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Latest IIT JAM Mathematics Exam Question (Objective Questions), MCQ in English

Subjects : Section C - Numerical Answer Type Questions (NATQs), Section B - Multiple Select Questions, Section - A Multiple Choice Question

Question Bank IIT JAM Mathematics Exam - English

Section - A Multiple Choice Question

Q 1 :

Consider the 2 × 2 matric M =   ∈ M² (R) . If the eighth power of M Satisfies M⁸ â€‹ ,then the value of x is

1. A.21
2. B.22
3. C.34
4. D.35

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Q 2 :

The rank of the 4 x 6 matrix â€‹ with entries in R, is

1. A.1
2. B.2
3. C.3
4. D.4

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Q 3 :

Let V be the real vector space consisting of all polynomials in one variable with real coefficient and having degree at most 6, together with the Zero polynomial. Then, which one of the following is true ?

1. A.​ is a subspace of V
2. B.​ is a subspace of V
3. C.​ is a subspace of V
4. D.​ is a subspace of V

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Q 4 :

Let G be a group of order 2022. Let H and K be subgroups of G order 337 and 674, respectively. If H ∪ K is also a subgroup of G, than which one of the following is FALSE ?

1. A.H is a normal subgroup of H ∪ K
2. B.The order of H ∪ K is 1011
3. C.The order of H ∪ K is 674
4. D.K is a normal subgroup of H ∪ K

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Q 5 :

The radius of convergence of the power series â€‹ is.

1. A.4
2. B. â€‹
3. C. â€‹
4. D. â€‹

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Q 6 :

Let (Xn) and (b) be sequences of real numbers dened by X_l = 1,  and â€‹ for all n ∈N. Then which one of the following is true ?

1. A.(Xn) is convergent, but (Yn }is neSt convergent
2. B.(Xn) is not convergent,but (Yn) b convergent 
3. C.Both (Xn) and (Yn) are convergent and lim_nx_n â€‹
4. D.Both (X_n) and (Y_n) are convergent and lim â€‹

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Q 7 :

Suppose  and bn = â€‹ for n = 2,3,4, …. Then. Which one of the following is true ?

1. A.Both â€‹ are convergent
2. B.Both â€‹ are divergent
3. C. is convergent and â€‹ divergent. b, is
4. D. an is divergent and â€‹ bn is convergent

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Q 8 :

Consider the series â€‹ where m and p are real numbers.
Under which of the following conditions does the above series converge ?

1. A.m > 1
2. B.0 < m < 1 and p > 1
3. C.0 < m ≥ 1 and op 1
4. D.m = 1 and p >1

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Q 9 :

Let c be a positive real number are u let R² → R be dened by u (x, t) = â€‹ ds for (x, t) ∈ R². Then which one of the following is true ?

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Q 10 :

Let  Consider the function u : R² – {(0,0)} →R and V : R² - {(0, 0)} → R given by  The value of the detmterminant â€‹ at the point (cos θ, sin θ) is equal to

1. A.4 sin θ
2. B.4 cos θ
3. C.4 sin² θ
4. D.4 cos² θ

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Q 11 :

Consider the open rectangle G = {(s, t)} ∈ R² : 0 < s < 1 and 0 < t < 1} and the map T : G → R² given by
​
Then, the area of the image T(G) of the map T is equal to

1. A.​
2. D.1

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Q 12 :

Let T denote the sum of the convergent series  and let S denote the sum of the convergent series  â€‹where â€‹ m∈ N  Then, which one of the following is true ?

1. A.T = S and S ≠ 0
2. B.2 : F = S and S ≠ 0
3. C.T = 2 S and S ≠ O
4. D.T = S = 0

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Q 13 :

Let u : R → R be a twice continuously differentiable function such that u (0) > 0 and u’ (0) > 0. Suppose u satisfies u  for all x ∈ R. Consider the following two statements:
I. The function uu’ is monotonically increasing on [0, ∞).
II. The function u is monotonically increasing on [0, ∞).
​Then, which one of the following is correct ?

1. A.Both I and II are false
2. B.Both I and II are true
3. C.I is false, but II is true
4. D.I is true , but II is false

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Q 14 :

The value of is equal to 

1. A.​∞
2. B.1
3. C.e
4. D.0

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Q 15 :

For f∈ R, let [f] denotes the greatest integer less than or equal to t. Define functions h : R(2 ) → R and g∶ R → R by
​
Then , which one of the following is fales ?

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Q 16 :

Let P ϵ M₄, (R) be such that P+ is the zero matrIx, but P³ is a non - zero matrix. Then, which one of the following is false ?

1. A.For every Non-zero vector v R⁴, the subset {v, Pv, p²v, p³v} of the real vector space IR⁴ is linearly Independent.
2. B.The rank of pkis 4 – K for every K ϵ {1, 2, 3, 4}
3. C.0 is an eigenvalue of P
4. D.If Q ϵ M⁴ (R) is such that Q4 is the zero matrix, But Q³ is a non - zero matrix, than there exists a non - singular matrix S c M4 (IR) such that S ^-1 QS = P

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Q 17 :

For X, Y, ϵ M₂, (R), define (X, Y) = XY– YX. Let 0 ϵ M2 (R₂) denotes the zero matrix, Consider the two statements :
P : (X, Y, Z)) + (’Y(Z, X)) + (Z(X, D) = 0 for all X, Y, Z c M₂ (R).
Q : (X, (E Z) = ((X, Y), Z) for all X, Y, Z e M₂ (R). Then, which one of the following is correct ?

1. A.Both P and Q are true
2. B.P is true, but Q is false
3. C.P is false, but Q is true
4. D.Both P and Q are false

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Q 18 :

Consider the system of linear equations
x + y + t = 4
2x - 4t = 7
x + y + z = 5
x - 3y - z - 10 t =  λ,
Where x, y, z, t are variables and λ is a constant. Then which one of the following is true ?

1. A.If λ = 1, then system has a unique solution
2. B.If λ = 2, then the system has infinitely many solutions
3. C.If λ = 1, then the system has infinitely many solutions
4. D.If λ = 2, then the system has a unique solution

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Q 19 :

Consider the group (Q,+) and its subgroup (z,+) For the quotient group Q/z, which one of the following is true ?

1. A.Q / Z contains a subgroup isomorphic to (z,+)
2. B.There is exactly one group homomorphism from (Q/ z, ) to (Q , +)
3. C.For all n ∈ N, there exists g ∈ Q/z such that the order of g is n
4. D.Q/z is not a cyclic group

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Q 20 :

For P ∈ M₅ R) and i, j ∈ {1, 2, ...., 5}, let, P _ii denotes the (i, D th entry. of P. Let S = {P ∈ M₅ (R) : P_ij = Prs, for tj, r, s e { 1, 2, ... 5} with i + r = j + s}. Then which one of the following is false ?

1. A.S is a subspace of the vector space over R of all 5 x 5 symmetric matrices
2. B.The dimension of S over R is 5
3. C.The dimension of S over R is 11
4. D.If P ∈ S and all the entries of P are integers, then 5 divides the sum of all the diagonal entries of P

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Q 21 :

On the open interval ( – c, c), where c, is a positive real number, y (x) is an infinitely differentiable solution of the differential equation.
​
with the initial condition y (0) = 0. Then, which one of the following is correct ?

1. A.y(x) has a local maximum at the origin
2. B.y(x) has a local minimum at the origin
3. C.y(x) is strictly increasing on the open interval (– 8, 8) for some positive real number 8.
4. D.y(x) is strictly decreasing on the open interval (– 8, 8) for some positive real number 8

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Q 22 :

Let H : R → R be the function given by 
Letf : R → Rbe defined by
​ (x sin 0) de for x ∈ R.
Then , which one of the following is true ?

1. A.xf" (x) + f (x) + xf(x) = 0 for all x∈R
2. B.xf" (x) - f (x) + xf(x) = 0 for all x∈R
3. C.xf" (x) + f (x) - xf(x) = 0 for all x∈R
4. D.xf" (x) - f (x) - xf(x) = 0 for all x∈R

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Q 23 :

Consider the differntial equation
y’ ’ + ay’ + y = sin I for x c R (**)
​Then, which one of the following is true ?

1. A.If a = 0 , then all the solutions of (**) are unbounded over R
2. B.If a = 1, then all the solutions of (**) are unbounded over (0, ∞).
3. C.If a = 1, then all the solutions of (**) tend to zero as x →∞
4. D.If a = 2, then all the solution of (**) are bounded over ( – ∞, 0).

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Q 24 :

For ge Z , let i c Z37 denotes the residue class of g modulo 37. Consider the group U37 = { i c Z37 : 1 Kg $ 37 with, GCD (g, 37) = 1} with respect to multiplication modulo 37.
​Then which one of the following is FALSE ?

1. A.The set ​contains exactly 2 elements.
2. B.The order of the element 10 in U,, is 36.
3. C.There is exactly one group homomorphism from U37 to Z, +)
4. D.There is exactly one group homomorphism from U37 to (Q, +)

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Q 25 :

For some real number c with 0 < c < 1, let ∅ : (1 – c, 1 + c) → (0, ∞) be a differentiable function such that ∅ (1) = 1 and y = ∅ (x) is a solution of the differential equation (x² + y²)dx – 4xy dy = 0.
​Then, which one of the following is true ?

1. A.(3(∅ (x))² + x 2 )² = 4 x
2. B.(3(∅ (x))² – x 2 )² = 4 x
3. C.(3(∅ (x))² + x 2 )² = 4 ∅ (x)
4. D.(3(∅ (x))² – x 2 )² = 4 ∅ (x)

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Q 26 :

For a 4 x 4 matrix M ϵ M4, (C). Let M denotes the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space H = {M ϵ M_⁴ (C) : M_T = M }, where MT denotes the transpose of M. The dimension of H as a vector space over is equal to

1. A.6
2. B.16
3. C.15
4. D.12

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Q 27 :

Let a, b be positive real number such that a < b. Given that : dt  the value of â€‹ dt equal to

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Q 28 :

For – 1 ≤ x ≤1, if f(x) is the sum of the convergent power series  is then â€‹ equal to

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Q 29 :

For nc N and I ∈ [1, ∞), let fn (x)
​
​Then, which one of the following is true ?

1. A.fn (x) is not a polynomial in x, if n is odd and n ≥ 3
2. B.fn (x) is not a polynomial in x, if n is ever and n ≥ 4
3. C.fn (x) is not a polynomial in x, for an n ∈N
4. D.fn (x) is not a polynomial in x, for any n ≥ 3

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Q 30 :

Let P be a 3 x 3 real matrix having eigenvalues λ₁= 0, λ₂ = 1 and λ₃ = –1.
​
are eigenvectors of the matrix P corresponding to the eigenvalues λ₁, λ₂ and λ₃ respectively . Then the entry in the first row and the third column of P is

1. A.0
2. B.- 1
3. C.1
4. D.2

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Section B - Multiple Select Questions

Q 31 :

Let (– c, c) be the largest open interval in R (where c is either a is positive real number or c = ∞ on which the solution y(x) of the differential equation  + 1 with initial condition y(0) = 0 exist and is unique.
Then, which of the following is/are true ?

1. A.y(x) is an odd function on (– c c)
2. B.y(X) is an even function on (– c c)
3. C.(y(x))2 has a local minimum at 0
4. D.(y(x))2 has a local maximum at 0

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Q 32 :

Let S be the set of all continuous functions f : [–1, 1] + → R satisfying the following three conditions
(i) fis infinitely differentiable on the open interval (– 1, 1),
(ii) The Taylor series​ at 0 converges tof(x) for each x∈ (– 1, 1),
(iii) = 0 for all n ∈N
Then , which of the following is/are true ?

1. A.f (0) = 0 for every/ ∈S.
2. B.f’ ​ = 0 for every f ∈ S
3. C.There exists f∈ S such that f' â€‹ ≠ 0
4. D.There exists/c S such that/(x) # 0 for some I e [–1, 1].

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Q 33 :

Define : [0,1] → [0, 1] by f (x) = if x = 0 ​
​
Then, which of the following is/are true ?

1. A.fis Riemann integrable on [0, 1]
2. B.g is Riemann integrable on [0, 1]
3. C.The composite function / og is Riemann integrable on [0, 1]
4. D.The composite function g of is Riemann integrable on [0, 1]

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Q 34 :

Let S be the set of all functionsf: R → R satisfying
​Then, which of the following is/are true ?

1. A.Every function in S is differentiable
2. B.There exists a function f∈ S such that f is differentiable, but fisnot twice differentiable
3. C.There exists a function f ∈ S such that f is twice differentiable, but f is not thrice differentiable
4. D.Every function in S is infinitely differentiable

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Q 35 :

A real - valued function y(x) defined on iR is said to be periodic, if there exists a real number T> 0 Such that 7(x + T) = y (x) for all x ∈R,
Consider the differential equation .d y + 47 = sin(aI), I ϵR, a
​Where a ϵR is a constant. 

1. A.All solutions of (*) are periodic for every choice of a.
2. B.All solutions of (*) are periodic for every choice of a ∈ R – {– 2,2}.
3. C.All solutions of (*) are periodic for every choice of a ∈ (Q – {– 2,2}.
4. D.if a ∈ R – Q, then there is a unique periodic solution of (*).

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Q 36 :

Let M be a positive real number and let u , v R² → R be continuous function satisfying 
​ for all (x , y) ∈ R²
Let F : R² → R² be given by
F(x, y) = (UCI, y), ϑ(x, y) for (x, y) c R²
Then, which of the following is/are true? ​

1. A.F is injective.
2. B.If K is closed R², then F(K) is open in ∈ R²
3. C.If K is open in R², then F (K) is closed h R².
4. D.If E is closed and bounded in R² (2, then F-1 ( E) is closed and bounded in R²

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Q 37 :

Let G be a finite group of order at least two and let denote the identity element of G. Let a : G → G be a bijective group homomorphism that satisfies the following two conditions:
(i) if a (g) = g for some g c G, then g = e
(ii) if (σ o σ) (g) = g for all g ∈ G.
Then , which of the following is/are correct ?

1. A.For each g ∈ G, there exists he G such than h–1 σ(h) = g
2. B.There exists x∈ G such that lo (x) ≠ e
3. C.The map σ satisfies σ (x) = x–1 for every x ∈ G
4. D.The order of the group G is an odd number

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Q 38 :

Let (xn) be a sequence of real numbers. Consider the set P = {n ∈ N: xn > Xm for all m e N with m > n} Then, which of the following is/are ?

1. A.If P is finite, than (xi) has a monotonically increasing subsequence
2. B.if P is finite, than no subsequence of (x_n) monotonically increasing
3. C.if P is infinite, then (h) has a monotonically decreasing subsequence
4. D.if P is infinite, than no subsequence of (b) is monotonically decreasing

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Q 39 :

Let V 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 T : Y → R be the linear map defined by T(1) = 1 and T(x(x – 1)...(x –k +1)) = 1 for 1 ≥ k ≤ 5
​Then , which of the following is/are true ?

1. A.T (x⁴ ) = 15
2. B.T (x⁴ ) = 5
3. C.T (x³ ) = 14
4. D.T (x³ ) = 3

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Q 40 :

Let P be a fixed 3 x 3 matrix with entries in R. Which of the following maps from M³ (IR) to M³ (R) is/are linear ?

1. A.T₁ : M₃ (R) → M^₃ (R) given by T1 (M ) = MP PM for M ∈ M₃ (R)
2. B.T₂ : M₃ (R) → M₃ (R) given by T2 (M ) = M²P P²M for M ∈ M₃ (R)
3. C.T₃ : M₃ (R) → M₃ (R) given by T₃ (M ) = MP² P²M for M ∈ M³ (R)
4. D.T₄ : M₃ (R) → M₃ (R) given by T₄ (M ) = MP₂ PM² for M ∈ M₃ (R)

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Section C - Numerical Answer Type Questions (NATQs)

Q 41 :

The value of the limit â€‹ the equal to ……. (Rounded off to two decimal places)

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Q 42 :

Consider the function   Such that â€‹ evaluated at the point (t, t2 t3) equal ctk for every t∈R Then the value of k is equal to ……

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Q 43 :

Let y(X) be the solution of the differential equation â€‹ + 3x² y = x², for x ϵ R, satisfying the initial condition y(0) = 4
Then, JT! y(X) is equal to..... . (Rounded off to two decimal places)

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Q 44 :

The sum of the series â€‹ is equal to (Rounded off to two decimal places)

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Q 45 :

The number of distinct subgroups of z₉₉₉ is... .

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Q 46 :

The number of elements of order 12 in the symmetric group S₇ is equal to ...... .

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Q 47 :

Let y(x) be the solution of the differential equation
​
Then, the value of y(5π/2)is equal to ..... (Rounded off to two decimal places)

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Q 48 :

Consider the region of G
Then , the volume of G is equal to ...... (Rounded off two decimal places)

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Q 49 :

Given that y(x) is a solution of the differential equation â€‹ on the interval (0 , ∞) such that exists and y(x) = 1. The value of y’(1) is equal to ....... (Rounded off to two decimal places)

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Q 50 :

Consider the family F₁ of curves lying in the region  and given by y  where is a positive real number Let F₂ be the family of or thogonal trajectories to Fl. Consider the curve C belonging to the family F₂ passing through the point ​ If a is a real number such â€‹ lies on C, then the value of a4 is equal to …… (Rounded off to two decimal places)

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