Page 1
JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
Page 2
JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Answer Key
Q1 (2)
Q2 (3)
Page 3
JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Answer Key
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)
Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2
?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
Page 4
JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Answer Key
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)
Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2
?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
?? '
( ?? )= 3 ×
1
3
· ?? '
(
?? 3
)- ?? '
( 3 - ?? )
= ?? '
(
?? 3
)- ?? '
( 3 - ?? )
If g is decreasing in ( 0, ?? )
g
'
( x)< 0
f
'
(
x
3
)+ f
'
( 3 - x)< 0
f
'
(
x
3
)< f
'
( 3 - x)
?
x
3
< 3 - x
? x <
9
4
Therefore ?? =
9
4
Then 8?? = 8 ×
9
4
= 18
Q3
?? ( ?? )= 2?? + 3( ?? )
2
3
?? '
( ?? )= 2 + 2?? -1
3
= 2 (1 +
1
?? 1
3
)
= 2 (
?? 1
3
+ 1
?? 1
3
)
So, maxima ( M) at x = -1&minima ( m) at x = 0
Q4
?? ( ?? )=
?? ?? 2
- 6?? - 16
Now,
Page 5
JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Answer Key
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)
Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2
?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
?? '
( ?? )= 3 ×
1
3
· ?? '
(
?? 3
)- ?? '
( 3 - ?? )
= ?? '
(
?? 3
)- ?? '
( 3 - ?? )
If g is decreasing in ( 0, ?? )
g
'
( x)< 0
f
'
(
x
3
)+ f
'
( 3 - x)< 0
f
'
(
x
3
)< f
'
( 3 - x)
?
x
3
< 3 - x
? x <
9
4
Therefore ?? =
9
4
Then 8?? = 8 ×
9
4
= 18
Q3
?? ( ?? )= 2?? + 3( ?? )
2
3
?? '
( ?? )= 2 + 2?? -1
3
= 2 (1 +
1
?? 1
3
)
= 2 (
?? 1
3
+ 1
?? 1
3
)
So, maxima ( M) at x = -1&minima ( m) at x = 0
Q4
?? ( ?? )=
?? ?? 2
- 6?? - 16
Now,
f
'
( x)=
-( x
2
+ 16)
( x
2
- 6x - 16)
2
f
'
( x)< 0
Thus ?? ( ?? ) is decreasing in
( -8, -2)? ( -2,8)? ( 8, 8)
Q5
?? '
( ?? )=
?? '
( ?? )- ?? '
( 2 - ?? )
2
, ?? '
(
3
2
)=
?? '
(
3
2
)- ?? '
(
1
2
)
2
= 0
Also f
'
(
1
2
)=
g
'
(
1
2
) -g
'
(
3
2
)
2
= 0, f
'
(
1
2
)= 0
? f
'
(
3
2
)= f
'
(
1
2
)= 0
? roots in (
1
2
, 1) and ( 1,
3
2
)
? f
''
( x) is zero at least twice in (
1
2
,
3
2
)
Q6
f
'
( x)= ( x + 3)
2
· 3( x - 2)
2
+ ( x - 2)
3
2( x + 3)
= 5( x + 3) ( x - 2)
2
( x + 1)
f
'
( x)= 0, x = -3, -1,2
?? ( -4)= -216
?? ( -3)= 0, ?? ( 4)= 49 × 8 = 392
?? = 392, ?? = -216
?? - ?? = 392 + 216 = 608
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