Linear Programming — CBSE Class 12 board PYQs

The objective function $Z=ax+by$ of an LPP has maximum value 42 at $(4,6)$ and minimum value 19 at $(3,2)$. Which of the following is true?

The corner points of the feasible region of a 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:

The corner points of the feasible region determined by the system of linear inequalities are $(0,0)$, $(4,0)$, $(2,4)$ and $(0,5)$. If the maximum value of $z=ax+by$, where $a,b>0$, occurs at both $(2,4)$ and $(4,0)$, then:

Solve the following Linear Programming problem graphically:
Maximize $Z=300x+600y$
subject to
$x+2y\le 12$
$2x+y\le 12$
$x+\frac{5}{4}y\ge 5$
$x\ge 0,y\ge 0$

Solve the following linear programming problem graphically:
Maximize $Z=x+2y$
subject to the constraints
$x+2y\ge 100$
$2x-y\le 0$
$2x+y\le 200$
$x\ge 0,y\ge 0$

A furniture trader deals in only two items — chairs and tables. He has Rs. 50,000 to invest and a space to store at most 35 items. A chair costs him Rs. 1000 and a table costs him Rs. 2000. The trader earns a profit of Rs. 150 and Rs. 250 on a chair and a table, respectively. For …

A company produces two types of goods A and B that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can use at most 9 g of silver and 8 g of gold. If each unit of type A br …

A factory manufactures two types of screws, A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machine to manufacture a packet of screws 'A', while it takes 6 minutes on th …

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on …

Solve the following L.P.P. graphically:
Minimize $Z=5x+10y$
subject to the constraints
$x+2y\le 120$
$x+y\ge 60$
$x-2y\ge 0$
$x\ge 0,y\ge 0$

A retired person wants to invest an amount of Rs. 50,000. His broker recommends investing in two types of bonds 'A' and 'B' yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond 'A' and at least Rs. 10,000 in bond 'B'. He …

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each un …

Find graphically the maximum value of $Z=2x+5y$, subject to the constraints
$2x+4y\le 8$
$3x+y\le 6$
$x+y\le 4$
$x\ge 0,y\ge 0$

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each piece of type A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of type B requires 12 labour hours for fabricating and 3 labour hours for finish …

A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated at Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres an …

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7 pe …

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman's time. In a day, the factory has the availability of not more …

A merchant plans to sell two types of personal computers — a desktop model and a portable model — that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type o …

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and tha …

A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods $F_1$ and $F_2$ are available. Food $F_1$ costs Rs. 4 per unit and $F_2$ costs Rs. 6 per unit. One unit of food $F_1$ contains 3 units of Vitamin A and 4 units of minerals. One unit of food …

A factory owner purchases two types of machines, A and B, for his factory. The requirements and the limitations for the machines are as follows:
Machine A: occupies $1000{\text{m}}^2$, needs 12 men, daily output 60 units.
Machine B: occupies $1200{\text{m}}^2$, needs 8 men, daily …

If a young man rides his motorcycle at 25 km/hour, he has to spend Rs. 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel withi …

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan and a sewing machine cost Rs. 360 and Rs. 240 respectively. He can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. …

Solve the following linear programming problem graphically:
Maximize $Z=60x+15y$
subject to the constraints
$x+y\le 50$
$3x+y\le 90$
$x\ge 0,y\ge 0$

Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

Solve the following linear programming problem graphically:
Minimize $Z=3x+5y$
subject to the constraints
$x+y\ge 2$
$x+3y\ge 3$
$x\ge 0,y\ge 0$