JAM 2026 Question Paper Mathematics (MA)

JAM 2026 Confidential Mathematics (MA)

Special Instructions / Useful Data

= The set of all integers
!"= The set of all rational numbers
#" ="The set of all real numbers
$" ="The set of all complex numbers
#% " = # × # × & × #, where # occurs ' times
$% " = $ × $ × & × $, where $ occurs ' times
(% ="The symmetric group of all permutations on 1,2,3, … , '
% =""The additive group of integers modulo '
)* ,% +#- " =" The set of all . × ' matrices with real entries
*,%

* ,% +$- " =" The set
)*,% et of all . × ' matrices with complex entries
)% +#- " =" The set of alla l ' × ' matrices
)% +$- " =" The set of all
aall ' × ' matrices
mat rices with
atri
ri with reall entries
entr
trie
ies
ie

) " =" The transpose ix )
mat atri
atrice
ricess with
ce with complex
ccom
ompl
omplex
plex entries
eent
ntri
ntries
ries
/

) " =" The conjugate ix )
se ooff th
the matrix
matrtrix
tr
0

4 5 6 = The set of all el ements of 4 which are not in 6, for sets 4 aand nd 6
transpose
te ttra
rans
ra nspose of thee ma
ns matrix
matr
trix
tr

[7] " =" The largest int or equal to 7 for any er 7
elements
elem
em
nteg
eger
eg
integer er less than
an o ny rreaeal nu
real number
8 9 +7 - " = "
:;
:<

8 +7 - = " ?
+>- :? ;
:<
@ A 8 is the composition
ionn function
compositi
itio
io func
ncti
nc on given
tion
ti ggiv
iven
iv by +@ A 8-+7 - = @B8
en by @BB8+7 -C

MA Page 2 of 38

JAM 2026 Confidential Mathematics (MA)

Section A: Q.1 – Q.10 Carry ONE mark each.

K%
For each positive integer ', let 7% = 1 D +D1-% E and G% = H1 E J L
Q.1 F F
% I%

Which ONE of the following statements about the sequences (7% -""and (G
G% -"is
TRUE?

(A) (7% - and (G
G% - are convergent.

(B) (7% - is not conve
convergent
co verg
vergen
rgentt an
en (G% - iiss no
andd (G nott convergent.
conv
conver
nverge
er gent
ge nt.
nt

(C) (7% - is converg
convergent
verg
rgen
rg G% - iss no
ent and (G
en nott convergent.
conv
conver
nverge
er gent
ge nt..
nt

(D) (7% - is not con
convergent
co
onve
on (G% - is convergent.
vergentt andd (G
ve

Q.2 positive
For each posi
osit
itiv
itivee in
iv teger '
integer
inte
te ',, d inee 7% = +D
define
def
efin
efin 1-% L
D1
Which ONE
E of
o the
t ffol
ollo
ollowi
lowing
wi ng ssta
following tate
ta tement
te nts abou
nt
statements outt th
about the se quence ++7
sequ
qu
sequence 7% - is FALSE?

ts M N O ssuc
(A) There exists such
uchh th
uc at ||7
that 1|| P M ffor
7% D 1 or aall
ll p
pos
positive
osit
os itiv
it ivee in
iv inte gers '.
integers
ntegers
tege
te ge

(B) There exists M N O such that for all ) N O there exists a positive integer
Q N ) for which |7R D 1| N M.

(C) For all M N O and ) N O there exists a positive integer Q such that
Q N ) and |7R D 1| P MM.

(D) For all M N O and ) N O there exists a positive integer Q such that Q N )
and |7R D 1| N M.

MA Page 3 of 38

JAM 2026 Confidential Mathematics (MA)

Q.3 Let 8S +1, T- U # be a differentiable function such that

8 9 +7- = +8 +7 - D V7-I E V for all 7 W +1, T-.

Which ONE of the following is a possible value of 8 +3- D 8+2-?

1
VE
X
(A)

1
VD
X
(B)

(C) V 1
E
2 3

(D) V 1
E
2 2

MA Page 4 of 38

JAM 2026 Confidential Mathematics (MA)

Q.4 The general solution of the differential equation

7 sin H J " = G sin H J E " ,"""""7" N "O
Y ZG Y <
< Z7 < I

is given by ______.

(A) cos H< J E log +7 I - = ^, where ^ is an arbitrary constant
Y \

(B) cos HYJ E log +7 I - = ^, where ^ is an arbitrary constant
< \

(C) cos H< J E log B_7
7CC = ^
^,, wheree ^ iis an aarb
Y \ wher
where
er arbitrary
rbit
rb itra
it rary
rary ccon
constant
onstant
on

(D) cos HYJ E log B_7
7CC = ^, wh
where ^ is an arbitrary
< \ ry cconstant

MA Page 5 of 38

JAM 2026 Confidential Mathematics (MA)

Q.5 Let 8S #K U # be a continuous function. Then

I < Y

` ` ` 8 +7, G, a-ZaZGZ7 "" = " ddddddL
b b b

I Y I

` ` ` 8+7, G, a-Z7ZaZG
(A)

b b Y

I < I

` ` ` 8 +7, G, a-ZGZaZ7
ZG
(B)

b b e

I I <

-Za
` ` ` 8+7, G, aa-ZaZ7ZG
ZaZ7
Z7ZG
ZG
(C)

b Y b

I I <

-ZG
` ` ` 8+7, G, aa-ZGZ7Za
ZGZ7
Z7Za
Z7 Za
(D)

b e b

MA Page 6 of 38

JAM 2026 Confidential Mathematics (MA)

Q.6 Let 8S +1, 2- U # be a continuous function satisfying
lim 8+7 - = 3""hnj"" lim 8 +7 - = 3L
<UFf <UIk

Then which ONE of the following is necessarily TRUE?

(A) 8 is bounded and 8 has a maximum or a minimum but not both.

(B) 8 is bounded and 8 has a maximum or a minimum or both.

(C) 8 is boundedd and 8 has neither a maximum nor a minimum.

(D) 8 is unbounde
unbounded an 8 has
dedd and
de as eeit
either
ithe
it herr no m
he max
maximum
axim
aximum
imum o
orr no minimum
minimum.
um.
um

Q.7 Let 8S # U # bbee a twicee differentiable function hat 8 +'- = 1 for all
n such that
tha
ha
'W" .
Which ONE
E of
o the
the ffol
ollo
ollowi
lowing
wi ng ssta
following tate
ta teme
te ment
me
statementsnts is necessarily
nt neces
essa
sari
sa rily
rily TRUE?
TRU
TRUE?
RU E?

(A) 8 9 +'- = O for all ' W
or all .

(B) 8 9 +7 - = O for iinfinitely
inf
nfin
nf init
in itel
it many 7 W # 5
elyy many
el .

(C) 8 99 +'- = O for all ' W .

(D) 8 99 +7 - = O for infinitely many 7 W # 5 .

MA Page 7 of 38

JAM 2026 Confidential Mathematics (MA)

Q.8 Let 4 be an invertible p × p real matrix. Let 6 be the matrix obtained by
interchanging the second and third columns of 4 and then adding 3 times the
third column to the fifth column.
Which ONE of the following statements is TRUE?

(A) 6kF is obtained by interchanging the second and third rows of 4kF , and then
adding 3 times the third row to the fifth row.

(B) 6kF is obtained
ined by interchanging the second and third rows of 4kF , and then
adding D3 times
times the fifth row to the third row.

(C) 6kF is obtaine
obtained
ined b adding D3
ed by 3 ttim
times
imes
im es the
the third
tthi
hird
hird row
row to the fifth
h row,
ro and then
interchanging thee second and third rows of 4kF .
ing th

(D) 6kF is obtaine
obtained
ined
ed by addi
ad ng 3 tim
by adding
ding
di times
imes
imes the
he fifth
ffif
ifth
th row to
to the
the third
thir
third
ir
rdd row,
row, and
a then
interchanging
ing th
thee se
second
nd aand third
nd tthi
hird
hi ows of 4kF .
rows
rd rrow
ow

Q.9 Which ONE
E of
o tthe
he ffol
following
ollo
ollowing sta
lo statements
tate
ta teme
te ment
ment
me nts is F
FAL
FALSE?
ALSE?
AL

(A) (K is isomorphic
rphic to a subgroup of (q .

(B) K is isomorphic to a subgroup of (q .

(C) (K is isomorphic to a quotient group of (q .

(D) r is isomorphic to a quotient group of (q .

MA Page 8 of 38

JAM 2026 Confidential Mathematics (MA)

Q.10 Let t be the triangular region in the plane with vertices at +O,O-, +1,O- and +1,1-.
Let 8 be a function defined from t to # given in polar coordinates.
Then the double integral of 8 over t is equal to ______.

_I vw _I cosD1 BFw}C
q
` `
u` 8 Zxy z Zz E ` `
{` 8 Zx~ zZz
(A)
b b F b

(B)
_I vw _
_II cosD1 BFw} C
q
` `
u` Zx y zZz"" D `
8 Zxy {`
{` 8 Zx
Zx~~ zZz
zZ
b b F b

_I •w _I
_ I sinD1 BFw}C
q
` `
u` 8 Z€y z Zz E `
Z€y `
{`
{ 8Z
Z€~
€~ zZ
zZzz
(C)
b b F b

_I •w _I
_ I sinD1 BFw}C
q
` `
u` 8 Z€y z Zz D `
Z€y `
{`
{ 8Z
Z€~
€~ zZ
zZz
(D)
b b F b

MA Page 9 of 38

JAM 2026 Confidential Mathematics (MA)

Section A: Q.11 – Q.30 Carry TWO marks each.

Q.11 Given a sequence of real numbers (•% -, define

• , i„""•% † O • , i„""•% P O
‚% = ƒ % and ‡% = ƒ %
O, i„""•% P O O, i„""•% † O

for each positive integer '.
Which ONE of the following statements is TRUE?

(A) If ˆ‰
%ŠF •% does not converge, then ˆ%ŠF ‚% does not converge.
‰

(B) If ˆ‰ verges bbut ˆ‰
%ŠF •% conve
converges
conv
ve %ŠF||•
•% | d
does not converge, th en ˆ‰
then ŠF ‚%
%ŠF
%Š

converges.

(C) If ˆ‰
%ŠF •% converges but ˆ‰
converges
conv
onve
onve %ŠF|•% | does not converge, then
en ˆ‰
% ŠF ‚% does not
%ŠF
Š

converge.

(D) If ˆ‰
%ŠF •% con
converges,
conv
onve
onverg
ve rges
rges, th en ˆ‰
then %ŠF
% F ‚% con
ŠF converges
onve
verg
rges orr ˆ‰
es o ŠF ‡% c
%ŠF
% converges.
con
onverges.
onve
on ve

MA Page 10 of 38

JAM 2026 Confidential Mathematics (MA)

Q.12 For which one of the following pairs of values of • and ‚, does the series
‰
converge for all 7 W +•, ‚-?
+K<-?
‹
>ŠF I +Fk<-
_? ?

(A) • = kF and ‚ = F
q I

(B) • = D1 and ‚ = F
Œ

(C) • = kF and ‚ = F
I q

(D) • = kF and ‚ = F
q K

Q.13 Let 8S # U # bbee a function
func
nction
on that
ttha
hat has derivatives of all
ha a orders.
ord
orders.
rd s.
Which ONE
E of
o the
he ffol
ollo
ollowi
lowing
ng stateme
following ments is FALSE?
me
statements FALSE
SE??

(A) If 8 is h •olŽnomi r- + -
•olŽnomihl
lŽno
nomi
nomihl
hl in 7 o„ j•
j•g•••
j•g•
g•••
g• •• h‘ mo
mos‘ p
p,, ‘’
‘’•n
•n 8 ++r- 7 = O ffor all 7 W #.

(B) If 8 +r- +7- = O ffor ll 7 W #
or aall #,, th en 8 iiss a po
then poly
polynomial
lyno
ly nomi
nomial
mi in 7 o
al in off de
degr
degree
egree
gree at most 5.
gr

(C) If 8 +>- +•- = O for all 1 “ ” “ p and 8 +r- +•- N O for some • W #, then 8+7-
has a local minimum at •.

(D) If 8 +>- +•- = O for all 1 “ ” “ X and 8 +•- +•- P O for some • W #, then 8+7 -
has a local maximum at •.

MA Page 11 of 38

JAM 2026 Confidential Mathematics (MA)

Q.14 Let 4 be a p × 3 real matrix of rank 2 and ‚ be a non-zero p × 1 ••hl column
1 T
vector. Suppose {2~ and {p~ are two solutions of the system of linear equations
3 X
47 = ‚.
Which ONE of the following is also a solution of 47 = ‚?

3
{3~
(A)

3

p
{–~
(B)

—

p
{X~
(C)

—

–
{˜~
(D)

—

MA Page 12 of 38

JAM 2026 Confidential Mathematics (MA)

Q.15 Let 8 ™ #I U # be the function given by

8 +7, G- = +7 I D 1-I E +G I D 1-I for all +7, G- W #I L

Which ONE of the following statements is TRUE?

(A) 8 has local maxima at exactly two points.

(B) 8 has local minima at exactly two points.

(C) 8 has local minim
minima
m ima at exactly three points.
im

(D) 8 has exactly
tly four
four saddle points.
poin
ints
ints.
ts

Q.16 The orthogona
orthogonal
onal
al ttraje
trajectories
ject
je ctoriess of tthe
ct he ffam
family
amil
amily of curves
il ccur
urve
vess
ve

G = " D3
D37 D 3 E .š < re parameter
for a real par
aram
ar amet
am er .
eter
et

____
____
__ ___.
__
are given by ______._.

(A) 7 = Y D F E ” š kKY , ” is a real parameter
K ›

(B) 7 = Y D F E ” š KY , ” is a real parameter
K ›

(C) 7 = kY E F E ” š kKY , ” is a real parameter
K ›

(D) 7 = kY E F E ” š KY , ” is a real parameter
K ›

MA Page 13 of 38

JAM 2026 Confidential Mathematics (MA)

Q.17 Let 8 ™ #I U # be the function defined by

p 2 ! " 4! #
, ( , !) $ (0,0)
1 2 + !2
8+7, G- = œ
0,%%%%%%%%%%%%%%%%%%%( , !) = (0,0)
Which ONE of the following statements is FALSE?

(A) & is continuous at (0,0)

(B) '*
'-
at (0,1) = 0

(C) '*
'.
at (1,0) = 5

(D) /&
ouss at (0 ,0)
0,0
/!
is continuou
continuous
nuou
ou

MA Page 14 of 38

JAM 2026 Confidential Mathematics (MA)

Let 3: 5" , 8 9 ; be the solution of the differential equation
Q.18 6 6
7 7

<!
(cos ) " ! = 2! 7 (sin " >) cos
<

satisfying 3(0) = @ Then 3 5 8 =______.
? 6
7 A

(A) 2

(B) B + 2C2
2

(C) >
2

(D) >
C2

MA Page 15 of 38

JAM 2026 Confidential Mathematics (MA)

Q.19 The value of the following expression
-

lim E F GH |sin I| <I
-9D
J

is __
_______K
____K

F6 + >
2(F 6 " >)
(A)

F6 " >
2(F 6 + >)
(B)

(C) 2(F 6 + >)
F6 " >

(D) 2(F 6 " >)
F6 + >

clee ! = C
CLL "M" 2 6
7
Q.20 The semi-circ
ircl
cl
semi-circle is rrot
rotated
otat
ot ated
ated clockwise
cclo
lock
lo ckwi
ckwise
wise by
by the
th angle in the

plane about th
the origin.
igin
What is the area of the planar region traced out by the semi-circle?

(A) LN

(B) 4N

(C) 2N

(D) N

MA Page 16 of 38

JAM 2026 Confidential Mathematics (MA)

Q.21 Let &: [>, O] 9 ; be a non-constant continuous function such that & (>) = & (O).

Which ONE of the following statements is necessarily TRUE?

(A) There exists P Q [4, R] such that & (P) = &(P + >).

(B) There exists P Q [B, S] such that & (P) = &(P + 2).

(C) There exists P Q [2, L] such that & (P) = &(P + B).

ts P Q [>, M] suc
(D) There exists such
uchh th
uc at & (P) = &(P + 4).
that

Q.22 Let T be the
he sol id of intersection of two solid spheres T?, T7 g
ssolid
olid
ol given
giv
iven by
iv
T? : 7
+ ! 7 + (U " >)7 V 4%%%%%%and%%%%%%T
4%%%%%%and%%%%%%T7 : 7
+ ! 7 + (U + >)7 V 4 .
Which ONE
E of the following
the ffol
ollo
ollowi
lowing ite
wi iterated
tera
rate
rated integrals
te integr als expresses
gral expr
pres sess the
esse
es se volu of T?
the volume
vo

C# C#G- X
C#G-
RE E W4 "
55W4 >88 <! <
7 " !7 " >
(A)

J J

7 CAG-
C
CA G- X
RE E 5W4 " 7 " ! 7 " >8 <! <
(B)

J J

7 CAG-
C
CA G- X
2E E 5W4 " 7 " ! 7 " >8 <! <
(C)

J J

C# C#G- X
2E E 5W4 " 7 " ! 7 " >8 <! <
(D)

J J

MA Page 17 of 38

JAM 2026 Confidential Mathematics (MA)

Q.23 Consider the differential equation

<7! <!
Y + YBY + 2yY = Ysin(F )K
< 7 <
Which ONE of the following is a particular solution of this differential
equation?

(A) "F G- sin(F - )

(B) "F G7- sin(F - )

(C) "F G- (sin(FF - ) + cos(F - ))))

(D) "F G7- (sin(
(sin(F
(F - ) + cos(F - ))

Q.24 Let F? , F7 , F# , FA , FZ dden
denote
enot
en ote the standard
ot stan
anda
an dard bas
da basis
asis
is vectors
rs o
off th
thee re
real
al vector
v space ;Z .
Let \ be the
the subspace
subs
subspa
bs pace
pa of ;Z spanned
ce of sspa
panned by
pa y the orss F? + F7 , F7 + F# ,
the vectors
vector
or
F# + FA + FZ . Consider
Consid
Co ider
id er the
he rea
eall ve
ea
real vect
ctor
ctor space
vector Z (;)K Let
s ce ^7,Z
7,Z Let

Z (;) b nullspace
{ Q ^7,Z
T = {` 7,Z nul lspace(`) = \}K
ulls
ul ls \}K

E of the following statements
Which ONE statement
nts is TRUE?
nt

(A) T is the empty set.

(B) T = {0}K

(C) T is non-empty and T is not a subspace of ^7,Z (;).

(D) T is a nonzero subspace of ^7,Z (;).

MA Page 18 of 38

JAM 2026 Confidential Mathematics (MA)

Q.25 Let ` be a nonzero complex f × f matrix such that g h `g j 0 for all column
vectors g in kq .
Which ONE of the following statements is necessarily TRUE?

(A) All eigenvalues of ` are negative real numbers.

(B) If g? , g7 are column vectors in kq such that `g? = g7 and `g7 = g? ,
then g? = g7 .

(C) g h `g $ 0 for ll nonzero column vectors g in kq .
or all

uess of `h are neg
(D) All eigenvalue
ue
eigenvalues negative
egat
egativ
ativee re
iv real
al n
num
numbers.
umbe
umbers
bers.
rs

Q.26 Let &: r t v bbee a surjective
surj
surjective
rj ve g
gro
group
roup
ro up homomorphism
hom
homom
omomor
orph
phis
ism
is m from
from an abel group r to
an abelian
ab
group
the additive gro
roup
ro rs vK L
integers
up ooff in
intege
gers
ge Let w denote th
thee ke off &K%K%
kernel o

Which ONE
E of following
o tthe follo
lowi
lowing
wi statements
ng ssta
tate
ta teme
te ment
me FALSE?
ntss is F
nt FAL
ALSE
ALSE??
SE

(A) There exists a group homomorphism 3: x t r such that z ~ 3 is the identity
homomorphism.

(B) w is isomorphic to a quotient group of rK

(C) For all non-zero homomorphisms •: x t r and €: • t w, the composition
€ ~ • is non-zero.

(D) r has a subgroup isomorphic to vK

MA Page 19 of 38

JAM 2026 Confidential Mathematics (MA)

Q.27 Which ONE of the following statements is FALSE?

(A) There is a surjective group homomorphism from the additive group of rational
numbers to the multiplicative group of all complex roots of unity.

(B) The multiplicative group of complex numbers of modulus one is isomorphic to
a quotient group of the additive group of real numbers.

(C) Any group homomorphism from the multiplicative group of nonzero complex
numbers into
to the group
he ggro
roup
ro up ooff al invertible
alll in
inve
vert
ve rtib
rt le 2 × 2 m
ible
ib matrices
mat
atri
atrice
ricess with
ce with
h real
r entries has
nontrivial kerne
erne
nel.
ne l.
kernel.

(D) There exists
ts a ggro
group
roup homomorphism from the symmetric gro
ro group
roup
ro o f symbols
up on
multiplicative
into the multi
ultipl
plic
pl oupp of all invertible f × f mat
group
icative grou
ic ou matrices
atri
atricess with
ri wi real entries,
trivial
which has a triv
tri
ivia
iviall ke
ia kernel.
kern
rnel
rnel.
el

MA Page 20 of 38

JAM 2026 Confidential Mathematics (MA)

Q.28 For any two points `, ‚ Q ;# , let gFPI(`, ‚) denote the vector from the point `
to the point ‚ and let <(`, ‚) denote the length of gFPI (`, ‚). Let ƒ: ;# t ;#
be a linear transformation such that

<„ƒ (`), ƒ (‚)… = < (`, ‚) for all `, ‚ Q ;# .

Which ONE of the following statements is FALSE ?

(A) ƒ is an injective map.

(B) ƒ is a surjecti
surjective
ective m
map.

(C) For any four oints `, ‚, †, T iin
points
ur ppoi
oint
nt n ;# , we h
hav
have
avee
av

gFPI„ƒ (`), ƒ (‚ )… @ gFPI„ƒ
gFPI„„ƒ († ), ƒ (T)… = gFPI(`,, ‚) @ gFPI (†,
† , T ),

where @ denot
denotes
otes
es the
the uusu
usual
sual
sual ddot pro
product
rodu
roduct
du in ;# K
ct in

(D) For any four oints `
ur ppoi
oi
points `,, ‚, †
†,, T in
n ;# , we have

gFPI„ƒ (`), ƒ (‚ )… × gFPI„ƒ
PI„„ƒ († ), ƒ (T)… = gFPI
gFPI
gF PI gFPI (`,
` , ‚) × gFPI
PI (†,
† , T),

where × denotes
ot the usual
al cross product i ;# K
duct in

MA Page 21 of 38

JAM 2026 Confidential Mathematics (MA)

Q.29 For each positive integer f, let
> >
q = %> + + ‡ + " loˆ‰ f
2 f
and
cos%I
q
!q = E %dI %K
? I7
Which ONE of the following statements about the sequences ( q ) and (!q ) is
TRUE?

(A) ( q ) and (!q ) are convergent.

(B) ( q ) is converg
nverge nt and (!
convergent
gent
ge !q ) iiss no
nott co
conv
convergent.
nver
nverge
ergent
ge nt.
nt

(C) ( q ) is not conv
convergent
nver
nv nd (!
ergent and
er !q ) is convergent.

(D) ( q ) is not conv
convergent
nver
nverge
gent
ge and (!q ) iiss not
nt and no convergent.
convergent
co nt..

Q.30 Let r be thee po
powe
ppower set of {{>,2,B,4}.
werr set
we >,2
2,B
B,4
4}. Le
Lett th
thee bi
bina
binary
nary
na on Š o
ry operation
ope
opera
perati
ra tion
ti on r be
‹%Š%Œ = (• Ž Œ) • (Œ Ž •)K
Which ONE of the following statements is TRUE?

(A) {>} is an iden•i•y elemen• oz (r, Š).

(B) (r, Š) is an abelian group but not cyclic.

(C) (r, Š) is a group and has an element of order 4.

(D) (r, Š) is a group and has an element of order R.

MA Page 22 of 38

JAM 2026 Confidential Mathematics (MA)

Section B: Q.31 – Q.40 Carry TWO marks each.

Q.31 Which of the following functions &: ; 9 ; has/have a local minimum at
= 0?

(A) & ( ) = sin | |

#
& ( ) = sin +
(B)
L

(C) & ( ) = A
+ 7
+B

(D) & ( ) = min
n { " [ ], > " +[ ]}

Q.32 Which of the
he ffol
following
ollowi
ol wing
wing sta
statements
tate
ta teme
te ment
me nts is/are
is/a
/are
/a re TRUE?
TRU
RUE?
E?

(A) There exists
ts a monotone
mon
monot
onoton
otonee sequence
on sequ
se quen
quence
ence that
ttha
hatt does
ha does not
not converge
ccon
onve
on verg
vergee but
rg but has
ha a convergent
subsequence.

(B) There exists a sequence that has a bounded subsequence but does not have any
convergent subsequence.

(C) There exists a sequence ( q ) such that given any positive integer ‘, ( q ) has a
subsequence converging to ‘.

(D) There exists a sequence ( q ) such that (| q’? " q |) converges to 0 but ( q )

does not converge.

MA Page 23 of 38

JAM 2026 Confidential Mathematics (MA)

Q.33 Which of the following functions is/are differentiable at = >?

(A) & ( ) = | " >|#

(B) & ((! )) = | #7 *
= |! " 1|
>|

! #7 ,F -.
G-
2, ||!|| 3
V1>
X

&((!)) = “
/

=+
(C)
,F ">
*1
2, | |>
|!| ”1>

(D) & ((! )) = [ ]
= [!]

Q.34 Let &:: 4
;594
; bbee the
the function
functi
tion
tion given
giv
given
iven by
by

cos
coss ! ,2 !Q
7–8
&((!)) =
= 6•
0%%%%,
09999292
09 !—
; –K
8
8<

Which of the following
he ffol
ollo
ol lowi
lowing statements
wi sta
tate
ta tement
te nts is/are
nt is/a
is /are
/a re TRUE?
TRU
TRUE?
RU E?

(A) (!) is continuous at 0.

"
(B) (!) is continuous at .
#

&
(C) (!) is Riemann integrable on [0, 1] and $' (!) %! = sin 1.

&
(D) (!) is Riemann integrable on [0, 1] and $' (!) %! = 0.

MA Page 24 of 38

JAM 2026 Confidential Mathematics (MA)

Q.35 Consider the differential equation
C
(?@ cos ! * !@ sin ! ) %! A ?! cos ! %@ = 09999for9! 7 B02 E <
D
Which of the following is/are integrating factor(s) of the differential equation?

(A) &
.F

(B) !@

(C) sec !

(D) Gsec !

!? @
n(1H
sin
si H!) 2 ! I 0
1H!
Let (!2 @)) = +1A
Q.36
1A!!?
0999999999999292 ! = 0

he following
Which of the ffol
ollo
ol lowi
lowing
wing statements
ssta
tate
ta teme
te ment
me ntss is/are
nt is/a
is /are
/a re TRUE?
TRU
TRUE?
RU E?

(A) lim lim (!2 @) exists.
@50 !50

J
(B)
J!
is continuous at the point (021)<

J
(C) is continuous at the point (021)<
J@

K(.2F)
(D) lim does not exist.
(.2F)5('2') L. / MF /

MA Page 25 of 38

JAM 2026 Confidential Mathematics (MA)

0 1 0 0 0
Q1 0T
Q.37
0 0 0
P S
Let N = P0 0 0 1 0S 7 UV (W).
P0 0 0 0 1S
O0 0 1 0 0R

Which of the following statements is/are TRUE?

(A) nullity(N * X) Y ?, where X is the Z × Z identity matrix.

(B) N has D distinct
tinct eigenvalues in W.
W

(C) N has \ distinc
distinct
tinc
nctt ei
nc uess in 4
eigenvalue
eigenvalues
ue 4..

(D) If ^ is an eigenva
enva
en lue of N, then there exists a positive integer
eigenvalue
igen valu
va er _ suc
such that
^` = 1.

MA Page 26 of 38

JAM 2026 Confidential Mathematics (MA)

Q.38 Let a`bd denote the number of ways of choosing g distinct objects out of
_ distinct objects.
Which of the following statements is/are TRUE?

v
(A) 1\
h(*1)j k q = *xD<
?p
jw'

v
(B) 1\
h(*1)j k q = 1?z<
?p A 1
jw'

v
(C) 1\
h(*1)j k q = xD<
?p
?p
jw'

v
(D) 1\
h(*1)j k q=*
*1?z<
1?z<
1?
?p A 1
?p
jw'

Q.39 ricess N ooff order D × x and
matrices
Consider matri
atri
rice
ce nd { of or der x × D w
orde
order
de with
th
h real
re entries such
that N{9 = 90 and {N = 90.
and {N9 90.
he following
Which of the ffol
ollo
ol lowi
lowing statements
wi sta
tate
ta tement
te nts is/are
nt is/a
is /are
/a re TRUE?
TRU
TRUE?
RU E?

(A) r}n~es•}ce(N) € nulls•}ce({) and r}n~es•}ce({) € nulls•}ce(N).

(B) r}n•(N) A r}n•({ ) 3 D.

(C) If r}n~es•}ce(N) = nulls•}ce({ ), then r}n•(N) A r}n•({) = D.

(D) r}n~es•}ce({) = nulls•}ce(N)<

MA Page 27 of 38

JAM 2026 Confidential Mathematics (MA)

Q.40 Let ‚ be the subset of 4 defined by

ƒA
? @ „G?
ABC
< = +>
‚ E: ?F
ƒ2 AF
„2 DF
…2 %
;G7 HF
82 D… !# A
@%;!# I 0†<
JKL
D… @
A %G?
;BC

Which of the following statements is/are FALSE?

(A) <
‚ is closed under the usual addition in M.
4.
M

(B) <
‚ is a subspac
subspace
pace
ce ooff the real
al vvec
vector
ecto
ec spacee M
torr sp
to 4..

(C) <
‚ is a two dime nsional subspace of the vector space M
dimensional
dim
mens
me ns 4 over H
8..

(D) <
‚ is a four dim
dimensional
imen
imensi
en sion
si onal
onal ssub
subspace
ubsp
ub spac
sp acee of the
ac acee M
he vector space
spac
sp ac 4o ver H
over
ove
ve 8..

Section C: Q.41 – Q.50
50 C
Carry
Car
arry
ry ONE
ONE mark
mar
mark
ar k each.
each
ea ch..
ch

*
((`‡))
("#)
ie /h (lo
log $ %)&'' , " is ______
Q.41
The radius off convergence of the series (!")#
"+'

(rounded off to one decimal place).

' *
(0123 ! )45 467
- ./ 9 ;, = ______(rounded off to one decimal place).
Q.42
(8&')#
: 8+'

MA Page 28 of 38

JAM 2026 Confidential Mathematics (MA)

" P' " P"
lim O @R@ S =______(rounded off to one decimal place).
Q.43
"N* B Q P'
B" B"Q P"
B

76cos V
' '
lim O T @U S = ______(
Q.44
V
5 sin 5
______(rounded
(ro
roun
unde
unded off to one decimal
de dec
decim
ec
cimal place).
5N:

Q.45 Let W and X be rrea
real
eall nu
ea numb
numbers
mber
mb ers such
er ch that
t thee differential
differenti
di tial
al equation
eequ
quat
quatio
ation
ion

(YZ @ W,Y
Y [ T \,
\ @ cos CY );, @ (Z,Y
co CY ( ,Y ! @ CJ
CJ,C Y ] @ X, sin
si CY);Y = J

is exact.
Then W @ X = _______(
______(rounded
_(rounde
_( ded
de d of
offf to one
one d
dec
decimal
ecim
ecimal
im al p
pla
place).
lace
la ce).
ce ).

` `
Q.46 Let ^ = cos(_, @ \Y)F where , = @ CbF Y = T O @ bS
bS.
! [

ef
Then j
eh h+k
= ______(rounded off to one decimal place).
p

MA Page 29 of 38

JAM 2026 Confidential Mathematics (MA)

Q.47 Let ! denote a cyclic group having " elements. If there is a surjective group

homomorphism
omorphism from ! to #$ , then the total number of such distinct surjective

homomorphisms is ______.

1 ' & [& ] ) *
%$ ( |& ' 1| *&
& = ______(rounded
Q.48 #
(ro
roun
unde
unded
de d off cimal place).
off to one decimal
dec
d ecim
ec

53
253
2
Let + , -# (.
((.).
.). Su torr / = 0 4
433 6 iin
n .# belongs
Q.49

&
Supp
Suppose
ppos
pp osee the
os the co
colu
column
lumn
lu mn vector
vec
vecto
ec to belo
be to the

intersection of nullspace(+) and rangespace(+7 ).
Then |&| = ______(rounded off to one decimal place).

Q.50 Let 8 be a 5 × 5 real matrix with det(8) = 4. Let 9 be the matrix of cofactors
of 8. Then det(9) = ________ (rounded off to one decimal place).

MA Page 30 of 38

JAM 2026 Confidential Mathematics (MA)

Section C: Q.51 – Q.60 Carry TWO marks each.

Q.51 Let :; < and > be fixed real numbers such that

? (& ) = @A
BC
D EA
EC
D :&A
:&A BC , wh
wher
whereF
ereF
ereF @ an
andF
dF E aare
re aarb
arbitrary
rbit
rb itrary real constants,
it

al ssol
is the general olut
ol ution of tthe
solution he ddif
iffe
if fere
fe rent
rentia
differentialiall eq
ia equa
uati
ua tion
ti
equationon

* E? *?
D< D >? = 'A BC G
*& E *&

Then :(< D >)= __
____
______(rounded
____
__ __(ro
__ roundedd of
ro offf to one
one decimal
decim
imal
al ppla
place).
lace
la ce).
ce )..

Q.52 Let the function H I JE K J be defined by

? &
M&4 tan'1 N O D ?P tan'1 Q 4 R ; &? S T
H (&; ?) = L 4& M?
T;FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFotherwise

Then UV? NV&OW at (T;T) is ______ (rounded off to two decimal places).
V VH

Q.53 The double integral of H (&; ?) = & over the triangular region with vertices at

N' ; O , (1;4) and (1; '1) is ______(rounded off to one decimal place).
@ @
E E

MA Page 31 of 38

JAM 2026 Confidential Mathematics (MA)

Q.54 Let
j kmq v ^_`b f Y
H(&) = X X A (1 ' \ E ) *\ *?G
BN O
2
2Z
$ $

Then H x (yzM) = ______(rounded off to one decimal place).

Q.55 The volume
me off th tetrahedron
thee te
tetr
trah
tr ahed
ah bounded
edron bo
ed boun
unde
un ness & = 1
planes
ded by the p
de pla
lane
ne 1;; ? = 4; { = P and
14& D }? D ~
~{{ = •T iiss __
____
______(rounded
____
__ __((ro
__ roun
unde
unded off
de off to one
one d
dec
decimal
ecimal
ec al p
pla
place).
lace).
lace
la

Q.56 Let - , -# (J). If

215 2P T 215 '215 214
€ •215‚ = • 24 ‚ ; -0 T 6 = •21T‚ ; - • FF215 ‚ = •'21}‚
T '5 2PT 25 FFFFFT FFFFFT

then det(-) = ______(rounded off to one decimal place).

MA Page 32 of 38

JAM 2026 Confidential Mathematics (MA)

Q.57 Let ƒ; „ !. If the system of linear equations
"# + #2$# + #2% = #1#
2"# + #3$# + #% = #2#
&"# + #5$# + #'% = '

has infinitely many solutions, then & + '# = ______ (rounded off to one
decimal place).

Q.58 A fruit shop has 4 ddif
has
op ha different
iffe
if fere
ferent
re nt types
es of
o bananas.
bana
nana
nas. The number
num
umbe
umberr of ways
be way in which 12
w
bananas can
n be bbou
bought
ough
ou ghtt wi
gh with
th aatt le
leas
least
astt on
as onee banana ffro
from
rom
ro m ea
each typ
type,
ype,
yp e, iis ______.

Q.59 Let ( be a 3 × 3 real matrix such that given any column vector " !) , the
column vector (" is the reflection of " about the plane
{*&, ', -& - '.: &, ' !}/
Then the sum of the diagonal elements of ( is ______(rounded off to one
decimal place).

MA Page 33 of 38

JAM 2026 Confidential Mathematics (MA)

Q.60 Let 0678 denote the number of ways of choosing 9 distinct objects out of
; distinct objects. Then,

0<>8 + 0?@8 + 0AB8 + 0C)8 + 0ED8 + 0@>
<
8 + 0@@
?
8 = ______.

MA Page 34 of 38