JEE Main Previous Year Questions (2025): Continuity and Differentiability

 Page 1


JEE Main Previous Year Questions 
(2025): Continuity and 
Differentiability 
Q1: Let the function, 
?? (?? )={
-?? ???? ?? -?? , ?? <?? ?? ?? +?? ?? , ?? ??? 
be differentiable for all ?? ??? , where ?? >?? , ?? ??? . If the area of the region enclosed 
by ?? =?? (?? ) and the line ?? =-???? is ?? +?? v?? ,?? ,?? ??? , then the value of ?? +?? is 
____ . 
JEE Main 2025 (Online) 22nd January Morning Shift 
Ans: 34 
Solution: 
f(x) is continuous and differentiable 
 at ?? =1; LHL=RHL,LHD=RHD
-3?? -2=?? 2
+?? ,-6?? =?? ?? =2,1;?? =-12
?? (?? )={
-6?? 2
-2, ?? <1
4-12?? , ?? =1
 
 
Area =? ?
1
-v3
?(-6?? 2
-2+20)???? +? ?
2
1
?(4-12?? +20)?? ?? ] 
=16+12v3+6=22+12v3 
? ?? +?? =34 
Page 2


JEE Main Previous Year Questions 
(2025): Continuity and 
Differentiability 
Q1: Let the function, 
?? (?? )={
-?? ???? ?? -?? , ?? <?? ?? ?? +?? ?? , ?? ??? 
be differentiable for all ?? ??? , where ?? >?? , ?? ??? . If the area of the region enclosed 
by ?? =?? (?? ) and the line ?? =-???? is ?? +?? v?? ,?? ,?? ??? , then the value of ?? +?? is 
____ . 
JEE Main 2025 (Online) 22nd January Morning Shift 
Ans: 34 
Solution: 
f(x) is continuous and differentiable 
 at ?? =1; LHL=RHL,LHD=RHD
-3?? -2=?? 2
+?? ,-6?? =?? ?? =2,1;?? =-12
?? (?? )={
-6?? 2
-2, ?? <1
4-12?? , ?? =1
 
 
Area =? ?
1
-v3
?(-6?? 2
-2+20)???? +? ?
2
1
?(4-12?? +20)?? ?? ] 
=16+12v3+6=22+12v3 
? ?? +?? =34 
Q2: Let ?? (?? )={
?? ?? , ?? <?? ?????? {?? +?? +[?? ],?? +?? [?? ]}, ?? =?? =?? ?? , ?? >?? 
where [.] denotes greatest integer function. If ?? and ?? are the number of points, 
where ?? is not continuous and is not differentiable, respectively, then ?? +?? equals 
____ . 
Ans: 5 
Solution: 
?? (?? )={
3?? ; ?? <0
min{1+?? ,?? } ; 0=?? <1
min{2+?? ,?? +2} ; 1=?? <2
?? (?? )={
3x ;?? <0
x ;
x+2 ;1=?? <1
5 ;?? >2
 
 
Not continuous at x?{1,2}??? =2 
Not diff. at ?? ?{0,1,2}??? =3 
?? +?? =5 
Q3: Let ?? (?? )=?????? ?? ?8
??
?? =?? ?? ?(
?????? (?? /?? ?? +?? )+??????
?? (?? /?? ?? +?? )
?? -??????
?? (?? /?? ?? +?? )
) Then ?????? ?? ??? ?
?? ?? -?? ?? (?? )
(?? -?? (?? ))
 is equal to 
____ . 
JEE Main 2025 (Online) 28th January Evening Shift 
Ans: 1 
Page 3


JEE Main Previous Year Questions 
(2025): Continuity and 
Differentiability 
Q1: Let the function, 
?? (?? )={
-?? ???? ?? -?? , ?? <?? ?? ?? +?? ?? , ?? ??? 
be differentiable for all ?? ??? , where ?? >?? , ?? ??? . If the area of the region enclosed 
by ?? =?? (?? ) and the line ?? =-???? is ?? +?? v?? ,?? ,?? ??? , then the value of ?? +?? is 
____ . 
JEE Main 2025 (Online) 22nd January Morning Shift 
Ans: 34 
Solution: 
f(x) is continuous and differentiable 
 at ?? =1; LHL=RHL,LHD=RHD
-3?? -2=?? 2
+?? ,-6?? =?? ?? =2,1;?? =-12
?? (?? )={
-6?? 2
-2, ?? <1
4-12?? , ?? =1
 
 
Area =? ?
1
-v3
?(-6?? 2
-2+20)???? +? ?
2
1
?(4-12?? +20)?? ?? ] 
=16+12v3+6=22+12v3 
? ?? +?? =34 
Q2: Let ?? (?? )={
?? ?? , ?? <?? ?????? {?? +?? +[?? ],?? +?? [?? ]}, ?? =?? =?? ?? , ?? >?? 
where [.] denotes greatest integer function. If ?? and ?? are the number of points, 
where ?? is not continuous and is not differentiable, respectively, then ?? +?? equals 
____ . 
Ans: 5 
Solution: 
?? (?? )={
3?? ; ?? <0
min{1+?? ,?? } ; 0=?? <1
min{2+?? ,?? +2} ; 1=?? <2
?? (?? )={
3x ;?? <0
x ;
x+2 ;1=?? <1
5 ;?? >2
 
 
Not continuous at x?{1,2}??? =2 
Not diff. at ?? ?{0,1,2}??? =3 
?? +?? =5 
Q3: Let ?? (?? )=?????? ?? ?8
??
?? =?? ?? ?(
?????? (?? /?? ?? +?? )+??????
?? (?? /?? ?? +?? )
?? -??????
?? (?? /?? ?? +?? )
) Then ?????? ?? ??? ?
?? ?? -?? ?? (?? )
(?? -?? (?? ))
 is equal to 
____ . 
JEE Main 2025 (Online) 28th January Evening Shift 
Ans: 1 
Solution: 
?? (?? )=lim
?? ?8
??
?? =0
?? ?(tan 
?? 2
?? -tan 
?? 2
?? +1
)=tan ?? 
lim
?? ?0
?(
?? ?? -?? tan ?? ?? -tan ?? )=lim
?? ?0
??? tan ?? (?? ?? -tan ?? -1)
(?? -tan ?? )
 
=1 
Q4: Let [?? ] be the greatest integer less than or equal to ?? . Then the least value of ?? ?
?? for which ?????? ?? ??? +?(?? ([
?? ?? ]+[
?? ?? ]+?+[
?? ?? ])-?? ?? ([
?? ?? ?? ]+[
?? ?? ?? ?? ]+?+[
?? ?? ?? ?? ])=?? is 
equal to ____ . 
JEE Main 2025 (Online) 29th January Morning Shift 
Ans: 24 
Solution: 
To find the least natural number ?? for which the following inequality holds: 
lim
?? ?0
+?(?? ([
1
?? ]+[
2
?? ]+?+[
?? ?? ])-?? 2
([
1
?? 2
]+[
2
2
?? 2
]+?+[
9
2
?? 2
]))=1 
we simplify the expression inside the limit. 
As ?? ?0
+
,[
?? ?? ] approximates to 
?? ?? . Thus, the problem becomes finding: 
(1+2+?+?? )-(1
2
+2
2
+?+9
2
)=1 
The sum of the first ?? natural numbers is given by: 
?? (?? +1)
2
 
And the sum of the squares of the first 9 natural numbers is: 
1
2
+2
2
+?+9
2
=
9·10·19
6
 
Thus, the inequality becomes: 
?? (?? +1)
2
-
9·10·19
6
=1 
Solving this, we rewrite: 
?? (?? +1)=572 
The least natural number ?? satisfying this condition is 24 . 
Q5: If ?????? ?? ??? ?(
?????? ?? ?? )
?? ?? ?? =?? , then ???? ?????? ?? ?? is equal to ____ 
JEE Main 2025 (Online) 3rd April Evening Shift 
Ans: 32 
Solution: 
Page 4


JEE Main Previous Year Questions 
(2025): Continuity and 
Differentiability 
Q1: Let the function, 
?? (?? )={
-?? ???? ?? -?? , ?? <?? ?? ?? +?? ?? , ?? ??? 
be differentiable for all ?? ??? , where ?? >?? , ?? ??? . If the area of the region enclosed 
by ?? =?? (?? ) and the line ?? =-???? is ?? +?? v?? ,?? ,?? ??? , then the value of ?? +?? is 
____ . 
JEE Main 2025 (Online) 22nd January Morning Shift 
Ans: 34 
Solution: 
f(x) is continuous and differentiable 
 at ?? =1; LHL=RHL,LHD=RHD
-3?? -2=?? 2
+?? ,-6?? =?? ?? =2,1;?? =-12
?? (?? )={
-6?? 2
-2, ?? <1
4-12?? , ?? =1
 
 
Area =? ?
1
-v3
?(-6?? 2
-2+20)???? +? ?
2
1
?(4-12?? +20)?? ?? ] 
=16+12v3+6=22+12v3 
? ?? +?? =34 
Q2: Let ?? (?? )={
?? ?? , ?? <?? ?????? {?? +?? +[?? ],?? +?? [?? ]}, ?? =?? =?? ?? , ?? >?? 
where [.] denotes greatest integer function. If ?? and ?? are the number of points, 
where ?? is not continuous and is not differentiable, respectively, then ?? +?? equals 
____ . 
Ans: 5 
Solution: 
?? (?? )={
3?? ; ?? <0
min{1+?? ,?? } ; 0=?? <1
min{2+?? ,?? +2} ; 1=?? <2
?? (?? )={
3x ;?? <0
x ;
x+2 ;1=?? <1
5 ;?? >2
 
 
Not continuous at x?{1,2}??? =2 
Not diff. at ?? ?{0,1,2}??? =3 
?? +?? =5 
Q3: Let ?? (?? )=?????? ?? ?8
??
?? =?? ?? ?(
?????? (?? /?? ?? +?? )+??????
?? (?? /?? ?? +?? )
?? -??????
?? (?? /?? ?? +?? )
) Then ?????? ?? ??? ?
?? ?? -?? ?? (?? )
(?? -?? (?? ))
 is equal to 
____ . 
JEE Main 2025 (Online) 28th January Evening Shift 
Ans: 1 
Solution: 
?? (?? )=lim
?? ?8
??
?? =0
?? ?(tan 
?? 2
?? -tan 
?? 2
?? +1
)=tan ?? 
lim
?? ?0
?(
?? ?? -?? tan ?? ?? -tan ?? )=lim
?? ?0
??? tan ?? (?? ?? -tan ?? -1)
(?? -tan ?? )
 
=1 
Q4: Let [?? ] be the greatest integer less than or equal to ?? . Then the least value of ?? ?
?? for which ?????? ?? ??? +?(?? ([
?? ?? ]+[
?? ?? ]+?+[
?? ?? ])-?? ?? ([
?? ?? ?? ]+[
?? ?? ?? ?? ]+?+[
?? ?? ?? ?? ])=?? is 
equal to ____ . 
JEE Main 2025 (Online) 29th January Morning Shift 
Ans: 24 
Solution: 
To find the least natural number ?? for which the following inequality holds: 
lim
?? ?0
+?(?? ([
1
?? ]+[
2
?? ]+?+[
?? ?? ])-?? 2
([
1
?? 2
]+[
2
2
?? 2
]+?+[
9
2
?? 2
]))=1 
we simplify the expression inside the limit. 
As ?? ?0
+
,[
?? ?? ] approximates to 
?? ?? . Thus, the problem becomes finding: 
(1+2+?+?? )-(1
2
+2
2
+?+9
2
)=1 
The sum of the first ?? natural numbers is given by: 
?? (?? +1)
2
 
And the sum of the squares of the first 9 natural numbers is: 
1
2
+2
2
+?+9
2
=
9·10·19
6
 
Thus, the inequality becomes: 
?? (?? +1)
2
-
9·10·19
6
=1 
Solving this, we rewrite: 
?? (?? +1)=572 
The least natural number ?? satisfying this condition is 24 . 
Q5: If ?????? ?? ??? ?(
?????? ?? ?? )
?? ?? ?? =?? , then ???? ?????? ?? ?? is equal to ____ 
JEE Main 2025 (Online) 3rd April Evening Shift 
Ans: 32 
Solution: 
To solve the given limit problem, we start by analyzing the expression: 
lim
?? ?0
?(
tan ?? ?? )
1
?? 2
=?? 
This limit exhibits the indeterminate form 1
8
. To handle this form, we use the transformation: 
?? =?? lim
?? ?0
?(
tan ?? ?? -1)
1
?? 2
 
Expanding tan ?? using its Taylor series near ?? =0, we have: 
tan ?? =?? +
?? 3
3
+
2
15
?? 5
+? 
Substituting this expansion into the limit, we get: 
tan ?? -?? ?? 3
=
(?? +
?? 3
3
+
2
15
?? 5
+?-?? )
?? 3
=
?? 3
3
+
2
15
?? 5
+?
?? 3
 
This simplifies to: 
?? 3
3?? 3
=
1
3
 
Thus, the limit becomes: 
?? =?? 1
3
 
log
?? ?? =
1
3
 
Finally, computing 96log
?? ?? : 
96log
?? ?? =96·
1
3
=32 
Q6: Let ?? and ?? be the number of points at which the function ?? (?? )=
?????? {?? ,?? ?? ,?? ?? ,…?? ????
},?? ?R, is not differentiable and not continuous, respectively. 
Then ?? +?? is equal to ____ . 
JEE Main 2025 (Online) 4th April Morning Shift 
Ans: 3 
Solution: 
for ?? =1,?? 21
=?? 19
=?=?? . 
?? (?? )={
?? ?? <-1
?? 21
-1=?? =0
?? 0<?? <1
?? 21
?? =1
 
Page 5


JEE Main Previous Year Questions 
(2025): Continuity and 
Differentiability 
Q1: Let the function, 
?? (?? )={
-?? ???? ?? -?? , ?? <?? ?? ?? +?? ?? , ?? ??? 
be differentiable for all ?? ??? , where ?? >?? , ?? ??? . If the area of the region enclosed 
by ?? =?? (?? ) and the line ?? =-???? is ?? +?? v?? ,?? ,?? ??? , then the value of ?? +?? is 
____ . 
JEE Main 2025 (Online) 22nd January Morning Shift 
Ans: 34 
Solution: 
f(x) is continuous and differentiable 
 at ?? =1; LHL=RHL,LHD=RHD
-3?? -2=?? 2
+?? ,-6?? =?? ?? =2,1;?? =-12
?? (?? )={
-6?? 2
-2, ?? <1
4-12?? , ?? =1
 
 
Area =? ?
1
-v3
?(-6?? 2
-2+20)???? +? ?
2
1
?(4-12?? +20)?? ?? ] 
=16+12v3+6=22+12v3 
? ?? +?? =34 
Q2: Let ?? (?? )={
?? ?? , ?? <?? ?????? {?? +?? +[?? ],?? +?? [?? ]}, ?? =?? =?? ?? , ?? >?? 
where [.] denotes greatest integer function. If ?? and ?? are the number of points, 
where ?? is not continuous and is not differentiable, respectively, then ?? +?? equals 
____ . 
Ans: 5 
Solution: 
?? (?? )={
3?? ; ?? <0
min{1+?? ,?? } ; 0=?? <1
min{2+?? ,?? +2} ; 1=?? <2
?? (?? )={
3x ;?? <0
x ;
x+2 ;1=?? <1
5 ;?? >2
 
 
Not continuous at x?{1,2}??? =2 
Not diff. at ?? ?{0,1,2}??? =3 
?? +?? =5 
Q3: Let ?? (?? )=?????? ?? ?8
??
?? =?? ?? ?(
?????? (?? /?? ?? +?? )+??????
?? (?? /?? ?? +?? )
?? -??????
?? (?? /?? ?? +?? )
) Then ?????? ?? ??? ?
?? ?? -?? ?? (?? )
(?? -?? (?? ))
 is equal to 
____ . 
JEE Main 2025 (Online) 28th January Evening Shift 
Ans: 1 
Solution: 
?? (?? )=lim
?? ?8
??
?? =0
?? ?(tan 
?? 2
?? -tan 
?? 2
?? +1
)=tan ?? 
lim
?? ?0
?(
?? ?? -?? tan ?? ?? -tan ?? )=lim
?? ?0
??? tan ?? (?? ?? -tan ?? -1)
(?? -tan ?? )
 
=1 
Q4: Let [?? ] be the greatest integer less than or equal to ?? . Then the least value of ?? ?
?? for which ?????? ?? ??? +?(?? ([
?? ?? ]+[
?? ?? ]+?+[
?? ?? ])-?? ?? ([
?? ?? ?? ]+[
?? ?? ?? ?? ]+?+[
?? ?? ?? ?? ])=?? is 
equal to ____ . 
JEE Main 2025 (Online) 29th January Morning Shift 
Ans: 24 
Solution: 
To find the least natural number ?? for which the following inequality holds: 
lim
?? ?0
+?(?? ([
1
?? ]+[
2
?? ]+?+[
?? ?? ])-?? 2
([
1
?? 2
]+[
2
2
?? 2
]+?+[
9
2
?? 2
]))=1 
we simplify the expression inside the limit. 
As ?? ?0
+
,[
?? ?? ] approximates to 
?? ?? . Thus, the problem becomes finding: 
(1+2+?+?? )-(1
2
+2
2
+?+9
2
)=1 
The sum of the first ?? natural numbers is given by: 
?? (?? +1)
2
 
And the sum of the squares of the first 9 natural numbers is: 
1
2
+2
2
+?+9
2
=
9·10·19
6
 
Thus, the inequality becomes: 
?? (?? +1)
2
-
9·10·19
6
=1 
Solving this, we rewrite: 
?? (?? +1)=572 
The least natural number ?? satisfying this condition is 24 . 
Q5: If ?????? ?? ??? ?(
?????? ?? ?? )
?? ?? ?? =?? , then ???? ?????? ?? ?? is equal to ____ 
JEE Main 2025 (Online) 3rd April Evening Shift 
Ans: 32 
Solution: 
To solve the given limit problem, we start by analyzing the expression: 
lim
?? ?0
?(
tan ?? ?? )
1
?? 2
=?? 
This limit exhibits the indeterminate form 1
8
. To handle this form, we use the transformation: 
?? =?? lim
?? ?0
?(
tan ?? ?? -1)
1
?? 2
 
Expanding tan ?? using its Taylor series near ?? =0, we have: 
tan ?? =?? +
?? 3
3
+
2
15
?? 5
+? 
Substituting this expansion into the limit, we get: 
tan ?? -?? ?? 3
=
(?? +
?? 3
3
+
2
15
?? 5
+?-?? )
?? 3
=
?? 3
3
+
2
15
?? 5
+?
?? 3
 
This simplifies to: 
?? 3
3?? 3
=
1
3
 
Thus, the limit becomes: 
?? =?? 1
3
 
log
?? ?? =
1
3
 
Finally, computing 96log
?? ?? : 
96log
?? ?? =96·
1
3
=32 
Q6: Let ?? and ?? be the number of points at which the function ?? (?? )=
?????? {?? ,?? ?? ,?? ?? ,…?? ????
},?? ?R, is not differentiable and not continuous, respectively. 
Then ?? +?? is equal to ____ . 
JEE Main 2025 (Online) 4th April Morning Shift 
Ans: 3 
Solution: 
for ?? =1,?? 21
=?? 19
=?=?? . 
?? (?? )={
?? ?? <-1
?? 21
-1=?? =0
?? 0<?? <1
?? 21
?? =1
 
 
Clearly, ?? (?? ) is continuous everywhere. 
??? =0{1 ;?? <-1
?? '
(?? )={
21?? 20
;-1=?? =0
1 ;0<?? <1
21·?? 20
;?? =1
 
??? =3 
??? +?? =3 
Q7: The number of points of discontinuity of the function ?? (?? )=[
?? ?? ?? ]-[v?? ],?? ?
[?? ,?? ], where [·] denotes the greatest integer function, is ____ . 
Solution: 
To determine the points of discontinuity of the function ?? (?? )=[
?? 2
2
]-[v?? ], where [.] denotes 
the greatest integer function, we need to identify possible values of ?? where discontinuities 
might occur within the interval [0,4]. 
Discontinuity Analysis 
For the term [
?? 2
2
] : 
The probable values of ?? that could cause discontinuities are the roots or specific values where 
the integer part changes between consecutive integers. The transitions happen when: 
=1,2,3,4,5,6,7,8 
??? =v2,2,v6,2v2,v10,2v3,v14,4 
For the term [v?? ] : 
The values of ?? where [v?? ] changes are straightforward. They occur at: 
?? =1,2 
Discontinuity Check