of the following sets are not convex
(a) \( \{(x, y) \mid 3 \le x^2 + y^2 \le 5\} \)
(b) \( \{(x, y) \mid 3x^2 + 2y^2 \le 6\} \)
(c) \( \{(x, y) \mid y^2 \le x\} \)
(d) \( \{(x, y) \mid x \ge 2, x \le 3\} \)
Answer: (a) \( \{(x, y) \mid 3 \le x^2 + y^2 \le 5\} \)

Question. Which of the following sets are convex
(a) \( \{(x, y) \mid x^2 + y^2 \ge 1\} \)
(b) \( \{(x, y) \mid y^2 \ge x\} \)
(c) \( \{(x, y) \mid 3x^2 + 4y^2 \ge 5\} \)
(d) \( \{(x, y) \mid y \ge 2, y \le 4\} \)
Answer: (d) \( \{(x, y) \mid y \ge 2, y \le 4\} \)

Question. For the following shaded area (boundary lines \( x + 2y = 8, x - y = 1, 2x + y = 2 \)), the linear constraints except \( x \ge 0 \) and \( y \ge 0 \), are
(a) \( 2x + y \le 2, x - y \le 1, x + 2y \le 8 \)
(b) \( 2x + y \ge 2, x - y \le 1, x + 2y \le 8 \)
(c) \( 2x + y \ge 2, x - y \ge 1, x + 2y \le 8 \)
(d) \( 2x + y \ge 2, x - y \ge 1, x + 2y \ge 8 \)
Answer: (b) \( 2x + y \ge 2, x - y \le 1, x + 2y \le 8 \)

Question. For the following feasible region (boundary lines \( 2x + y = 600, x = 250, y = 350 \)), the linear constraints except \( x \ge 0 \) and \( y \ge 0 \), are
(a) \( x \ge 250, y \le 350, 2x + y = 600 \)
(b) \( x \le 250, y \le 350, 2x + y = 600 \)
(c) \( x \le 250, y \le 350, 2x + y \ge 600 \)
(d) \( x \le 250, y \le 350, 2x + y \le 600 \)
Answer: (d) \( x \le 250, y \le 350, 2x + y \le 600 \)

Question. Which of the following is not a vertex of the positive region bounded by the inequalities \( 2x + 3y \le 6, 5x + 3y \le 15 \) and \( x, y \ge 0 \),
(a) \( (0, 2) \)
(b) \( (0, 0) \)
(c) \( (3, 0) \)
(d) None of the options
Answer: (d) None of the options

Question. The vertex of common graph of inequalities \( 2x + y \ge 2 \) and \( x - y \le 3 \), is
(a) \( (0, 0) \)
(b) \( \left(\frac{5}{3}, -\frac{4}{3}\right) \)
(c) \( \left(\frac{5}{3}, \frac{4}{3}\right) \)
(d) \( \left(-\frac{5}{3}, -\frac{4}{3}\right) \)
Answer: (b) \( \left(\frac{5}{3}, -\frac{4}{3}\right) \)

Question. A vertex of bounded region of inequalities \( x \ge 0, x + 2y \ge 0 \) and \( 2x + y \le 4, y \ge 0 \) is
(a) \( (1, 1) \)
(b) \( (0, 1) \)
(c) \( (3, 0) \)
(d) \( (0, 0) \)
Answer: (d) \( (0, 0) \)

Question. A vertex of the linear inequalities \( 2x + 3y \le 6, x + 4y \le 4 \) and \( x, y \ge 0 \), is
(a) \( (1, 0) \)
(b) \( (1, 1) \)
(c) \( \left(\frac{12}{5}, \frac{2}{5}\right) \)
(d) \( \left(\frac{2}{5}, \frac{12}{5}\right) \)
Answer: (c) \( \left(\frac{12}{5}, \frac{2}{5}\right) \)

Question. Consider the inequalities \( x_1 + x_2 \le 3, 2x_1 + 5x_2 \ge 10, x_1, x_2 \ge 0 \), which of the following points lies in the feasible region
(a) \( (2, 2) \)
(b) \( (1, 2) \)
(c) \( (2, 1) \)
(d) \( (4, 2) \)
Answer: (b) \( (1, 2) \)

Question. The region represented by the inequalities \( x + y \ge 2, 2x + y \ge 6, y \ge 2, x + y \le 10, x \ge 0, y \ge 0 \) is
(a) Unbounded
(b) A polygon
(c) Exterior of a triangle
(d) None of the options
Answer: (a) Unbounded

Question. A wholesale merchant wants to start the business of cereal with Rs. 24,000. Wheat is Rs. 400 per quintal and rice is Rs. 600 per quintal. He has capacity to store 200 quintal cereal. He earns the profit Rs. 25 per quintal on wheat and Rs. 40 on rice. If he store x quintal rice and y quintal wheat, then for maximum profit the objective function is
(a) \( 25x + 40y \)
(b) \( 40x + 25y \)
(c) \( 400x + 600y \)
(d) \( \frac{400}{40}x + \frac{600}{25}y \)
Answer: (b) \( 40x + 25y \)

Question. A firm produces two types of product A and B. The profit on both is Rs. 2 per item. Every product need processing on machines \( M_1 \) and \( M_2 \). For A, machines \( M_1 \) and \( M_2 \) takes 1 minute and 2 minute respectively and that of for B, machines \( M_1 \) and \( M_2 \) takes the time 1 minute and 1 minute. The machines \( M_1, M_2 \) are not available more than 8 hours and 10 hours any of day respectively. If the products made x of A and y of B, then the linear constraints for the L.P.P. except \( x \ge 0, y \ge 0 \) are
(a) \( x + y \le 480, 2x + y \le 600 \)
(b) \( x + y \le 8, 2x + y \le 10 \)
(c) \( x + y \ge 480, 2x + y \ge 600 \)
(d) \( x + y \ge 8, 2x + y \ge 10 \)
Answer: (a) \( x + y \le 480, 2x + y \le 600 \)

Question. In a test of Mathematics, there are two types of questions to be answered, short answered and long answered. The relevant data are given below:
Time taken to solve: Short answered questions = 5 minutes, Long answered questions = 10 minutes
Marks: Short answered questions = 3, Long answered questions = 5
Number of questions: Short answered questions = 10, Long answered questions = 14
The total marks are 100. Student can solve all the questions. To secure maximum marks, student solve x short answered and y long answered questions in three hours, then the linear constraints except \( x \ge 0, y \ge 0 \), are
(a) \( 5x + 10y \le 180, x \le 10, y \le 14 \)
(b) \( x + 10y \ge 180, x \le 10, y \le 14 \)
(c) \( 5x + 10y \ge 180, x \ge 10, y \ge 14 \)
(d) \( 5x + 10y \le 180, x \ge 10, y \ge 14 \)
Answer: (a) \( 5x + 10y \le 180, x \le 10, y \le 14 \)

Question. A company manufactures two types of telephone sets A and B. The A type telephone set requires 2 hour and B type telephone requires 4 hour to make. The company has 800 work hour per day. 300 telephone can pack in a day. The selling prices of A and B type telephones are Rs. 300 and 400 respectively. For maximum profit company produces x telephones of A type and y telephones of B type. Then except \( x \ge 0 \) and \( y \ge 0 \), linear constraints are
(a) \( x + 2y \le 400; x + y \le 300 \)
(b) \( 2x + y \le 400; x + y \ge 300 \)
(c) \( 2x + y \ge 400; x + y \ge 300 \)
(d) \( x + 2y \ge 400; x + y \le 300 \)
Answer: (a) \( x + 2y \le 400; x + y \le 300 \)

Question. In a factory which produces two products A and B, in manufacturing product A, the machine and the carpenter requires 3 hours each and in manufacturing product B, the machine and carpenter requires 5 hour and 3 hour respectively. The machine and carpenter works at most 80 hour and 50 hour per week respectively. The profit on A and B are Rs. 6 and Rs. 8 respectively. If profit is maximum by manufacturing x and y units of A and B type products respectively, then for the function \( 6x + 8y \), the constraints are
(a) \( x \ge 0, y \ge 0, 5x + 3y \le 80, 3x + 2y \le 50 \)
(b) \( x \ge 0, y \ge 0, 3x + 5y \le 80, 3x + 3y \le 50 \)
(c) \( x \ge 0, y \ge 0, 3x + 5y \ge 80, 2x + 3y \ge 50 \)
(d) \( x \ge 0, y \ge 0, 5x + 3y \ge 80, 3x + 2y \ge 50 \)
Answer: (b) \( x \ge 0, y \ge 0, 3x + 5y \le 80, 3x + 3y \le 50 \)