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Page 1
JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
N= the set of all positive integers .
Z= the set of all integers .
Q=the set of all rational numbers .
R= the set of all real numbers .
R
?? =the ?? -dimensional Euclidean space.
C=the set of all complex numbers .
?? ?? (R) = the real vector space of all ?? ×?? matrices with entries in R.
?? ?? (C) = the complex vector space of all ?? ×?? matrices with entries in C.
gcd(?? ,?? ) = the greatest common divisor of the integers ?? and ?? .
?? ?
=the transpose of the matrix ?? .
?? -?? =the complement of the set ?? in the set ?? ,that is,{?? ??? :?? ??? }.
ln?? =the natural logarithm of ?? (to the base ?? ) .
|?? |=the absolute value of ?? .
?? '
,?? ''
,?? '''
=the first,second and the third derivatives of the function ?? ,respectively .
?? ?? =the symmetric group consisting of all permutations of {1,2,…,?? }.
Z
?? =the additive group of integers modulo ?? .
?? °?? is the composite function defined by (?? °?? )(?? )=?? (?? (?? )) .
The phrase ‘real vector space’ refers to a vector space over R.
Page 2
JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
N= the set of all positive integers .
Z= the set of all integers .
Q=the set of all rational numbers .
R= the set of all real numbers .
R
?? =the ?? -dimensional Euclidean space.
C=the set of all complex numbers .
?? ?? (R) = the real vector space of all ?? ×?? matrices with entries in R.
?? ?? (C) = the complex vector space of all ?? ×?? matrices with entries in C.
gcd(?? ,?? ) = the greatest common divisor of the integers ?? and ?? .
?? ?
=the transpose of the matrix ?? .
?? -?? =the complement of the set ?? in the set ?? ,that is,{?? ??? :?? ??? }.
ln?? =the natural logarithm of ?? (to the base ?? ) .
|?? |=the absolute value of ?? .
?? '
,?? ''
,?? '''
=the first,second and the third derivatives of the function ?? ,respectively .
?? ?? =the symmetric group consisting of all permutations of {1,2,…,?? }.
Z
?? =the additive group of integers modulo ?? .
?? °?? is the composite function defined by (?? °?? )(?? )=?? (?? (?? )) .
The phrase ‘real vector space’ refers to a vector space over R.
JAM 2022 MATHEMATICS - MA
MA 2/42
Section A: Q.1 – Q.10 carry ONE mark each.
Q.1
Consider the 2×2 matrix ?? =(
1 1
1 0
)??? 2
(R) . If the eighth power of ??
satisfies ?? 8
(
1
0
)= (
?? ?? ) , then the value of ?? is
(A) 21
(B) 22
(C) 34
(D) 35
Q.2
The rank of the 4 ×6 matrix (
1 1
1 0
1 0
0 1
0 0
1 0
0 1
0 0
0 1
1 0
0 1
1 1
) with entries in R, is
(A) 1
(B) 2
(C) 3
(D) 4
Page 3
JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
N= the set of all positive integers .
Z= the set of all integers .
Q=the set of all rational numbers .
R= the set of all real numbers .
R
?? =the ?? -dimensional Euclidean space.
C=the set of all complex numbers .
?? ?? (R) = the real vector space of all ?? ×?? matrices with entries in R.
?? ?? (C) = the complex vector space of all ?? ×?? matrices with entries in C.
gcd(?? ,?? ) = the greatest common divisor of the integers ?? and ?? .
?? ?
=the transpose of the matrix ?? .
?? -?? =the complement of the set ?? in the set ?? ,that is,{?? ??? :?? ??? }.
ln?? =the natural logarithm of ?? (to the base ?? ) .
|?? |=the absolute value of ?? .
?? '
,?? ''
,?? '''
=the first,second and the third derivatives of the function ?? ,respectively .
?? ?? =the symmetric group consisting of all permutations of {1,2,…,?? }.
Z
?? =the additive group of integers modulo ?? .
?? °?? is the composite function defined by (?? °?? )(?? )=?? (?? (?? )) .
The phrase ‘real vector space’ refers to a vector space over R.
JAM 2022 MATHEMATICS - MA
MA 2/42
Section A: Q.1 – Q.10 carry ONE mark each.
Q.1
Consider the 2×2 matrix ?? =(
1 1
1 0
)??? 2
(R) . If the eighth power of ??
satisfies ?? 8
(
1
0
)= (
?? ?? ) , then the value of ?? is
(A) 21
(B) 22
(C) 34
(D) 35
Q.2
The rank of the 4 ×6 matrix (
1 1
1 0
1 0
0 1
0 0
1 0
0 1
0 0
0 1
1 0
0 1
1 1
) with entries in R, is
(A) 1
(B) 2
(C) 3
(D) 4
JAM 2022 MATHEMATICS - MA
MA 3/42
Q.3 Let ?? be the real vector space consisting of all polynomials in one variable with
real coefficients and having degree at most 6, together with the zero polynomial.
Then which one of the following is true?
(A) {?? ??? : ?? (1 2)?Q} / is a subspace of ?? .
(B) {?? ??? : ?? (1 2)=1} / is a subspace of ?? .
(C) {?? ??? : ?? (1 2)=?? (1)} / is a subspace of ?? .
(D) {?? ??? : ?? '(1 2)=1} / is a subspace of ?? .
Q.4 Let ?? be a group of order 2022. Let ?? and ?? be subgroups of ?? of order 337 and
674, respectively. If ?? ??? is also a subgroup of ?? , then which one of the
following is FALSE?
(A) ?? is a normal subgroup of ?? ??? .
(B) The order of ?? ??? is 1011 .
(C) The order of ?? ??? is 674.
(D) ?? is a normal subgroup of ?? ??? .
Page 4
JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
N= the set of all positive integers .
Z= the set of all integers .
Q=the set of all rational numbers .
R= the set of all real numbers .
R
?? =the ?? -dimensional Euclidean space.
C=the set of all complex numbers .
?? ?? (R) = the real vector space of all ?? ×?? matrices with entries in R.
?? ?? (C) = the complex vector space of all ?? ×?? matrices with entries in C.
gcd(?? ,?? ) = the greatest common divisor of the integers ?? and ?? .
?? ?
=the transpose of the matrix ?? .
?? -?? =the complement of the set ?? in the set ?? ,that is,{?? ??? :?? ??? }.
ln?? =the natural logarithm of ?? (to the base ?? ) .
|?? |=the absolute value of ?? .
?? '
,?? ''
,?? '''
=the first,second and the third derivatives of the function ?? ,respectively .
?? ?? =the symmetric group consisting of all permutations of {1,2,…,?? }.
Z
?? =the additive group of integers modulo ?? .
?? °?? is the composite function defined by (?? °?? )(?? )=?? (?? (?? )) .
The phrase ‘real vector space’ refers to a vector space over R.
JAM 2022 MATHEMATICS - MA
MA 2/42
Section A: Q.1 – Q.10 carry ONE mark each.
Q.1
Consider the 2×2 matrix ?? =(
1 1
1 0
)??? 2
(R) . If the eighth power of ??
satisfies ?? 8
(
1
0
)= (
?? ?? ) , then the value of ?? is
(A) 21
(B) 22
(C) 34
(D) 35
Q.2
The rank of the 4 ×6 matrix (
1 1
1 0
1 0
0 1
0 0
1 0
0 1
0 0
0 1
1 0
0 1
1 1
) with entries in R, is
(A) 1
(B) 2
(C) 3
(D) 4
JAM 2022 MATHEMATICS - MA
MA 3/42
Q.3 Let ?? be the real vector space consisting of all polynomials in one variable with
real coefficients and having degree at most 6, together with the zero polynomial.
Then which one of the following is true?
(A) {?? ??? : ?? (1 2)?Q} / is a subspace of ?? .
(B) {?? ??? : ?? (1 2)=1} / is a subspace of ?? .
(C) {?? ??? : ?? (1 2)=?? (1)} / is a subspace of ?? .
(D) {?? ??? : ?? '(1 2)=1} / is a subspace of ?? .
Q.4 Let ?? be a group of order 2022. Let ?? and ?? be subgroups of ?? of order 337 and
674, respectively. If ?? ??? is also a subgroup of ?? , then which one of the
following is FALSE?
(A) ?? is a normal subgroup of ?? ??? .
(B) The order of ?? ??? is 1011 .
(C) The order of ?? ??? is 674.
(D) ?? is a normal subgroup of ?? ??? .
JAM 2022 MATHEMATICS - MA
MA 4/42
Q.5 The radius of convergence of the power series
?(
?? 3
4
?? )?? 5?? 8
?? =1
is
(A) 4
(B)
v4
5
(C)
1
4
(D)
1
v4
5
Page 5
JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
N= the set of all positive integers .
Z= the set of all integers .
Q=the set of all rational numbers .
R= the set of all real numbers .
R
?? =the ?? -dimensional Euclidean space.
C=the set of all complex numbers .
?? ?? (R) = the real vector space of all ?? ×?? matrices with entries in R.
?? ?? (C) = the complex vector space of all ?? ×?? matrices with entries in C.
gcd(?? ,?? ) = the greatest common divisor of the integers ?? and ?? .
?? ?
=the transpose of the matrix ?? .
?? -?? =the complement of the set ?? in the set ?? ,that is,{?? ??? :?? ??? }.
ln?? =the natural logarithm of ?? (to the base ?? ) .
|?? |=the absolute value of ?? .
?? '
,?? ''
,?? '''
=the first,second and the third derivatives of the function ?? ,respectively .
?? ?? =the symmetric group consisting of all permutations of {1,2,…,?? }.
Z
?? =the additive group of integers modulo ?? .
?? °?? is the composite function defined by (?? °?? )(?? )=?? (?? (?? )) .
The phrase ‘real vector space’ refers to a vector space over R.
JAM 2022 MATHEMATICS - MA
MA 2/42
Section A: Q.1 – Q.10 carry ONE mark each.
Q.1
Consider the 2×2 matrix ?? =(
1 1
1 0
)??? 2
(R) . If the eighth power of ??
satisfies ?? 8
(
1
0
)= (
?? ?? ) , then the value of ?? is
(A) 21
(B) 22
(C) 34
(D) 35
Q.2
The rank of the 4 ×6 matrix (
1 1
1 0
1 0
0 1
0 0
1 0
0 1
0 0
0 1
1 0
0 1
1 1
) with entries in R, is
(A) 1
(B) 2
(C) 3
(D) 4
JAM 2022 MATHEMATICS - MA
MA 3/42
Q.3 Let ?? be the real vector space consisting of all polynomials in one variable with
real coefficients and having degree at most 6, together with the zero polynomial.
Then which one of the following is true?
(A) {?? ??? : ?? (1 2)?Q} / is a subspace of ?? .
(B) {?? ??? : ?? (1 2)=1} / is a subspace of ?? .
(C) {?? ??? : ?? (1 2)=?? (1)} / is a subspace of ?? .
(D) {?? ??? : ?? '(1 2)=1} / is a subspace of ?? .
Q.4 Let ?? be a group of order 2022. Let ?? and ?? be subgroups of ?? of order 337 and
674, respectively. If ?? ??? is also a subgroup of ?? , then which one of the
following is FALSE?
(A) ?? is a normal subgroup of ?? ??? .
(B) The order of ?? ??? is 1011 .
(C) The order of ?? ??? is 674.
(D) ?? is a normal subgroup of ?? ??? .
JAM 2022 MATHEMATICS - MA
MA 4/42
Q.5 The radius of convergence of the power series
?(
?? 3
4
?? )?? 5?? 8
?? =1
is
(A) 4
(B)
v4
5
(C)
1
4
(D)
1
v4
5
JAM 2022 MATHEMATICS - MA
MA 5/42
Q.6 Let (?? ?? ) and (?? ?? ) be sequences of real numbers defined by
?? 1
=1, ?? 1
=
1
2
, ?? ?? +1
=
?? ?? +?? ?? 2
, and ?? ?? +1
= v?? ?? ?? ?? for all ?? ?N.
Then which one of the following is true?
(A) (?? ?? ) is convergent, but (?? ?? ) is not convergent.
(B) (?? ?? ) is not convergent, but (?? ?? ) is convergent.
(C) Both (?? ?? ) and (?? ?? ) are convergent and lim
?? ?8
?? ??
> lim
?? ?8
?? ?? .
(D) Both (?? ?? ) and (?? ?? ) are convergent and lim
?? ?8
?? ??
= lim
?? ?8
?? ?? .
Q.7 Suppose
?? ?? =
3
?? +3
5
?? -5
and ?? ?? =
1
(1+?? 2
)
1
4
for ?? =2,3,4,….
Then which one of the following is true?
(A) Both ? ?? ?? 8
?? =2
and ? ?? ?? 8
?? =2
are convergent.
(B) Both ? ?? ?? 8
?? =2
and ? ?? ?? 8
?? =2
are divergent.
(C) ? ?? ?? 8
?? =2
is convergent and ? ?? ?? 8
?? =2
is divergent.
(D) ? ?? ?? 8
?? =2
is divergent and ? ?? ?? 8
?? =2
is convergent.
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