CBSE Class 12 Mathematics Linear Programming MCQs Set 02

Question. The feasible solution of a L.P.P. belongs to
(a) Only first quadrant
(b) First and third quadrant
(c) Second quadrant
(d) Any quadrant
Answer: d

Question. Region represented by x ≥ 0, y ≥ 0 is
(a) first quadrant
(b) second quadrant
(c) third quadrant
(d) fourth quadrant
Answer: a

Question. The number of corner points of the L.P.P.
Max Z = 20x + 3y subject to the constraints x + y ≤ 5, 2x + 3y ≤ 12, x ≥ 0, y ≥ 0 are
(a) 4
(b) 3
(c) 2
(d) 1
Answer: a

Question. Objective function of a L.P.P. is
(a) a constant
(b) a function to be optimised
(c) a relation between the variables
(d) None of these
Answer: b

Question. For the constraint of a linear optimizing function z = x1 + x2, given by x1+ x2 ≤1, 3x1+ x2 ≥ 3 and x1, x2 ≥ 0,
(a) There are two feasible regions
(b) There are infinite feasible regions
(c) There is no feasible region
(d) None of these.
Answer: c

Question. The maximum value of z = 5x + 2y, subject to the constraints x + y ≤ 7, x + 2y ≤ 10, x, y ≥ 0 is
(a) 10
(b) 26
(c) 35
(d) 70
Answer: c

Question. The maximum value of z = 2x + 5y subject to the constraints 2x + 5y ≤ 10, x + 2y ≥1, x – y ≤ 4, x ≥ y ≥ 0, occurs at
(a) exactly one point
(b) exactly two points
(c) infinitely many points
(d) None of these
Answer: c

Question. If a point (h, k) satisfies an inequation ax + by ≥ 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 these
Answer: b

Question. Maximize Z = 4x + 6y, subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0, is
(a) 16 at (4,0)
(b) 24 at (0, 4)
(c) 24 at (6, 0)
(d) 36 at (0, 6)
Answer: d

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

Question. A printing company prints two types of magazines A and B.
The company earns `10 and `15 on each magazine A and B respectively. These are processed on three machines I, II & III and total time in hours available per week on each machine is as follows: 56
The number of constraints is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: c

Question. Inequation y – x ≤ 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 these
Answer: a

Question. Which of the following cannot be considered as the objective function of a linear programming problem?
(a) Maximize z = 3x + 2y
(b) Minimize z = 6x + 7y + 9z
(c) Maximize z = 2x
(d) Minimize z = x2 + 2xy + y²
Answer: d

Question. Maximize Z = 3x + 5y, subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y, ≤ 9, x ≥ 0, y ≥ 0, is
(a) 20 at (1, 0)
(b) 30 at (0, 6)
(c) 37 at (4, 5)
(d) 33 at (6, 3)
Answer: c

Question. Shamli wants to invest `50,000 in saving certificates and PPE. She wants to invest atleast `15,000 in saving certificates and at least `20,000 in PPF. The rate of interest on saving certificates is 8% p.a. and that on PPF is 9% p.a.
Formulation of the above problem as LPP to determine maximum yearly income, is
(a) Maximize Z = 0.08x + 0.09y Subject to, x + y ≤ 50,000, x ≥ 15000, y ≥ 20,000
(b) Maximize Z = 0.08x + 0.09y Subject to, x + y ≤ 50,000, x ≥ 15000, y ≤ 20,000
(c) Maximize Z = 0.08x + 0.09y Subject to, x + y ≤ 50,000, x ≤ 15000, y ≥ 20,000
(d) Maximize Z = 0.08x + 0.09y Subject to, x + y ≤ 50,000, x ≤ 15000, y ≤ 20,000
Answer: a

Question. A company manufactures two types of products A and B.
The storage capacity of its godown is 100 units. Total investment amount is ₹ 30,000. The cost price of A and B are ₹ 400 and ₹ 900 respectively. Suppose all the products have sold and per unit profit is ₹ 100 and ₹ 120 through A and B respectively. If x units of A and y units of B be produced, then two linear constraints and iso-profit line are respectively
(a) x + y =100; 4x + 9y = 300, 100x + 120y = c
(b) x + y ≤100; 4x + 9y ≤ 300, x + 2y = c
(c) x + y ≤100; 4x + 9y ≤ 300,100x +120y = c
(d) x + y ≤100; 9x + 4y ≤ 300, x + 2y = c
Answer: c

Question. A linear programming of linear functions deals with
(a) Minimizing
(b) Optimizing
(c) Maximizing
(d) None of these
Answer: b

Question. The lines 5x + 4y ≥ 20, x ≤ 6, y ≤ 4 form
(a) A square
(b) A rhombus
(c) A triangle
(d) A quadrilateral
Answer: d

Question. Corner points of feasible region of inequalities gives
(a) optional solution of L.P.P.
(b) objective function
(c) constraints.
(d) linear assumption
Answer: a

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 of 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

Question. The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is
(a) 90
(b) 120
(c) 96
(d) 240
Answer: b

Question. Z = 6x + 21y, subject to x + 2y ≥ 3, x + 4y ≥ 4, 3x + y ≥ 3, x ≥ 0, y ≥ 0. The minimum value of Z occurs at
(a) (4, 0)
(b) (28, 8)
(c) (2 , 1/2)
(d) (0, 3)
Answer: c 

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 = 2 q
(c) q = 2 p
(d) q = 3 p
Answer: d

Question. A wholesale merchant wants to start the business of cereal with ₹ 24000. Wheat is ₹ 400 per quintal and rice is ₹ 600 per quintal. He has capacity to store 200 quintal cereal. He earns the profit ₹ 25 per quintal on wheat and ₹ 40 per quintal on rice. If he store x quintal rice and y quintal wheat, then for maximum profit, the objective function is
(a) 25 x + 40 y
(b) 40x + 25 y
(c) 400x + 600y
(d) 400/40 x + 600/5 y
Answer: b

Question. The maximum value of P = x + 3y such that 2x + y ≤ 20, x + 2y ≤ 20, x ≥0, y ≥0 is
(a) 10
(b) 60
(c) 30
(d) None of these
Answer: c

Question. The maximum value of z = 4x + 2y subject to constraints 2x + 3y ≤ 18, x + y ≥10 and x, y ≥ 0, is
(a) 36
(b) 40
(c) 20
(d) None
Answer: d

Question. The no. of convex polygon formed bounding the feasible region of the L.P.P. Max. Z = 30x + 60y subject to the constraints 5x + 2y ≤ 10, x + y ≤ 4, x ≥ 0, y ≥ 0 are
(a) 2
(b) 3
(c) 4
(d) 1
Answer: d

Question. The optimal value of the objective function is attained at the points
(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 these.
Answer: c

Question. The solution set of the following system of inequations: x + 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥1, is
(a) bounded region
(b) unbounded region
(c) only one point
(d) empty set
Answer: d

Question. L.P.P is a process of finding
(a) Maximum value of objective function
(b) Minimum value of objective function
(c) Optimum value of objective function
(d) None of these
Answer: c

Question. Children have been invited to a birthday party. It is necessary to give them return gifts. For the purpose, it was decided that they would be given pens and pencils in a bag. It was also decided that the number of items in a bag would be atleast 5. If the cost of a pen is `10 and cost of a pencil is `5, minimize the cost of a bag containing pens and pencils. Formulation of LPP for this problem is
(a) Minimize C = 5x + 10y subject to x + y ≤ 10, x ≥ 0, y ≥ 0
(b) Minimize C = 5x + 10y subject to x + y ≥10, x ≥ 0, y ≥ 0
(c) Minimize C = 5x + 10y subject to x + y ≥ 5, x ≥ 0, y ≥ 0
(d) Minimize C = 5x + 10y subject to x + y ≤ 5, x ≥ 0, y ≥ 0
Answer: a

Question. The solution set of the inequation \( 3x + 2y > 3 \) is
(a) half plane not containing the origin
(b) half plane containing the origin
(c) the point being on the line \( 3x + 2y = 3 \)
(d) None of the options
Answer: (a) half plane not containing the origin
Solution: Since the co-ordinate of origin \( (0, 0) \) does not satisfy the inequation \( 3x + 2y > 3 \) i.e.,
\( 3 \times 0 + 2 \times 0 = 0 + 0 = 0 \ngtr 3 \)
Therefore, answer is half plane not containing the origin.

Question. The optimal value of the objective function is attained at the points
(a) given by intersection of inequation with y-axis 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.
Solution: The fact given under (c) is used to find out optimal value of the objective function, when feasible region is known.

Question. If the constraints in a linear programming problem are changed
(a) solution is not defined
(b) the objective function has to be modified
(c) the problems is to be re-evaluated
(d) None of the options
Answer: (c) the problems is to be re-evaluated
Solution: If constraint in LPP changed then feasible region will also change and so solution, which is also defined, will changed although objective function remain same.
Therefore, problem is to be re-evaluated.

Question. Which of the following statement is correct?
(a) Every LPP admits an optimal solution.
(b) Every LPP admits unique optimal solution.
(c) If a LPP gives two optimal solutions it has infinite number of solutions.
(d) None of the options
Answer: (c) If a LPP gives two optimal solutions it has infinite number of solutions.
Solution: (a) and (b) are incorrect because optimal value may exist or may not exist also they may unique or may not be.
Only (c) is correct because two optimal means there exist two corner points in feasible region having same value of objective function, which are maximum or minimum in this case all points of feasible region lying on the line joining these points are optimal.

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
 

Question. Assertion (A) : The maximum value of \( Z = 5x + 3y \), satisfying the conditions \( x \geq 0 \), \( y \geq 0 \) and \( 5x + 2y \leq 10 \), is 15.
Reason (R) : A feasible region may be bounded or unbounded.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: We have, corner points \( (0, 0) \), \( (2, 0) \), \( (0, 5) \).
\( \therefore Z_{\text{max}} = 5 \times 0 + 3 \times 5 = 15 \) at \( (0, 5) \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : The maximum value of \( Z = x + 3y \). Such that \( 2x + y \leq 20 \), \( x + y \leq 20 \), \( x, y \geq 0 \) is 30.
Reason (R) : The variables that enter into the problem are called decision variables.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: We have, corner points be \( (0, 0) \), \( (10, 0) \), \( \left(\frac{20}{3}, \frac{20}{3}\right) \), \( (0, 10) \).
\( \therefore Z_{\text{max}} = x + 3y = 0 + 3 \times 10 = 30 \) at \( (0, 10) \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : Shaded region represented by \( 2x + 5y \geq 80 \), \( x + y \leq 20 \), \( x \geq 0 \), \( y \geq 0 \) is
Reason (R) : A region or a set of points is said to be convex if the line joining any two of its points lies completely in the region.
Answer: (d) A is false but R is true.
Solution: Clearly, Assertion (A) is false and Reason (R) is true.
Hence, (d) is the correct option.