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Page 1
JEE Mains Previous Year Questions
(2021-2024): Differential Equations
2024
Q1 - 2024 (01 Feb Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) -
1 , ?? ( 0 ) = 1.
Then, (
1
v 2
+ y (
1
v 2
) )
2
equals :
(1)
4
4 + v e
(2)
3
3 - v e
(3)
2
1 + v e
(4)
1
2 - v e
Q2 - 2024 (01 Feb Shift 1)
If ?? = ?? ( ?? ) is the solution of the differential equation
( ?? + 1 ) ???? = ( 2 ?? + ( ?? + 1 )
4
) ???? , ?? ( 0 ) = 2, then, ?? ( 1 ) equals
Q3 - 2024 (01 Feb Shift 2)
Let ?? be a non-zero real number. Suppose ?? : R ? R is a differentiable function such that
?? ( 0 ) = 2 and l i m
x ? - 8
? f ( x ) = 1. If ?? '
( x ) = ???? ( ?? ) + 3, for all x ? R, then ?? ( - log
?? ? 2 ) is equal
to
(1) 3
(2) 5
(3) 9
(4) 7
Q4 - 2024 (01 Feb Shift 2)
Page 2
JEE Mains Previous Year Questions
(2021-2024): Differential Equations
2024
Q1 - 2024 (01 Feb Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) -
1 , ?? ( 0 ) = 1.
Then, (
1
v 2
+ y (
1
v 2
) )
2
equals :
(1)
4
4 + v e
(2)
3
3 - v e
(3)
2
1 + v e
(4)
1
2 - v e
Q2 - 2024 (01 Feb Shift 1)
If ?? = ?? ( ?? ) is the solution of the differential equation
( ?? + 1 ) ???? = ( 2 ?? + ( ?? + 1 )
4
) ???? , ?? ( 0 ) = 2, then, ?? ( 1 ) equals
Q3 - 2024 (01 Feb Shift 2)
Let ?? be a non-zero real number. Suppose ?? : R ? R is a differentiable function such that
?? ( 0 ) = 2 and l i m
x ? - 8
? f ( x ) = 1. If ?? '
( x ) = ???? ( ?? ) + 3, for all x ? R, then ?? ( - log
?? ? 2 ) is equal
to
(1) 3
(2) 5
(3) 9
(4) 7
Q4 - 2024 (01 Feb Shift 2)
If
????
????
=
1 + ?? - ?? 2
?? , ?? ( 1 ) = 1, then 5 ?? ( 2 ) is equal to :
Q5 - 2024 (27 Jan Shift 1)
Let ?? = ?? ( ?? ) and ?? = ?? ( ?? ) be solutions of the differential equations
dx
dt
+ ax = 0 and
dy
dt
+ by = 0 respectively, a , b ? R. Given that ?? ( 0 ) = 2 ; ?? ( 0 ) = 1 and 3 ?? ( 1 ) = 2 ?? ( 1 ) , the
value of ?? , for which x ( t ) = y ( t ) , is :
(1) log 2
3
? 2
(2) log
4
? 3
(3) log
3
? 4
(4) log 4
3
? 2
Q6 - 2024 (27 Jan Shift 1)
If the solution of the differential equation ( 2 ?? + 3 ?? - 2 ) ???? + ( 4 ?? + 6 ?? - 7 ) ???? = 0 , ?? ( 0 ) =
3, is ???? + ???? + 3 log
?? ? | 2 ?? + 3 ?? - ?? | = 6, then ?? + 2 ?? + 3 ?? is equal to
Q7 - 2024 (27 Jan Shift 2)
If ?? = ?? ( ?? ) is the solution curve of the differential equation ( ?? 2
- 4 ) ???? - ( ?? 2
- 3 ?? ) ???? =
0, ?? > 2 , ?? ( 4 ) =
3
2
and the slope of the curve is never zero, then the value of ?? ( 10 )
equals :
(1)
3
1 + ( 8 )
1 / 4
(2)
3
1 + 2 v 2
(3)
3
1 - 2 v 2
(4)
3
1 - ( 8 )
1 / 4
Q8 - 2024 (27 Jan Shift 2)
If the solution curve, of the differential equation
????
????
=
?? + ?? - 2
?? - ?? passing through the point
( 2 , 1 ) is t a n
- 1
? (
?? - 1
?? - 1
) -
1
?? log
?? ? ( ?? + (
?? - 1
?? - 1
)
2
) = log
?? ? | ?? - 1 |, then 5 ?? + ?? is equal to
Q9 - 2024 (29 Jan Shift 1)
Page 3
JEE Mains Previous Year Questions
(2021-2024): Differential Equations
2024
Q1 - 2024 (01 Feb Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) -
1 , ?? ( 0 ) = 1.
Then, (
1
v 2
+ y (
1
v 2
) )
2
equals :
(1)
4
4 + v e
(2)
3
3 - v e
(3)
2
1 + v e
(4)
1
2 - v e
Q2 - 2024 (01 Feb Shift 1)
If ?? = ?? ( ?? ) is the solution of the differential equation
( ?? + 1 ) ???? = ( 2 ?? + ( ?? + 1 )
4
) ???? , ?? ( 0 ) = 2, then, ?? ( 1 ) equals
Q3 - 2024 (01 Feb Shift 2)
Let ?? be a non-zero real number. Suppose ?? : R ? R is a differentiable function such that
?? ( 0 ) = 2 and l i m
x ? - 8
? f ( x ) = 1. If ?? '
( x ) = ???? ( ?? ) + 3, for all x ? R, then ?? ( - log
?? ? 2 ) is equal
to
(1) 3
(2) 5
(3) 9
(4) 7
Q4 - 2024 (01 Feb Shift 2)
If
????
????
=
1 + ?? - ?? 2
?? , ?? ( 1 ) = 1, then 5 ?? ( 2 ) is equal to :
Q5 - 2024 (27 Jan Shift 1)
Let ?? = ?? ( ?? ) and ?? = ?? ( ?? ) be solutions of the differential equations
dx
dt
+ ax = 0 and
dy
dt
+ by = 0 respectively, a , b ? R. Given that ?? ( 0 ) = 2 ; ?? ( 0 ) = 1 and 3 ?? ( 1 ) = 2 ?? ( 1 ) , the
value of ?? , for which x ( t ) = y ( t ) , is :
(1) log 2
3
? 2
(2) log
4
? 3
(3) log
3
? 4
(4) log 4
3
? 2
Q6 - 2024 (27 Jan Shift 1)
If the solution of the differential equation ( 2 ?? + 3 ?? - 2 ) ???? + ( 4 ?? + 6 ?? - 7 ) ???? = 0 , ?? ( 0 ) =
3, is ???? + ???? + 3 log
?? ? | 2 ?? + 3 ?? - ?? | = 6, then ?? + 2 ?? + 3 ?? is equal to
Q7 - 2024 (27 Jan Shift 2)
If ?? = ?? ( ?? ) is the solution curve of the differential equation ( ?? 2
- 4 ) ???? - ( ?? 2
- 3 ?? ) ???? =
0, ?? > 2 , ?? ( 4 ) =
3
2
and the slope of the curve is never zero, then the value of ?? ( 10 )
equals :
(1)
3
1 + ( 8 )
1 / 4
(2)
3
1 + 2 v 2
(3)
3
1 - 2 v 2
(4)
3
1 - ( 8 )
1 / 4
Q8 - 2024 (27 Jan Shift 2)
If the solution curve, of the differential equation
????
????
=
?? + ?? - 2
?? - ?? passing through the point
( 2 , 1 ) is t a n
- 1
? (
?? - 1
?? - 1
) -
1
?? log
?? ? ( ?? + (
?? - 1
?? - 1
)
2
) = log
?? ? | ?? - 1 |, then 5 ?? + ?? is equal to
Q9 - 2024 (29 Jan Shift 1)
A function ?? = ?? ( ?? ) satisfies
?? ( ?? ) sin ? 2 ?? + sin ? ?? - ( 1 + c o s
2
? ?? ) ?? '
( ?? ) = 0 with condition ?? ( 0 ) = 0. Then ?? (
?? 2
) is equal to
(1) 1
(2) 0
(3) -1
(4) 2
Q10 - 2024 (29 Jan Shift 1)
If the solution curve ?? = ?? ( ?? ) of the differential equation ( 1 + ?? 2
) ( 1 + log
?? ? ?? ) ???? + ?? ?? ?? =
0 , ?? > 0 passes through the point ( 1 , 1 ) and ?? ( ?? ) =
?? - ta n ? (
3
2
)
?? + ta n ? (
3
2
)
, then ?? + 2 ?? is
Q11 - 2024 (29 Jan Shift 2)
If sin ? (
?? ?? ) = log
?? ? | ?? | +
?? 2
is the solution of the differential equation ?? c o s ? (
?? ?? )
????
????
= ?? c o s ? (
?? ?? ) +
?? and ?? ( 1 ) =
?? 3
, then ?? 2
is equal to
(1) 3
(2) 12
(3) 4
(4) 9
Q12 - 2024 (30 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation s e c ? x ?? y + { 2 ( 1 - x ) t a n ? x + x ( 2 -
x ) } dx = 0 such that ?? ( 0 ) = 2. Then ?? ( 2 ) is equal to :
(1) 2
(2) 2 { 1 - sin ? ( 2 ) }
(3) 2 { sin ? ( 2 ) + 1 }
(4) 1
Q13 - 2024 (30 Jan Shift 1)
Page 4
JEE Mains Previous Year Questions
(2021-2024): Differential Equations
2024
Q1 - 2024 (01 Feb Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) -
1 , ?? ( 0 ) = 1.
Then, (
1
v 2
+ y (
1
v 2
) )
2
equals :
(1)
4
4 + v e
(2)
3
3 - v e
(3)
2
1 + v e
(4)
1
2 - v e
Q2 - 2024 (01 Feb Shift 1)
If ?? = ?? ( ?? ) is the solution of the differential equation
( ?? + 1 ) ???? = ( 2 ?? + ( ?? + 1 )
4
) ???? , ?? ( 0 ) = 2, then, ?? ( 1 ) equals
Q3 - 2024 (01 Feb Shift 2)
Let ?? be a non-zero real number. Suppose ?? : R ? R is a differentiable function such that
?? ( 0 ) = 2 and l i m
x ? - 8
? f ( x ) = 1. If ?? '
( x ) = ???? ( ?? ) + 3, for all x ? R, then ?? ( - log
?? ? 2 ) is equal
to
(1) 3
(2) 5
(3) 9
(4) 7
Q4 - 2024 (01 Feb Shift 2)
If
????
????
=
1 + ?? - ?? 2
?? , ?? ( 1 ) = 1, then 5 ?? ( 2 ) is equal to :
Q5 - 2024 (27 Jan Shift 1)
Let ?? = ?? ( ?? ) and ?? = ?? ( ?? ) be solutions of the differential equations
dx
dt
+ ax = 0 and
dy
dt
+ by = 0 respectively, a , b ? R. Given that ?? ( 0 ) = 2 ; ?? ( 0 ) = 1 and 3 ?? ( 1 ) = 2 ?? ( 1 ) , the
value of ?? , for which x ( t ) = y ( t ) , is :
(1) log 2
3
? 2
(2) log
4
? 3
(3) log
3
? 4
(4) log 4
3
? 2
Q6 - 2024 (27 Jan Shift 1)
If the solution of the differential equation ( 2 ?? + 3 ?? - 2 ) ???? + ( 4 ?? + 6 ?? - 7 ) ???? = 0 , ?? ( 0 ) =
3, is ???? + ???? + 3 log
?? ? | 2 ?? + 3 ?? - ?? | = 6, then ?? + 2 ?? + 3 ?? is equal to
Q7 - 2024 (27 Jan Shift 2)
If ?? = ?? ( ?? ) is the solution curve of the differential equation ( ?? 2
- 4 ) ???? - ( ?? 2
- 3 ?? ) ???? =
0, ?? > 2 , ?? ( 4 ) =
3
2
and the slope of the curve is never zero, then the value of ?? ( 10 )
equals :
(1)
3
1 + ( 8 )
1 / 4
(2)
3
1 + 2 v 2
(3)
3
1 - 2 v 2
(4)
3
1 - ( 8 )
1 / 4
Q8 - 2024 (27 Jan Shift 2)
If the solution curve, of the differential equation
????
????
=
?? + ?? - 2
?? - ?? passing through the point
( 2 , 1 ) is t a n
- 1
? (
?? - 1
?? - 1
) -
1
?? log
?? ? ( ?? + (
?? - 1
?? - 1
)
2
) = log
?? ? | ?? - 1 |, then 5 ?? + ?? is equal to
Q9 - 2024 (29 Jan Shift 1)
A function ?? = ?? ( ?? ) satisfies
?? ( ?? ) sin ? 2 ?? + sin ? ?? - ( 1 + c o s
2
? ?? ) ?? '
( ?? ) = 0 with condition ?? ( 0 ) = 0. Then ?? (
?? 2
) is equal to
(1) 1
(2) 0
(3) -1
(4) 2
Q10 - 2024 (29 Jan Shift 1)
If the solution curve ?? = ?? ( ?? ) of the differential equation ( 1 + ?? 2
) ( 1 + log
?? ? ?? ) ???? + ?? ?? ?? =
0 , ?? > 0 passes through the point ( 1 , 1 ) and ?? ( ?? ) =
?? - ta n ? (
3
2
)
?? + ta n ? (
3
2
)
, then ?? + 2 ?? is
Q11 - 2024 (29 Jan Shift 2)
If sin ? (
?? ?? ) = log
?? ? | ?? | +
?? 2
is the solution of the differential equation ?? c o s ? (
?? ?? )
????
????
= ?? c o s ? (
?? ?? ) +
?? and ?? ( 1 ) =
?? 3
, then ?? 2
is equal to
(1) 3
(2) 12
(3) 4
(4) 9
Q12 - 2024 (30 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation s e c ? x ?? y + { 2 ( 1 - x ) t a n ? x + x ( 2 -
x ) } dx = 0 such that ?? ( 0 ) = 2. Then ?? ( 2 ) is equal to :
(1) 2
(2) 2 { 1 - sin ? ( 2 ) }
(3) 2 { sin ? ( 2 ) + 1 }
(4) 1
Q13 - 2024 (30 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation ( 1 - ?? 2
) ???? = [ ???? + ( ?? 3
+
2 ) v 3 ( 1 - ?? 2
) ] ???? , - 1 < ?? < 1 , ?? ( 0 ) = 0. If ?? (
1
2
) =
?? ?? , ?? and ?? are coprime numbers,
then m + n is equal to
Q14 - 2024 (31 Jan Shift 1)
The solution curve of the differential equation
?? ????
????
= ?? ( log
?? ? ?? - log
?? ? ?? + 1 ) , ?? > 0 , ?? > 0 passing
through the point ( ?? , 1 ) is
(1) | log
?? ?
?? ?? | = ??
(2) | log
?? ?
?? ?? | = ?? 2
(3) | log
?? ?
?? ?? | = ??
(4) 2 | log
?? ?
?? ?? | = ?? + 1
Q15 - 2024 (31 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
=
( ta n ? ?? ) + ?? s i n ? ?? ( s e c ? ?? - s i n ? ?? ta n ? ?? )
,
x ? ( 0 ,
?? 2
) satisfying the condition y (
?? 4
) = 2.
Then, ?? (
?? 3
) is
(1) v 3 ( 2 + log
e
? v 3 )
(2)
v 3
2
( 2 + log
?? ? 3 )
(3) v 3 ( 1 + 2 log
?? ? 3 )
(4) v 3 ( 2 + log
e
? 3 )
Page 5
JEE Mains Previous Year Questions
(2021-2024): Differential Equations
2024
Q1 - 2024 (01 Feb Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) -
1 , ?? ( 0 ) = 1.
Then, (
1
v 2
+ y (
1
v 2
) )
2
equals :
(1)
4
4 + v e
(2)
3
3 - v e
(3)
2
1 + v e
(4)
1
2 - v e
Q2 - 2024 (01 Feb Shift 1)
If ?? = ?? ( ?? ) is the solution of the differential equation
( ?? + 1 ) ???? = ( 2 ?? + ( ?? + 1 )
4
) ???? , ?? ( 0 ) = 2, then, ?? ( 1 ) equals
Q3 - 2024 (01 Feb Shift 2)
Let ?? be a non-zero real number. Suppose ?? : R ? R is a differentiable function such that
?? ( 0 ) = 2 and l i m
x ? - 8
? f ( x ) = 1. If ?? '
( x ) = ???? ( ?? ) + 3, for all x ? R, then ?? ( - log
?? ? 2 ) is equal
to
(1) 3
(2) 5
(3) 9
(4) 7
Q4 - 2024 (01 Feb Shift 2)
If
????
????
=
1 + ?? - ?? 2
?? , ?? ( 1 ) = 1, then 5 ?? ( 2 ) is equal to :
Q5 - 2024 (27 Jan Shift 1)
Let ?? = ?? ( ?? ) and ?? = ?? ( ?? ) be solutions of the differential equations
dx
dt
+ ax = 0 and
dy
dt
+ by = 0 respectively, a , b ? R. Given that ?? ( 0 ) = 2 ; ?? ( 0 ) = 1 and 3 ?? ( 1 ) = 2 ?? ( 1 ) , the
value of ?? , for which x ( t ) = y ( t ) , is :
(1) log 2
3
? 2
(2) log
4
? 3
(3) log
3
? 4
(4) log 4
3
? 2
Q6 - 2024 (27 Jan Shift 1)
If the solution of the differential equation ( 2 ?? + 3 ?? - 2 ) ???? + ( 4 ?? + 6 ?? - 7 ) ???? = 0 , ?? ( 0 ) =
3, is ???? + ???? + 3 log
?? ? | 2 ?? + 3 ?? - ?? | = 6, then ?? + 2 ?? + 3 ?? is equal to
Q7 - 2024 (27 Jan Shift 2)
If ?? = ?? ( ?? ) is the solution curve of the differential equation ( ?? 2
- 4 ) ???? - ( ?? 2
- 3 ?? ) ???? =
0, ?? > 2 , ?? ( 4 ) =
3
2
and the slope of the curve is never zero, then the value of ?? ( 10 )
equals :
(1)
3
1 + ( 8 )
1 / 4
(2)
3
1 + 2 v 2
(3)
3
1 - 2 v 2
(4)
3
1 - ( 8 )
1 / 4
Q8 - 2024 (27 Jan Shift 2)
If the solution curve, of the differential equation
????
????
=
?? + ?? - 2
?? - ?? passing through the point
( 2 , 1 ) is t a n
- 1
? (
?? - 1
?? - 1
) -
1
?? log
?? ? ( ?? + (
?? - 1
?? - 1
)
2
) = log
?? ? | ?? - 1 |, then 5 ?? + ?? is equal to
Q9 - 2024 (29 Jan Shift 1)
A function ?? = ?? ( ?? ) satisfies
?? ( ?? ) sin ? 2 ?? + sin ? ?? - ( 1 + c o s
2
? ?? ) ?? '
( ?? ) = 0 with condition ?? ( 0 ) = 0. Then ?? (
?? 2
) is equal to
(1) 1
(2) 0
(3) -1
(4) 2
Q10 - 2024 (29 Jan Shift 1)
If the solution curve ?? = ?? ( ?? ) of the differential equation ( 1 + ?? 2
) ( 1 + log
?? ? ?? ) ???? + ?? ?? ?? =
0 , ?? > 0 passes through the point ( 1 , 1 ) and ?? ( ?? ) =
?? - ta n ? (
3
2
)
?? + ta n ? (
3
2
)
, then ?? + 2 ?? is
Q11 - 2024 (29 Jan Shift 2)
If sin ? (
?? ?? ) = log
?? ? | ?? | +
?? 2
is the solution of the differential equation ?? c o s ? (
?? ?? )
????
????
= ?? c o s ? (
?? ?? ) +
?? and ?? ( 1 ) =
?? 3
, then ?? 2
is equal to
(1) 3
(2) 12
(3) 4
(4) 9
Q12 - 2024 (30 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation s e c ? x ?? y + { 2 ( 1 - x ) t a n ? x + x ( 2 -
x ) } dx = 0 such that ?? ( 0 ) = 2. Then ?? ( 2 ) is equal to :
(1) 2
(2) 2 { 1 - sin ? ( 2 ) }
(3) 2 { sin ? ( 2 ) + 1 }
(4) 1
Q13 - 2024 (30 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation ( 1 - ?? 2
) ???? = [ ???? + ( ?? 3
+
2 ) v 3 ( 1 - ?? 2
) ] ???? , - 1 < ?? < 1 , ?? ( 0 ) = 0. If ?? (
1
2
) =
?? ?? , ?? and ?? are coprime numbers,
then m + n is equal to
Q14 - 2024 (31 Jan Shift 1)
The solution curve of the differential equation
?? ????
????
= ?? ( log
?? ? ?? - log
?? ? ?? + 1 ) , ?? > 0 , ?? > 0 passing
through the point ( ?? , 1 ) is
(1) | log
?? ?
?? ?? | = ??
(2) | log
?? ?
?? ?? | = ?? 2
(3) | log
?? ?
?? ?? | = ??
(4) 2 | log
?? ?
?? ?? | = ?? + 1
Q15 - 2024 (31 Jan Shift 1)
Let ?? = ?? ( ?? ) be the solution of the differential equation
????
????
=
( ta n ? ?? ) + ?? s i n ? ?? ( s e c ? ?? - s i n ? ?? ta n ? ?? )
,
x ? ( 0 ,
?? 2
) satisfying the condition y (
?? 4
) = 2.
Then, ?? (
?? 3
) is
(1) v 3 ( 2 + log
e
? v 3 )
(2)
v 3
2
( 2 + log
?? ? 3 )
(3) v 3 ( 1 + 2 log
?? ? 3 )
(4) v 3 ( 2 + log
e
? 3 )
Q16 - 2024 (31 Jan Shift 2)
The temperature T ( t ) of a body at time t = 0 is 160
°
F and it decreases continuously as
per the differential equation
dT
dt
= - K ( T - 80 ) , where K is positive constant. If T ( 15 ) =
120
°
F, then T ( 45 ) is equal to
(1) 85
°
F
(2) 95
°
F
(3) 90
°
F
(4) 80
°
F
Q17 - 2024 (31 Jan Shift 2)
Let ?? = ?? ( ?? ) be the solution of the differential equation
s e c
2
? ?? ?? ?? + ( ?? 2 ?? t a n
2
? ?? + t a n ? ?? ) ???? = 0
0 < ?? <
?? 2
, ?? (
?? 4
) = 0. If ?? (
?? 6
) = ?? ,
Then e
8 ?? is equal to
Answer Key
Q1 (4) Q2 (14) Q3 (3) ???? ( ?? )
Q5 (4) Q6 (29) Q7 (1) Q8 (11)
Q9 (1) Q10 (3) Q11 (1) ?? ?? ?? ( ?? )
Q ???? ( ???? ) Q14 (3) Q15 (1) Q16 (3)
Solutions
Q1
????
????
= 2 ?? ( ?? + ?? )
3
- ?? ( ?? + ?? ) - 1
?? + ?? = ??
????
????
- 1 = 2 ?? ?? 3
- ???? - 1
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Additional Information about Differential Equations: JEE Mains Previous Year Questions (2021-2024) for JEE Preparation
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Differential Equations: JEE Mains Previous Year Questions (2021-2024) JEE
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The content of Differential Equations: JEE Mains Previous Year Questions (2021-2024) is prepared as per the latest JEE syllabus.
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