JEE Main > Linear Programming

Explore popular questions from Linear Programming for JEE Main. This collection covers Linear Programming previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

For the following shaded area, the linear constraints except {tex} x \geq 0 {/tex} and {tex} y \geq 0 {/tex}, are

A

{tex} 2 x + y \leq 2 ,\ x - y \leq 1 ,\ x + 2 y \leq 8 {/tex}

{tex} 2 x + y \geq 2 ,\ x - y \leq 1 ,\ x + 2 y \leq 8 {/tex}

C

{tex} 2 x + y \geq 2 ,\ x - y \geq 1 ,\ x + 2 y \leq 8 {/tex}

D

{tex} 2 x + y \geq 2 ,\ x - y \geq 1 ,\ x + 2 y \geq 8 {/tex}

Explanation

To test the origin for {tex}2x+y =2,\ x-y=1{/tex} and {tex}x+2y=8{/tex} in reference to shaded area, {tex}0+0<2{/tex} is true for {tex}2x+y =2{/tex}. So for the region does not include origin {tex}(0,0){/tex}, {tex}2x+y \geq 2{/tex}. Again for {tex}x-y = 1, 0-0<1, {/tex}
{tex} \therefore \ x-y \leq 1{/tex}
Similarly for {tex} x+ 2y = 8, 0+0 < 8 ;{/tex}
{tex}\therefore x+2y\leq8{/tex}

Q 2.    

Correct4

Incorrect-1

Inequations {tex} 3 x - y \geq 3 {/tex} and {tex} 4 x - y > 4 {/tex}

Have solution for positive {tex} x {/tex} and {tex} y {/tex}

B

Have no solution for positive {tex} x {/tex} and {tex} y {/tex}

C

Have solution for all {tex} x {/tex}

D

Have solution for all {tex} y {/tex}

Explanation


Q 3.    

Correct4

Incorrect-1

Shaded region is represented by

A

{tex} 4 x - 2 y \leq 3 {/tex}

{tex} 4 x - 2 y \leq - 3 {/tex}

C

{tex} 4 x - 2 y \geq 3 {/tex}

D

{tex} 4 x - 2 y \geq - 3 {/tex}

Explanation

Q 4.    

Correct4

Incorrect-1

A Firm makes pants and shirts. A shirt takes 2 hour on machine and 3 hour of man labour while a pant takes 3 hour on machine and 2 hour of man labour. In a week there are 70 hour machine and 75 hour of man labour available. If the firm determine to make {tex} x {/tex} shirts and {tex} y {/tex} pants per week, then for this the linear constraints are

A

{tex} x \geq 0 , y \geq 0,\ 2 x + 3 y \geq 70,\ 3 x + 2 y \geq 75 {/tex}

B

{tex} x \geq 0 ,\ y \geq 0,\ 2 x + 3 y \leq 70,\ 3 x + 2 y \geq 75 {/tex}

C

{tex} x \geq 0 ,\ y \geq 0,\ 2 x + 3 y \geq 70,\ 3 x + 2 y \leq 75 {/tex}

{tex} x \geq 0 ,\ y \geq 0,\ 2 x + 3 y \leq 70, \ 3 x + 2 y \leq 75 {/tex}

Explanation


Linear constraints are {tex}2x +3y \leq 70, 3x+2y \geq 75{/tex}

Q 5.    

Correct4

Incorrect-1

For the L.P. problem {tex} \operatorname { Min } z = 2 x _ { 1 } + 3 x _ { 2 } {/tex} such that {tex} - x _ { 1 } + 2 x _ { 2 } \leq 4 , \quad x _ { 1 } + x _ { 2 } \leq 6 ,\ x _ { 1 } + 3 x _ { 2 } \geq 9 {/tex} and {tex} x _ { 1 } ,\ x _ { 2 } \geq 0 {/tex}

A

{tex} x _ { 1 } = 1.2 {/tex}

B

{tex} x _ { 2 } = 2.6 {/tex}

C

{tex} z = 10.2 {/tex}

All the above

Explanation

The graph of linear programming problem is as given below

Hence the required feasible region is given by the graph whose vertices are {tex}A (1.2, 2.6),\ B(4.5, 1.5){/tex} and {tex} C (\frac {8}{3}, \ \frac {10}{3}){/tex}.
Thus objective function is minimum at {tex}A (1.2, 2.6){/tex}.
So {tex}x_1 = 1.2,{/tex} {tex}x_2= 2.6{/tex} and {tex}z = 2 \times 1.2 + 3 \times 2.6 = 10.2{/tex}

Q 6.    

Correct4

Incorrect-1

A company manufactures two types of products {tex} A {/tex} and {tex} B {/tex}. The storage capacity of its godown is 100 units. Total investment amount is Rs. 30,000. The cost price of {tex} A {/tex} and {tex} B {/tex} are Rs. 400 and Rs. 900 respectively. If all the products have sold and per unit profit is Rs. 100 and Rs. 120 through {tex} A {/tex} and {tex} B {/tex} respectively. If {tex} x {/tex} units of {tex} A {/tex} and {tex} y {/tex} units of {tex} B {/tex} be produced, then two linear constraints and iso-profit line are respectively

A

{tex} x + y = 100,\ 4 x + 9 y = 300,\ 100 x + 120 y = c {/tex}

B

{tex} x + y \leq 100,\ 4 x + 9 y \leq 300,\ x + 2 y = c {/tex}

{tex} x + y \leq 100,\ 4 x + 9 y \leq 300,\ 100 x + 120y = c {/tex}

D

{tex} x + y \leq 100,\ 9 x + 4 y \leq 300, \ x + 2 y = c {/tex}

Explanation

{tex}x+y \leq100,\ 400x +900y \leq 3000c \ \mathrm{OR}{/tex}
{tex}4x+9y \leq300,\ \mathrm{and }\ 100x + 120y \leq c{/tex}

Q 7.    

Correct4

Incorrect-1

The L.P. problem {tex} \mathrm { Max }\ z = x _ { 1 } + x _ { 2 } {/tex} such that {tex} - 2 x _ { 1 } + x _ { 2 } \leq 1 ,\ x _ { 1 } \leq 2 ,\ x _ { 1 } + x _ { 2 } \leq 3 {/tex} and {tex} x _ { 1 } ,\ x _ { 2 } \geq 0 {/tex} has

A

One solution

B

Three solution

An infinite no. of solution

D

None of these

Explanation

Q 8.    

Correct4

Incorrect-1

The minimum value of the objective function {tex} z = 2 x + 10 y {/tex} for linear constraints {tex} x \geq 0 ,\ y \geq 0 {/tex}, {tex} x - y \geq 0 ,\ x - 5 y \leq - 5 {/tex}, is

A

10

15

C

12

D

8

Explanation



Q 9.    

Correct4

Incorrect-1

The maximum value of {tex} z = 4 x + 3 y {/tex} subject to the constraints {tex} 3 x + 2 y \geq 160, \ 5x + 2 y \geq 200 ,\ x + 2 y \geq 80 {/tex}; {tex} x ,\ y \geq 0 {/tex} is

A

320

B

300

C

230

None of these

Explanation


Q 10.    

Correct4

Incorrect-1

Minimize {tex} z = \sum \limits _ { j = 1 } ^ { n } \sum \limits_ { i = 1 } ^ { m } c _ { i j } x _ { i j } {/tex}
Subject to: {tex} \sum \limits _ { j = 1 } ^ { n } x _ { i j } \leq a _ { i } , i = 1 , \ldots \ldots m {/tex}
{tex}\quad \quad \quad \quad \sum \limits _ { i = 1 } ^ { m } x _ { i j } = b _ { j } , j = 1 , \ldots \ldots n {/tex}
is a (LPP) with number of constraints

{tex} m + n {/tex}

B

{tex} m - n {/tex}

C

{tex} m n {/tex}

D

{tex} \frac { m } { n } {/tex}

Explanation

Q 11.    

Correct4

Incorrect-1

For the following feasible region, the linear constraints except {tex} x \geq 0 {/tex} and {tex} y \geq 0 , {/tex} are

A

{tex} x \geq 250 , y \leq 350,2 x + y = 600 {/tex}

B

{tex} x \leq 250 , y \leq 350,2 x + y = 600 {/tex}

C

{tex} x \leq 250 , y \leq 350,2 x + y \geq 600 {/tex}

{tex} x \leq 250 , y \leq 350,2 x + y \leq 600 {/tex}

Explanation

Q 12.    

Correct4

Incorrect-1

Let {tex} X _ { 1 } {/tex} and {tex} X _ { 2 } {/tex} are optimal solutions of a L.P.P., then

A

{tex} X = \lambda X _ { 1 } + ( 1 - \lambda ) X _ { 2 } , \lambda \in R {/tex} is also an optimal solution

{tex} X = \lambda X _ { 1 } + ( 1 - \lambda ) X _ { 2 } , 0 \leq \lambda \leq 1 {/tex} gives an optimal solution

C

{tex} X = \lambda X _ { 1 } + ( 1 + \lambda ) X _ { 2 } , 0 \leq \lambda \leq 1 {/tex} gives an optimal solution

D

{tex} X = \lambda X _ { 1 } + ( 1 + \lambda ) X _ { 2 } , \lambda \in R {/tex} gives an optimal solution

Explanation

Q 13.    

Correct4

Incorrect-1

The points which provides the solution to the linear programming problem: {tex} { Max }\ ( 2 x + 3 y ) {/tex} subject to constraints: {tex} x \geq 0 ,\ y \geq 0,\ 2 x + 2 y \leq 9,\ 2 x + y \leq 7 {/tex}, {tex} x + 2 y \leq 8 {/tex}, is

A

{tex} ( 3,2.5 ) {/tex}

B

{tex} ( 2,3.5 ) {/tex}

C

{tex} ( 2,2.5 ) {/tex}

{tex} ( 1,3.5 ) {/tex}

Explanation



Q 14.    

Correct4

Incorrect-1

Two tailors {tex} A {/tex} and {tex} B {/tex} earns Rs. 15 and Rs. 20 per day respectively. {tex} A {/tex} can make 6 shirts and 4 pants in a day while {tex} B {/tex} can make 10 shirts and 3 pants. To spend minimum on 60 shirts and 40 pants, {tex} A {/tex} and {tex} B {/tex} working {tex} x {/tex} and {tex} y {/tex} days respectively. Then linear constraints except {tex} x \geq 0 ,\ y \geq 0 , {/tex} are and objective function are respectively

A

{tex} 15 x + 20 y \geq 0,\ 60 x + 40 y \geq 0 ,\ z = 4 x + 3 y {/tex}

B

{tex} 15 x + 20 y \geq 0,\ 6 x + 10 y = 10 ,\ z = 60 x + 60 y {/tex}

{tex} 6 x + 10 y \geq 60,\ 4 x + 3 y \geq 40 ,\ z = 60 x + 40 y {/tex}

D

{tex} 6 x + 10 y \leq 60,\ 4 x + 3 y \leq 40 ,\ z = 15 x + 20 y {/tex}

Explanation

Q 15.    

Correct4

Incorrect-1

A company manufactures two types of telephone sets {tex} A {/tex} and {tex} B {/tex}. The {tex} A {/tex} type telephone set requires 2 hour and {tex} B {/tex} 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 {tex} A {/tex} and {tex} B {/tex} type telephones are Rs. 300 and 400 respectively. For maximum profits company produces {tex} x {/tex} telephones of {tex} A {/tex} type and {tex} y {/tex} telephones of {tex} B {/tex} types. Then except {tex} x \geq 0 {/tex} and {tex} y \geq 0 {/tex}, linear constraints and the probable region of the L.P.P is of the type

{tex} x + 2 y \leq 400 ;\ x + y \leq 300 ;\ {/tex} {tex}Max\ z = 300 x + 400 y {/tex}, bounded

B

{tex} 2 x + y \leq 400 ;\ x + y \geq 300 ;\ { Max } \ z = 400 x + 300 y {/tex}, unbounded

C

{tex} 2 x + y \geq 400 ;\ x + y \geq 300 ;\ { Max }\ { z } = 300 x + 400 y {/tex}, parallelogram

D

{tex} x + 2 y \leq 400 ;\ x + y \geq 300 ;\ { Max }\ z = 300 x + 400 y {/tex}, square

Explanation



Q 16.    

Correct4

Incorrect-1

We have to purchase two articles {tex} A {/tex} and {tex} B {/tex} of cost Rs. 45 and Rs. 25 respectively, I can purchase total article maximum of Rs. 1000 . After selling the articles {tex} A {/tex} and {tex} B {/tex}, the profit per unit is Rs. 5 and 3 respectively. If I purchase the {tex} x {/tex} and {tex} y {/tex} numbers of articles {tex} A {/tex} and {tex} B {/tex} respectively, then the mathematical formulation of problem is

A

{tex} x \geq 0 ,\ y \geq 0,45 x + 25 y \geq 1000 ,\ 5 x + 3 y = c {/tex}

{tex} x \geq 0 ,\ y \geq 0,\ 45 x + 25 y \leq 1000 , \ 5 x + 3 y = c {/tex}

C

{tex} x \geq 0 ,\ y \geq 0,\ 45 x + 25 y \leq 1000 , \ 3 x + 5 y = c {/tex}

D

None of these

Explanation

(b) {tex} x \geq 0 ,\ y \geq 0,\ 45 x + 25 y \leq 1000 , \ 5 x + 3 y = c {/tex}

Q 17.    

Correct4

Incorrect-1

For the L.P. problem {tex} { Max }\ z = 3 x + 2 y {/tex} subject to {tex} x + y \geq 1 , \ y - 5 x \leq 0 , \ x - y \geq - 1 , \ x + y \leq 6 , \ x \leq 3 {/tex} and {tex} x , y \geq 0 {/tex}

A

{tex} x = 3 {/tex}

B

{tex} y = 3 {/tex}

C

{tex} z = 15 {/tex}

All the above

Explanation



Q 18.    

Correct4

Incorrect-1

The maximum value of objective function {tex} c = 2 x + 3 y {/tex} in the given feasible region, is

A

29

18

C

14

D

15

Explanation

Q 19.    

Correct4

Incorrect-1

The maximum value of {tex} 4 x + 5 y {/tex} subject to the constraints {tex} x + y \leq 20 ,\ x + 2 y \leq 35 ,\ x - 3 y \leq 12 {/tex} is

A

84

95

C

100

D

96

Explanation


Q 20.    

Correct4

Incorrect-1

For the following linear programming problem : minimize {tex} z = 4 x + 6 y {/tex} subject to the constraints {tex} 2 x + 3 y \geq 6 {/tex}, {tex} x + y \leq 8 ,\ y \geq 1 ,\ x \geq 0 {/tex}, the solution is

A

{tex} ( 0,2 ) {/tex} and {tex} ( 1,1 ) {/tex}

{tex} ( 0,2 ) {/tex} and {tex} ( 3 / 2,1 ) {/tex}

C

{tex} ( 0,2 ) {/tex} and {tex} ( 1,6 ) {/tex}

D

{tex} ( 0,2 ) {/tex} and {tex} ( 1,5 ) {/tex}

Explanation


Q 21.    

Correct4

Incorrect-1

For the constraint of a linear optimizing function {tex} z = x _ { 1 } + x _ { 2 } , {/tex} given by {tex} x _ { 1 } + x _ { 2 } \leq 1,3 x _ { 1 } + x _ { 2 } \geq 3 {/tex} and {tex} x _ { 1 } , x _ { 2 } \geq 0 {/tex}

A

There are two feasible regions

B

There are infinite feasible regions

There is no feasible region

D

None of these

Explanation

Q 22.    

Correct4

Incorrect-1

Which of the following is not a vertex of the positive region bounded by the inequalities {tex} 2 x + 3 y \leq 6 , {/tex} {tex} 5 x + 3 y \leq 15 {/tex} and {tex} x , y \geq 0 {/tex}

A

{tex} ( 0,2 ) {/tex}

B

{tex} ( 0,0 ) {/tex}

C

{tex} ( 3,0 ) {/tex}

None of these

Explanation

Q 23.    

Correct4

Incorrect-1

The intermediate solutions of constraints must be checked by substituting them back into

A

Objective function

Constraint equations

C

Not required

D

None of these

Explanation

Q 24.    

Correct4

Incorrect-1

For the constraints of a L.P. problem given by {tex} x _ { 1 } + 2 x _ { 2 } \leq 2000 , x _ { 1 } + x _ { 2 } \leq 1500 , x _ { 2 } \leq 600 {/tex} and {tex} x _ { 1 } , x _ { 2 } \geq 0 {/tex} which one of the following points does not lie in the positive bounded region

A

{tex} ( 1000,0 ) {/tex}

B

{tex} ( 0,500 ) {/tex}

C

{tex} ( 2,0 ) {/tex}

{tex} ( 2000,0 ) {/tex}

Explanation

Q 25.    

Correct4

Incorrect-1

A basic solution is called non-degenerate, if

A

All the basic variables are zero

None of the basic variables is zero

C

At least one of the basic variables is zero

D

None of these

Explanation