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JEE Main

Explore popular questions from Differential Equations for JEE Main. This collection covers Differential Equations previous year JEE Main questions hand picked by experienced teachers.

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Differential Equations

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Q 1. The differential equation which represents the family of curves {tex} y = c _ { 1 } e ^ { c _ { 2 } x } {/tex} where {tex} c _ { 1 } {/tex} and {tex} c _ { 2 } {/tex} are arbitrary constants is

A

{tex} y ^ { \prime } = y ^ { 2 } {/tex}

B

{tex} y ^ { \prime \prime } = y ^ { \prime } y {/tex}

C

{tex} y y ^ { \prime \prime } = y ^ { \prime } {/tex}

{tex} y y ^ { \prime \prime } = \left( y ^ { \prime } \right) ^ { 2 } {/tex}

Explanation


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Q 2. Solution of the differential equation {tex} \cos x d y = y ( \sin x - y ) d x , 0 < x < \frac { \pi } { 2 } {/tex} is

A

{tex} y \sec x = \tan x + c {/tex}

B

{tex} y \tan x = \sec x + c {/tex}

C

{tex} \tan x = ( \sec x + c ) y {/tex}

{tex} \sec x = ( \tan x + c ) y {/tex}

Explanation



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Q 3. Let {tex} f :( - 1,1 ) \rightarrow R {/tex} be a differentiable function with {tex} f ( 0 ) = - 1 {/tex} and {tex} f ^ { \prime } ( 0 ) = 1 . {/tex} Let {tex} g ( x ) = [ f ( 2 f ( x ) + 2 ) ] ^ { 2 } . {/tex} Then {tex} g ^ { \prime } ( 0 ) = {/tex}

-4

B

0

C

-2

D

4

Explanation


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Q 4. If {tex} \frac { d y } { d x } = y + 3 > 0 {/tex} and {tex} y ( 0 ) = 2 , {/tex} then {tex} y ( \ln 2 ) {/tex} is equal to

A

5

B

13

C

-2

7

Explanation


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Q 5. Let {tex} I {/tex} be the purchase value of an equipment and {tex} V ( t ) {/tex} be the value after it has been used for {tex} t {/tex} years. The value {tex} V ( t ) {/tex} depreciates at a rate given by differential equation {tex} \frac { d V ( t ) } { d t } = - k ( T - t ) , {/tex} where {tex} k > {/tex} 0 is a constant and {tex} T {/tex} is the total life in years of the equipment. Then the scrap value {tex} V ( T ) {/tex} of the equipment is

{tex} I - \frac { k T ^ { 2 } } { 2 } {/tex}

B

{tex} I - \frac { k ( T - t ) ^ { 2 } } { 2 } {/tex}

C

{tex} e ^ { - k T } {/tex}

D

{tex} T ^ { 2 } - \frac { 1 } { k } {/tex}

Explanation



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Q 6. At present, a firm is manufacturing {tex}2000{/tex} items. It is estimated that the rate of change of production {tex} P {/tex} w.r.t. additional number of workers {tex} x {/tex} is given by {tex} \frac { d P } { d x } = 100 - 12 \sqrt { x } {/tex} . If the firm employs {tex}25{/tex} more workers, then the new level of production of items is

A

3000

3500

C

4500

D

2500

Explanation

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Q 7. Let the population of rabbits surviving at a time {tex} t {/tex} be governed by the differential equation {tex} \frac { d p ( t ) } { d t } = \frac { 1 } { 2 } p ( t ) - 200 . {/tex} If {tex} p ( 0 ) = 100 {/tex} , then {tex} p ( t ) {/tex} equals

A

{tex} 600 - 500 e ^ { t / 2 } {/tex}

B

{tex} 400 - 300 e ^ { - t / 2 } {/tex}

{tex} 400 - 300 e ^ { t / 2 } {/tex}

D

{tex} 300 - 200 e ^ { - t / 2 } {/tex}

Explanation



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Q 8. If the differential equation representing the family of all circles touching {tex} x {/tex} -axis at the origin is {tex} \left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } = g ( x ) y , {/tex} then {tex} g ( x ) {/tex} equals

A

{tex} \frac { 1 } { 2 } x {/tex}

B

2{tex} x ^ { 2 } {/tex}

2{tex} x {/tex}

D

{tex} \frac { 1 } { 2 } x ^ { 2 } {/tex}

Explanation


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Q 9. If the general solution of the differential equation {tex}y^{\prime} = \frac{y}{x} + \phi \left(\frac{x}{y}\right){/tex} for some function {tex}\phi{/tex}, is given by y In |cx| = x, where c is an arbitary constant, then {tex}\phi(2){/tex} is equal to

A

{tex}4{/tex}

B

{tex} \frac { 1 } { 4 } {/tex}

C

{tex} - 4 {/tex}

{tex} - \frac { 1 } { 4 } {/tex}

Explanation



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Q 10. The general solution of the differential equation {tex} \sin 2 x \left( \frac { d y } { d x } - \sqrt { \tan x } \right) - y = 0 {/tex} is

A

{tex} y \sqrt { \tan x } = x + c {/tex}

B

{tex} y \sqrt { \cot x } = \tan x + c {/tex}

C

{tex} y \sqrt { \tan x } = \cot x + c {/tex}

{tex} y \sqrt { \cot x } = x + c {/tex}

Explanation


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Q 11. If {tex} \frac { d y } { d x } + y \tan x = \sin 2 x {/tex} and {tex} y ( 0 ) = 1 , {/tex} then {tex} y ( \pi ) {/tex} is equal to

A

1

B

- 1

-5

D

5

Explanation



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Q 12. If {tex} y ( x ) {/tex} is the solution of the differential equation {tex} ( x + 2 ) \frac { d y } { d x } = x ^ { 2 } + 4 x - 9 , x \neq - 2 {/tex} and {tex} y ( 0 ) = 0 , {/tex} then {tex} y ( - 4 ) {/tex} is equal to

0

B

1

C

-1

D

2

Explanation


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Q 13. The solution of the differential equation {tex} y d x - \left( x + 2 y ^ { 2 } \right) d y = 0 {/tex} is {tex} x = f ( y ) . {/tex} If {tex} f ( - 1 ) = 1 , {/tex} then {tex} f ( 1 ) {/tex} is equal to

A

4

3

C

2

D

1

Explanation


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Q 14. If a curve {tex} y = f ( x ) {/tex} passes through the point {tex} ( 1 , - 1 ) {/tex} and satisfies the differential equation, {tex} y ( 1 + x y ) d x = x d y , {/tex} then {tex} f \left( - \frac { 1 } { 2 } \right) {/tex} is equal to

{tex} \frac { 4 } { 5 } {/tex}

B

{tex}- \frac { 2 } { 5 } {/tex}

C

{tex} - \frac { 4 } { 5 } {/tex}

D

{tex} \frac { 2 } { 5 } {/tex}

Explanation


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Q 15. If the tangent at a point {tex} P , {/tex} with parameter {tex} t , {/tex} on the curve {tex} x = 4 t ^ { 2 } + 3 , y = 8 t ^ { 3 } - 1 , t \in \mathbb { R } {/tex} meets the curve again at a point {tex} Q , {/tex} then the coordinates of {tex} Q {/tex} are

A

{tex} \left( 16 t ^ { 2 } + 3 , - 64 t ^ { 3 } - 1 \right) {/tex}

B

{tex} \left( 4 t ^ { 2 } + 3 , - 8 t ^ { 3 } - 1 \right) {/tex}

C

{tex} \left( t ^ { 2 } + 3 , t ^ { 3 } - 1 \right) {/tex}

{tex} \left( t ^ { 2 } + 3 , - t ^ { 3 } - 1 \right) {/tex}

Explanation



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Q 16. For {tex}x \in R, x \neq 0,{/tex} if y(x) is a differentiable function such that {tex}x\int\limits^{x}_{1} y(t)dt = (x+1)\int\limits^{x}_{1}ty(t)dt{/tex} then y(x) equals (where C is a constant)

A

{tex} Cx ^ { 3 } \frac { 1 } { e ^ { x } } {/tex}

B

{tex} \frac { C } { x ^ { 2 } } e ^ { - \frac { 1 } { x } } {/tex}

C

{tex} \frac { c } { x } e ^ { - \frac { 1 } { x } } {/tex}

{tex} \frac { c } { x ^ { 3 } } e ^ { - \frac { 1 } { x } } {/tex}

Explanation


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Q 17. The differential equation of all non-vertical lines in a plane is

{tex} \frac { d ^ { 2 } y } { d x ^ { 2 } } = 0 {/tex}

B

{tex} \frac { d ^ { 2 } x } { d y ^ { 2 } } = 0 {/tex}

C

{tex} \frac { d y } { d x } = 0 {/tex}

D

{tex} \frac { d x } { d y } = 0 {/tex}

Explanation


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Q 18. The differential equation of all non-horizontal lines in a plane is

A

{tex} \frac { d ^ { 2 } y } { d x ^ { 2 } } = 0 {/tex}

{tex} \frac { d ^ { 2 } x } { d y ^ { 2 } } = 0 {/tex}

C

{tex} \frac { d y } { d x } = 0 {/tex}

D

{tex} \frac { d x } { d y } = 0 {/tex}

Explanation


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Q 19. A solution of the differential equation {tex} \left( \frac { d y } { d x } \right) ^ { 2 } - x \frac { d y } { d x } + y = 0 {/tex} is

A

{tex} y = 2 {/tex}

B

{tex} y = 2 x {/tex}

{tex} y = 2 x - 4 {/tex}

D

{tex} y = 2 x ^ { 2 } - 4 {/tex}

Explanation