Model Call Girl in Tilak Nagar Delhi reach out to us at π9953056974π
Β
IIT JAM Math 2022 Question Paper | Sourav Sir's Classes
1. JAM 2022 MATHEMATICS - MA
MA 1/42
Notation and Terminology
β = the set of all positive integers.
β€ = the set of all integers.
β = the set of all rational numbers.
β = the set of all real numbers.
βπ
= the π-dimensional Euclidean space.
β = the set of all complex numbers.
ππ(β) = the real vector space of all π Γ π matrices with entries in β.
ππ(β) = the complex vector space of all π Γ π matrices with entries in β.
gcd(π, π) = the greatest common divisor of the integers π and π.
πβ€
= the transpose of the matrix π.
π΄ β π΅ = the complement of the set π΅ in the set π΄, that is, {π₯ β π΄ βΆ π₯ β π΅}.
ln π₯ = the natural logarithm of π₯ (to the base π).
|π₯| = the absolute value of π₯.
π¦β²
, π¦β²β²
, π¦β²β²β²
= the first, second and the third derivatives of the function π¦, respectively.
ππ = the symmetric group consisting of all permutations of {1,2, β¦ , π}.
β€π = the additive group of integers modulo π.
π β π is the composite function defined by (π β π)(π₯) = π(π(π₯)).
The phrase βreal vector spaceβ refers to a vector space over β.
2. JAM 2022 MATHEMATICS - MA
MA 2/42
Section A: Q.1 β Q.10 carry ONE mark each.
Q.1 Consider the 2 Γ 2 matrix π = (
1 1
1 0
) β π2(β). If the eighth power of π
satisfies π8
(
1
0
) = (
π₯
π¦), then the value of π₯ is
(A) 21
(B) 22
(C) 34
(D) 35
Q.2
The rank of the 4 Γ 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 β, is
(A) 1
(B) 2
(C) 3
(D) 4
3. JAM 2022 MATHEMATICS - MA
MA 3/42
Q.3 Let π 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) {π β π βΆ π(1 2) β β}
β is a subspace of π.
(B) {π β π βΆ π(1 2) = 1}
β is a subspace of π.
(C) {π β π βΆ π(1 2) = π(1)}
β is a subspace of π.
(D) {π β π βΆ πβ²(1 2) = 1}
β is a subspace of π.
Q.4 Let πΊ be a group of order 2022. Let π» and πΎ be subgroups of πΊ of order 337 and
674, respectively. If π» βͺ πΎ is also a subgroup of πΊ, then which one of the
following is FALSE?
(A) π» is a normal subgroup of π» βͺ πΎ.
(B) The order of π» βͺ πΎ is 1011.
(C) The order of π» βͺ πΎ is 674.
(D) πΎ is a normal subgroup of π» βͺ πΎ.
4. JAM 2022 MATHEMATICS - MA
MA 4/42
Q.5 The radius of convergence of the power series
β (
π3
4π
) π₯5π
β
π=1
is
(A) 4
(B) β4
5
(C) 1
4
(D) 1
β4
5
5. JAM 2022 MATHEMATICS - MA
MA 5/42
Q.6 Let (π₯π) and (π¦π) be sequences of real numbers defined by
π₯1 = 1, π¦1 =
1
2
, π₯π+1 =
π₯π + π¦π
2
, and π¦π+1 = βπ₯ππ¦π for all π β β.
Then which one of the following is true?
(A) (π₯π) is convergent, but (π¦π) is not convergent.
(B) (π₯π) is not convergent, but (π¦π) is convergent.
(C) Both (π₯π) and (π¦π) are convergent and lim
πββ
π₯π > lim
πββ
π¦π .
(D) Both (π₯π) and (π¦π) are convergent and lim
πββ
π₯π = lim
πββ
π¦π .
Q.7 Suppose
ππ =
3π
+ 3
5π β 5
and ππ =
1
(1 + π2)
1
4
for π = 2,3,4, β¦ .
Then which one of the following is true?
(A) Both β ππ
β
π=2 and β ππ
β
π=2 are convergent.
(B) Both β ππ
β
π=2 and β ππ
β
π=2 are divergent.
(C) β ππ
β
π=2 is convergent and β ππ
β
π=2 is divergent.
(D) β ππ
β
π=2 is divergent and β ππ
β
π=2 is convergent.
6. JAM 2022 MATHEMATICS - MA
MA 6/42
Q.8 Consider the series
β
1
ππ (1 +
1
ππ
)
β
π=1
where π and π are real numbers.
Under which of the following conditions does the above series converge?
(A) π > 1.
(B) 0 < π < 1 and π > 1.
(C) 0 < π β€ 1 and 0 β€ π β€ 1.
(D) π = 1 and π > 1.
7. JAM 2022 MATHEMATICS - MA
MA 7/42
Q.9 Let π be a positive real number and let π’: β2
β β be defined by
π’(π₯, π‘) =
1
2π
β« ππ 2
ππ for (π₯, π‘) β β2
.
π₯+ππ‘
π₯βππ‘
Then which one of the following is true?
(A) π2
π’
ππ‘2
= π2
π2
π’
ππ₯2
on β2
.
(B) ππ’
ππ‘
= π2
π2
π’
ππ₯2
on β2
.
(C) ππ’
ππ‘
ππ’
ππ₯
= 0 on β2
.
(D) π2
π’
ππ‘ ππ₯
= 0 on β2
.
8. JAM 2022 MATHEMATICS - MA
MA 8/42
Q.10 Let π β (
π
4
,
π
2
). Consider the functions
π’ βΆ β2
β {(0, 0)} β β and π£ βΆ β2
β {(0, 0)} β β
given by
π’(π₯, π¦) = π₯ β
π₯
π₯2 + π¦2
and π£(π₯, π¦) = π¦ +
π¦
π₯2 + π¦2
.
The value of the determinant |
ππ’
ππ₯
ππ’
ππ¦
ππ£
ππ₯
ππ£
ππ¦
| at the point (cos π, sin π) is equal to
(A) 4 sin π.
(B) 4 cos π.
(C) 4 sin2
π.
(D) 4 cos2
π.
9. JAM 2022 MATHEMATICS - MA
MA 9/42
Section A: Q.11 β Q.30 Carry TWO marks each.
Q.11 Consider the open rectangle πΊ = {(π , π‘) β β2
βΆ 0 < π < 1 and 0 < π‘ < 1} and
the map π: πΊ β β2
given by
π(π , π‘) = (
ππ (1 β π‘)
2
,
π(1 β π )
2
) for (π , π‘) β πΊ .
Then the area of the image π(πΊ) of the map π is equal to
(A) π
4
(B) π2
4
(C) π2
8
(D) 1
10. JAM 2022 MATHEMATICS - MA
MA 10/42
Q.12 Let π denote the sum of the convergent series
1 β
1
2
+
1
3
β
1
4
+
1
5
β
1
6
+ β― +
(β1)π+1
π
+ β―
and let π denote the sum of the convergent series
1 β
1
2
β
1
4
+
1
3
β
1
6
β
1
8
+
1
5
β
1
10
β
1
12
+ β― = β ππ
β
π=1
,
where
π3πβ2 =
1
2π β 1
, π3πβ1 =
β1
4π β 2
and π3π =
β1
4π
for π β β.
Then which one of the following is true?
(A) π = π and π β 0.
(B) 2π = π and π β 0.
(C) π = 2π and π β 0.
(D) π = π = 0.
11. JAM 2022 MATHEMATICS - MA
MA 11/42
Q.13 Let π’: β β β be a twice continuously differentiable function such that π’(0) > 0
and π’β²(0) > 0. Suppose π’ satisfies
π’β²β²(π₯) =
π’(π₯)
1 + π₯2
for all π₯ β β.
Consider the following two statements:
I. The function π’π’β² is monotonically increasing on [0, β).
II. The function π’ is monotonically increasing on [0, β).
Then which one of the following is correct?
(A) Both I and II are false.
(B) Both I and II are true.
(C) I is false, but II is true.
(D) I is true, but II is false.
12. JAM 2022 MATHEMATICS - MA
MA 12/42
Q.14 The value of
lim
πββ
β
βπ + 1 β βπ
π(ln π)2
π
π=2
is equal to
(A) β
(B) 1
(C) π
(D) 0
13. JAM 2022 MATHEMATICS - MA
MA 13/42
Q.15 For π‘ β β, let [π‘] denote the greatest integer less than or equal to π‘. Define
functions β: β2
β β and π: β β β by
β(π₯, π¦) = {
β1
π₯2βπ¦
if π₯2
β π¦,
0 if π₯2
= π¦
and π(π₯) = {
sin π₯
π₯
if π₯ β 0,
0 if π₯ = 0.
Then which one of the following is FALSE?
(A)
lim
(π₯,π¦)β(β2,π)
cos (
π₯2
π¦
π₯2 + 1
) =
β1
2
.
(B) lim
(π₯,π¦)β(β2,2)
πβ(π₯,π¦)
= 0.
(C) lim
(π₯,π¦)β(π,π)
ln(π₯π¦β[π¦]
) = π β 2.
(D) lim
(π₯,π¦)β(0,0)
π2π¦
π(π₯) = 1.
14. JAM 2022 MATHEMATICS - MA
MA 14/42
Q.16 Let π β π4(β) be such that π4
is the zero matrix, but π3
is a nonzero matrix.
Then which one of the following is FALSE?
(A) For every nonzero vector π£ β β4
, the subset {π£, ππ£, π2
π£, π3
π£} of the real vector
space β4
is linearly independent.
(B) The rank of ππ
is 4 β π for every π β {1,2,3,4}.
(C) 0 is an eigenvalue of π.
(D) If π β π4(β) is such that π4
is the zero matrix, but π3
is a nonzero matrix, then
there exists a nonsingular matrix π β π4(β) such that πβ1
ππ = π.
15. JAM 2022 MATHEMATICS - MA
MA 15/42
Q.17 For π, π β π2(β), define (π, π) = ππ β ππ. Let π β π2(β) denote the zero
matrix. Consider the two statements:
π βΆ (π, (π, π)) + (π, (π, π)) + (π, (π, π)) = π for all π, π, π β π2(β).
π βΆ (π, (π, π)) = ((π, π), π) for all π, π, π β π2(β).
Then which one of the following is correct?
(A) Both π and π are true.
(B) π is true, but π is false.
(C) π is false, but π is true.
(D) Both π and π are false.
16. JAM 2022 MATHEMATICS - MA
MA 16/42
Q.18 Consider the system of linear equations
π₯ + π¦ + π‘ = 4,
2π₯ β 4π‘ = 7,
π₯ + π¦ + π§ = 5,
π₯ β 3π¦ β π§ β 10π‘ = π,
where π₯, π¦, π§, π‘ are variables and π is a constant. Then which one of the following
is true?
(A) If π = 1, then the system has a unique solution.
(B) If π = 2, then the system has infinitely many solutions.
(C) If π = 1, then the system has infinitely many solutions.
(D) If π = 2, then the system has a unique solution.
Q.19 Consider the group (β, +) and its subgroup (β€, +).
For the quotient group β/β€, which one of the following is FALSE?
(A) β/β€ contains a subgroup isomorphic to (β€, +).
(B) There is exactly one group homomorphism from β/β€ to (β, +).
(C) For all π β β, there exists π β β/β€ such that the order of π is π.
(D) β/β€ is not a cyclic group.
17. JAM 2022 MATHEMATICS - MA
MA 17/42
Q.20 For π β π5(β) and π, π β {1,2, β¦ ,5}, let πππ denote the (π, π)th
entry of π. Let
π = {π β π5(β) βΆ πππ = πππ for π, π, π, π β {1,2, β¦ ,5} with π + π = π + π }.
Then which one of the following is FALSE?
(A) π is a subspace of the vector space over β of all 5 Γ 5 symmetric matrices.
(B) The dimension of π over β is 5.
(C) The dimension of π over β is 11.
(D) If π β π and all the entries of π are integers, then 5 divides the sum of all the
diagonal entries of π.
18. JAM 2022 MATHEMATICS - MA
MA 18/42
Q.21 On the open interval (βπ, π), where π is a positive real number, π¦(π₯) is an
infinitely differentiable solution of the differential equation
ππ¦
ππ₯
= π¦2
β 1 + cos π₯,
with the initial condition π¦(0) = 0. Then which one of the following is correct?
(A) π¦(π₯) has a local maximum at the origin.
(B) π¦(π₯) has a local minimum at the origin.
(C) π¦(π₯) is strictly increasing on the open interval (βπΏ, πΏ) for some positive real
number πΏ.
(D) π¦(π₯) is strictly decreasing on the open interval (βπΏ, πΏ) for some positive real
number πΏ.
19. JAM 2022 MATHEMATICS - MA
MA 19/42
Q.22 Let π» βΆ β β β be the function given by π»(π₯) =
1
2
(ππ₯
+ πβπ₯) for π₯ β β.
Let π βΆ β β β be defined by
π(π₯) = β« π»(π₯ sin π)ππ
π
0
for π₯ β β.
Then which one of the following is true?
(A) π₯πβ²β²(π₯) + πβ²(π₯) + π₯π(π₯) = 0 for all π₯ β β.
(B) π₯πβ²β²(π₯) β πβ²(π₯) + π₯π(π₯) = 0 for all π₯ β β.
(C) π₯πβ²β²(π₯) + πβ²(π₯) β π₯π(π₯) = 0 for all π₯ β β.
(D) π₯πβ²β²(π₯) β πβ²(π₯) β π₯π(π₯) = 0 for all π₯ β β.
20. JAM 2022 MATHEMATICS - MA
MA 20/42
Q.23 Consider the differential equation
π¦β²β²
+ ππ¦β²
+ π¦ = sin π₯ for π₯ β β. (ββ)
Then which one of the following is true?
(A) If π = 0, then all the solutions of (ββ) are unbounded over β.
(B) If π = 1, then all the solutions of (ββ) are unbounded over (0, β).
(C) If π = 1, then all the solutions of (ββ) tend to zero as π₯ β β.
(D) If π = 2, then all the solutions of (ββ) are bounded over (ββ, 0).
Q.24 For π β β€, let π
Μ β β€37 denote the residue class of π modulo 37. Consider the
group π37 = {π
Μ β β€37 βΆ 1 β€ π β€ 37 with gcd(π, 37) = 1} with respect to
multiplication modulo 37. Then which one of the following is FALSE?
(A) The set {π
Μ β π37 βΆ π
Μ = (π
Μ )β1} contains exactly 2 elements.
(B) The order of the element 10
Μ
Μ
Μ
Μ in π37 is 36.
(C) There is exactly one group homomorphism from π37 to (β€, +).
(D) There is exactly one group homomorphism from π37 to (β, +).
21. JAM 2022 MATHEMATICS - MA
MA 21/42
Q.25 For some real number π with 0 < π < 1, let π: (1 β π, 1 + π) β (0, β) be a
differentiable function such that π(1) = 1 and π¦ = π(π₯) is a solution of the
differential equation
(π₯2
+ π¦2)ππ₯ β 4π₯π¦ ππ¦ = 0.
Then which one of the following is true?
(A)
(3(π(π₯))
2
+ π₯2
)
2
= 4π₯.
(B)
(3(π(π₯))
2
β π₯2
)
2
= 4π₯.
(C)
(3(π(π₯))
2
+ π₯2
)
2
= 4π(π₯).
(D)
(3(π(π₯))
2
β π₯2
)
2
= 4π(π₯).
22. JAM 2022 MATHEMATICS - MA
MA 22/42
Q.26 For a 4 Γ 4 matrix π β π4(β), let π
Μ denote the matrix obtained from π by
replacing each entry of π by its complex conjugate. Consider the real vector
space
π» = {π β π4(β) βΆ πβ€
= π
Μ }
where πβ€
denotes the transpose of π. The dimension of π» as a vector space over
β is equal to
(A) 6
(B) 16
(C) 15
(D) 12
23. JAM 2022 MATHEMATICS - MA
MA 23/42
Q.27 Let π, π be positive real numbers such that π < π. Given that
lim
πββ
β« πβπ‘2
ππ‘
π
0
=
βπ
2
,
the value of
lim
πββ
β«
1
π‘2
(πβππ‘2
β πβππ‘2
)ππ‘
π
0
is equal to
(A) βπ(βπ β βπ).
(B) βπ(βπ + βπ).
(C) ββπ(βπ + βπ).
(D) βπ(βπ β βπ).
24. JAM 2022 MATHEMATICS - MA
MA 24/42
Q.28 For β1 β€ π₯ β€ 1, if π(π₯) is the sum of the convergent power series
π₯ +
π₯2
22
+
π₯3
32
+ β― +
π₯π
π2
+ β―
then π (
1
2
) is equal to
(A)
β«
ln(1 β π‘)
π‘
1
2
0
ππ‘.
(B)
β β«
ln(1 β π‘)
π‘
1
2
0
ππ‘.
(C)
β« t ln(1 + π‘)
1
2
0
ππ‘.
(D)
β« t ln(1 β π‘)
1
2
0
ππ‘.
25. JAM 2022 MATHEMATICS - MA
MA 25/42
Q.29 For π β β and π₯ β [1, β), let
π
π(π₯) = β« (π₯2
+ (cos π)βπ₯2 β 1)
π
π
0
ππ.
Then which one of the following is true?
(A) ππ(π₯) is not a polynomial in π₯ if π is odd and π β₯ 3.
(B) ππ(π₯) is not a polynomial in π₯ if π is even and π β₯ 4.
(C) ππ(π₯) is a polynomial in π₯ for all π β β.
(D) ππ(π₯) is not a polynomial in π₯ for any π β₯ 3.
Q.30 Let π be a 3 Γ 3 real matrix having eigenvalues π1 = 0, π2 = 1 and π3 = β1.
Further, π£1 = (
1
0
0
) , π£2 = (
1
1
0
) and π£3 = (
1
0
1
) are eigenvectors of the matrix π
corresponding to the eigenvalues π1, π2 and π3, respectively. Then the entry in
the first row and the third column of π is
(A) 0
(B) 1
(C) β1
(D) 2
26. JAM 2022 MATHEMATICS - MA
MA 26/42
Section B: Q.31 β Q.40 Carry TWO marks each.
Q.31 Let (βπ, π) be the largest open interval in β (where π is either a positive real
number or π = β) on which the solution π¦(π₯) of the differential equation
ππ¦
ππ₯
= π₯2
+ π¦2
+ 1 with initial condition π¦(0) = 0
exists and is unique. Then which of the following is/are true?
(A) π¦(π₯) is an odd function on (βπ, π).
(B) π¦(π₯) is an even function on (βπ, π).
(C) (π¦(π₯))
2
has a local minimum at 0.
(D) (π¦(π₯))
2
has a local maximum at 0.
27. JAM 2022 MATHEMATICS - MA
MA 27/42
Q.32 Let π be the set of all continuous functions π: [β1,1] β β satisfying the following
three conditions:
(i) π is infinitely differentiable on the open interval (β1,1),
(ii) the Taylor series
π(0) + πβ²(0)π₯ +
πβ²β²(0)
2!
π₯2
+ β―
of π at 0 converges to π(π₯) for each π₯ β (β1,1),
(iii) π (
1
π
) = 0 for all π β β.
Then which of the following is/are true?
(A) π(0) = 0 for every π β π.
(B) πβ² (
1
2
) = 0 for every π β π.
(C) There exists π β π such that πβ² (
1
2
) β 0.
(D) There exists π β π such that π(π₯) β 0 for some π₯ β [β1,1].
28. JAM 2022 MATHEMATICS - MA
MA 28/42
Q.33 Define π: [0,1] β [0,1] by
π(π₯) =
{
1 if π₯ = 0,
1
π
if π₯ =
π
π
for some π, π β β with π β€ π and gcd(π, π) = 1,
0 if π₯ β [0,1] is irrational.
and define π: [0,1] β [0,1] by
π(π₯) = {
0 if π₯ = 0,
1 if π₯ β (0,1].
Then which of the following is/are true?
(A) π is Riemann integrable on [0,1].
(B) π is Riemann integrable on [0,1].
(C) The composite function π β π is Riemann integrable on [0,1].
(D) The composite function π β π is Riemann integrable on [0,1].
29. JAM 2022 MATHEMATICS - MA
MA 29/42
Q.34 Let π be the set of all functions π: β β β satisfying
|π(π₯) β π(π¦)|π
β€ |π₯ β π¦|3
for all π₯, π¦ β β.
Then which of the following is/are true?
(A) Every function in π is differentiable.
(B) There exists a function π β π such that π is differentiable, but π is not twice
differentiable.
(C) There exists a function π β π such that π is twice differentiable, but π is not
thrice differentiable.
(D) Every function in π is infinitely differentiable.
30. JAM 2022 MATHEMATICS - MA
MA 30/42
Q.35 A real-valued function π¦(π₯) defined on β is said to be periodic if there exists a
real number π > 0 such that π¦(π₯ + π) = π¦(π₯) for all π₯ β β.
Consider the differential equation
π2
π¦
ππ₯2
+ 4π¦ = sin(ππ₯), π₯ β β, (β)
where π β β is a constant.
Then which of the following is/are true?
(A) All solutions of (β) are periodic for every choice of π.
(B) All solutions of (β) are periodic for every choice of π β β β {β2, 2}.
(C) All solutions of (β) are periodic for every choice of π β β β {β2, 2}.
(D) If π β β β β, then there is a unique periodic solution of (β).
31. JAM 2022 MATHEMATICS - MA
MA 31/42
Q.36 Let π be a positive real number and let π’, π£ βΆ β2
β β be continuous functions
satisfying
βπ’(π₯, π¦)2 + π£(π₯, π¦)2 β₯ πβπ₯2 + π¦2 for all (π₯, π¦) β β2
.
Let πΉ: β2
β β2
be given by
πΉ(π₯, π¦) = (π’(π₯, π¦), π£(π₯, π¦)) for (π₯, π¦) β β2
.
Then which of the following is/are true?
(A) πΉ is injective.
(B) If πΎ is open in β2
, then πΉ(πΎ) is open in β2
.
(C) If πΎ is closed in β2
, then πΉ(πΎ) is closed in β2
.
(D) If πΈ is closed and bounded in β2
, then πΉβ1(πΈ) is closed and bounded in β2
.
32. JAM 2022 MATHEMATICS - MA
MA 32/42
Q.37 Let πΊ be a finite group of order at least two and let π denote the identity element
of πΊ. Let π: πΊ β πΊ be a bijective group homomorphism that satisfies the following
two conditions:
(i) If π(π) = π for some π β πΊ, then π = π,
(ii) (π β π)(π) = π for all π β πΊ.
Then which of the following is/are correct?
(A) For each π β πΊ, there exists β β πΊ such that ββ1
π(β) = π.
(B) There exists π₯ β πΊ such that π₯π(π₯) β π.
(C) The map π satisfies π(π₯) = π₯β1
for every π₯ β πΊ.
(D) The order of the group πΊ is an odd number.
33. JAM 2022 MATHEMATICS - MA
MA 33/42
Q.38 Let (π₯π) be a sequence of real numbers. Consider the set
π = {π β β βΆ π₯π > π₯π for all π β β with π > π}.
Then which of the following is/are true?
(A) If π is finite, then (π₯π) has a monotonically increasing subsequence.
(B) If π is finite, then no subsequence of (π₯π) is monotonically increasing.
(C) If π is infinite, then (π₯π) has a monotonically decreasing subsequence.
(D) If π is infinite, then no subsequence of (π₯π) is monotonically decreasing.
Q.39 Let π be the real vector space consisting of all polynomials in one variable with
real coefficients and having degree at most 5, together with the zero polynomial.
Let π: π β β be the linear map defined by π(1) = 1 and
π(π₯(π₯ β 1) β― (π₯ β π + 1)) = 1 for 1 β€ π β€ 5.
Then which of the following is/are true?
(A) π(π₯4) = 15.
(B) π(π₯3) = 5.
(C) π(π₯4) = 14.
(D) π(π₯3) = 3.
34. JAM 2022 MATHEMATICS - MA
MA 34/42
Q.40 Let π be a fixed 3 Γ 3 matrix with entries in β. Which of the following maps
from π3(β) to π3(β) is/are linear?
(A) π1: π3(β) β π3(β) given by π1(π) = ππ β ππ for π β π3(β).
(B) π2: π3(β) β π3(β) given by π2(π) = π2
π β π2
π for π β π3(β).
(C) π3: π3(β) β π3(β) given by π3(π) = ππ2
+ π2
π for π β π3(β).
(D) π4: π3(β) β π3(β) given by π4(π) = ππ2
β ππ2
for π β π3(β).
Section C: Q.41 β Q.50 Carry ONE mark each.
Q.41 The value of the limit
lim
πββ
(
(14
+ 24
+ β― + π4
)
π5
+
1
βπ
(
1
βπ + 1
+
1
βπ + 2
+ β― +
1
β4π
))
is equal to _________. (Rounded off to two decimal places)
35. JAM 2022 MATHEMATICS - MA
MA 35/42
Q.42 Consider the function π’: β3
β β given by
π’(π₯1, π₯2, π₯3) = π₯1π₯2
4
π₯3
2
β π₯1
3
π₯3
4
β 26π₯1
2
π₯2
2
π₯3
3
.
Let π β β and π β β be such that
π₯1
ππ’
ππ₯2
+ 2π₯2
ππ’
ππ₯3
evaluated at the point (π‘, π‘2
, π‘3
), equals ππ‘π
for every π‘ β β. Then the value of
π is equal to _________.
Q.43 Let π¦(π₯) be the solution of the differential equation
ππ¦
ππ₯
+ 3π₯2
π¦ = π₯2
, for π₯ β β,
satisfying the initial condition π¦(0) = 4.
Then lim
π₯ββ
π¦(π₯) is equal to _________. (Rounded off to two decimal places)
Q.44 The sum of the series
β
1
(4π β 3)(4π + 1)
β
π=1
is equal to _________. (Rounded off to two decimal places)
Q.45 The number of distinct subgroups of β€999 is _________.
36. JAM 2022 MATHEMATICS - MA
MA 36/42
Q.46 The number of elements of order 12 in the symmetric group S7 is equal to
_________.
Q.47 Let π¦(π₯) be the solution of the differential equation
π₯π¦2
π¦β²
+ π¦3
=
sin π₯
π₯
for π₯ > 0,
satisfying π¦ (
π
2
) = 0.
Then the value of π¦ (
5π
2
) is equal to _________. (Rounded off to two decimal
places)
Q.48 Consider the region
πΊ = {(π₯, π¦, π§ ) β β3
βΆ 0 < π§ < π₯2
β π¦2
, π₯2
+ π¦2
< 1}.
Then the volume of πΊ is equal to _________. (Rounded off to two decimal
places)
37. JAM 2022 MATHEMATICS - MA
MA 37/42
Q.49 Given that π¦(π₯) is a solution of the differential equation
π₯2
π¦β²β²
+ π₯π¦β²
β 4π¦ = π₯2
on the interval (0, β) such that lim
π₯β0+
π¦(π₯) exists and π¦(1) = 1. The value of
π¦β²
(1) is equal to _________. (Rounded off to two decimal places)
Q.50 Consider the family β±1 of curves lying in the region
{(π₯, π¦) β β2
βΆ π¦ > 0 and 0 < π₯ < π}
and given by
π¦ =
π(1 β cos π₯)
sin π₯
, where π is a positive real number.
Let β±2 be the family of orthogonal trajectories to β±1. Consider the curve π
belonging to the family β±2 passing through the point (
π
3
, 1). If π is a real number
such that (
π
4
, π) lies on π, then the value of π4
is equal to _________.
(Rounded off to two decimal places)
38. JAM 2022 MATHEMATICS - MA
MA 38/42
Section C: Q.51 β Q.60 Carry TWO marks each.
Q.51 For π‘ β β, let [π‘] denote the greatest integer less than or equal to π‘.
Let π· = {(π₯, π¦) β β2
βΆ π₯2
+ π¦2
< 4}. Let π: π· β β and π: π· β β be defined
by π(0, 0) = π(0, 0) = 0 and
π(π₯, π¦) = [π₯2
+ π¦2]
π₯2
π¦2
π₯4 + π¦4
, π(π₯, π¦) = [π¦2]
π₯π¦
π₯2 + π¦2
for (π₯, π¦) β (0, 0). Let πΈ be the set of points of π· at which both π and π are
discontinuous. The number of elements in the set πΈ is _________.
Q.52 If πΊ is the region in β2
given by
πΊ = {(π₯, π¦) β β2
βΆ π₯2
+ π¦2
< 1,
π₯
β3
< π¦ < β3π₯, π₯ > 0, π¦ > 0}
then the value of
200
π
β¬π₯2
ππ₯ ππ¦
πΊ
is equal to _________. (Rounded off to two decimal places)
39. JAM 2022 MATHEMATICS - MA
MA 39/42
Q.53
Let π΄ = (
1 1
0 1
β1 1
) and let π΄β€
denote the transpose of π΄. Let π’ = (
π’1
π’2
) and
π£ = (
π£1
π£2
π£3
) be column vectors with entries in β such that π’1
2
+ π’2
2
= 1 and
π£1
2
+ π£2
2
+ π£3
2
= 1. Suppose
π΄π’ = β2 π£ and π΄β€
π£ = β2 π’.
Then |π’1 + 2 β2 π£1| is equal to _________. (Rounded off to two decimal places)
Q.54 Let π: [0, π] β β be the function defined by
π(π₯) = {
(π₯ β π)πsin π₯
if 0 β€ π₯ β€
π
2
,
π₯πsin π₯
+
4
π
if
π
2
< π₯ β€ π.
Then the value of
β« π(π₯)ππ₯
π
0
is equal to _________. (Rounded off to two decimal places)
40. JAM 2022 MATHEMATICS - MA
MA 40/42
Q.55 Let π be the radius of convergence of the power series
1
3
+
π₯
5
+
π₯2
32
+
π₯3
52
+
π₯4
33
+
π₯5
53
+
π₯6
34
+
π₯7
54
+ β―.
Then the value of π2
is equal to _________. (Rounded off to two decimal
places)
Q.56 Define π: β2
β β by
π(π₯, π¦) = π₯2
+ 2π¦2
β π₯ for (π₯, π¦) β β2
.
Let π· = {(π₯, π¦) β β2
βΆ π₯2
+ π¦2
β€ 1} and πΈ = {(π₯, π¦) β β2
βΆ
π₯2
4
+
π¦2
9
β€ 1}.
Consider the sets
π·max = {(π, π) β π· βΆ π has absolute maximum on π· at (π, π)},
π·min = {(π, π) β π· βΆ π has absolute minimum on π· at (π, π)},
πΈmax = {(π, π) β πΈ βΆ π has absolute maximum on πΈ at (π, π)},
πΈmin = {(π, π) β πΈ βΆ π has absolute minimum on πΈ at (π, π)}.
Then the total number of elements in the set
π·max βͺ π·min βͺ πΈmax βͺ πΈmin
is equal to _________.
41. JAM 2022 MATHEMATICS - MA
MA 41/42
Q.57
Consider the 4 Γ 4 matrix π = (
11 10
10 11
10 10
10 10
10 10
10 10
11 10
10 11
) . Then the value of the
determinant of π is equal to _________.
Q.58 Let π be the permutation in the symmetric group S5 given by
π(1) = 2, Ο(2) = 3, Ο(3) = 1, Ο(4) = 5, Ο(5) = 4.
Define
π(π) = {π β π5 βΆ π β π = π β π}.
Then the number of elements in π(π) is equal to _________.
Q.59 Let π: (β1, 1) β β and π: (β1, 1) β β be thrice continuously differentiable
functions such that π(π₯) β π(π₯) for every nonzero π₯ β (β1, 1). Suppose
π(0) = ln 2, πβ²(0) = π, πβ²β²(0) = π2
, and πβ²β²β²(0) = π9
and
π(0) = ln 2, πβ²(0) = π, πβ²β²
(0) = π2
, and πβ²β²β²(0) = π3
.
Then the value of the limit
lim
π₯β0
ππ(π₯)
β ππ(π₯)
π(π₯) β π(π₯)
is equal to _________. (Rounded off to two decimal places)
42. JAM 2022 MATHEMATICS - MA
MA 42/42
Q.60 If π: [0, β) β β and π: [0, β) β [0, β) are continuous functions such that
β« π(π‘)ππ‘
π₯3+π₯2
0
= π₯2
and β« π‘2
ππ‘ = 9(π₯ + 1)3
π(π₯)
0
for all π₯ β [0, β),
then the value of
π(2) + π(2) + 16 π(12)
is equal to _________. (Rounded off to two decimal places)