phd mathematics entrance · m.phil./ph.d. (mathematics) entrance examination 2016-17 max time: 2...

PhD Mathematics Entrance

Contents(M.Phil/PhD Que, Paper):

· DU M.Phil/PhD Entrance Exam: 2016 · DU M.Phil/PhD Entrance Exam: 2017 · DU M.Phil/PhD Entrance Exam: 2018 · DU M.Phil/PhD Entrance Exam: 2019

No. of Pages: 55

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University of Delhi

[ 1 ]

M.Phil./Ph.D. (Mathematics) Entrance Examination 2016-17

Max Time: 2 hours Max Marks: 150

Instructions: There are 50 questions. Every question has four choices of which exactly

one is correct. For correct answer, 3 marks will be given. For wrong answer, 1 mark will

be deducted. Scientific calculators are allowed.

In the following R, N, Q, Z and C denote the set of all real numbers, natural numbers,

rational numbers, integers and complex numbers respectively.

(1) We know that 11−x =

∑∞n=0 x

n, for |x| < 1. The series∞∑n=1

n

(1

2

)nconverges to

A) 1. B) 2. C) 0. D) 12.

(2) Which of the following statements is correct?

A) Intersection of two connected sets in a topological space is connected.

B) Countable union of compact sets in a topological space is compact.

C) Closure of a compact set in a topological space is compact.

D) None of A), B), C) is correct.

(3) An uncountable set with co-countable topology is

A) second countable. B) first countable but not second countable.

C) not first countable. D) separable.

(4) Which of the following statements is true?

A) A convergent net in a topological space has a unique limit.

B) Only eventually constant nets are convergent in an infinite space with cofinite

topology.

C) If X and Y are topological spaces and f : X → Y be a function. f is continuous

on X if and only if f(limnxn) = lim

nf(xn) for every convergent sequence (xn)

in X.

D) None of A), B), C) is true.

(5) Let τ be the topology on R generated by the basis consisting of all open intervals

(a, b) and the sets (a, b) ∼ A, where A = {1/n : n ∈ N}. Then

A) τ is not comparable with lower limit topology on R.

B) τ is strictly coarser than the usual topology on R.

C) τ is not comparable with co-finite topology on R.

D) τ is strictly finer than upper limit topology on R.

1

[ 2 ]

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2 M.Phil./Ph.D Admission Entrance Test — 2016-17

(6) The value of the integral∫∞

0cosx

(1+x2)2dx is

A) π/e. B) π/2e. C) 2πe. D) 2π/e.

(7) If∞∑

n=−∞

an(z − 1)n is the Laurent series expansion of g(z) = z2

(z−1)2, then a−1 is

given by

A) 1. B) −1. C) 2. D) 32.

(8) If∞∑n=1

a2nz2n is the power series of z2

(1−z2)2, then a2 is given by

A) 2. B) 3. C) 6. D) 4.

(9) Let f(z) = z2− 2z and γ(t) = (cos t, sin t), 0 ≤ t ≤ π. Then∫γf(z)dz is equal to

A) −2/3. B) 1/3. C) −1/3. D) 5/3.

(10) Let H be a Hilbert space and {ek : k ∈ N} an orthonormal set in H. Which of

the following is true?

A)∑

k |〈x, ek〉|2 = ‖x‖2 for every x ∈ H.

B) S = {ek : k ∈ N} is a Hamel basis for H if S is total in H.

C) 〈x, ek〉 = 0 for all k then x = 0.

D) H is separable if {ek : k ∈ N} is total in H .

(11) For normed spaces X and Y , let T : X → Y be a bounded linear operator and

dim (Range T ) <∞. Which of the following is not true?

A) Ker T is closed in X.

B) X/Ker T is finite dimensional.

C) dim (X/Ker T ) > dim (Range T ).

D) Range T is closed.

(12) Let X = C00, the linear space of all complex sequences with only a finite number

of non-zero entries. For x = (xn), y = (yn) ∈ X, define an inner product on X by

〈x, y〉 =∞∑n=1

xnyn

and f : X → C be defined by

f(x) =∞∑n=1

xnn.

[ 3 ]


M.Phil./Ph.D. Admission Entrance Test — 2016-17 3

Then which of the following is not true?

A) f is bounded with ‖f‖ ≤ π.

B) there exists y ∈ X such that f(x) = 〈x, y〉 for all x ∈ X.

C) Ker f is a proper closed subspace of X.

D) (Ker f)⊥ = {0}.

(13) Let A ={

1n

: n ∈ N}

and B = [−1, 1] \ A. If f : [−1, 1]→ R be defined as

f(x) =

{1, if x ∈ A;

−1, if x ∈ B,then f is

A) Riemann integrable.

B) Lebesgue integrable but not Riemann integrable.

C) not Lebesgue integrable.

D) not measurable.

(14) Let A = {(x, 0) : x ∈ R} and B = {(x, e−x) : x ∈ R} be subsets of R2 with

Euclidean metric. If d(A,B) denotes the distance between A and B, then A and

B are

A) disjoint closed sets in R2 with d(A,B) > 0.

B) disjoint closed sets in R2 with d(A,B) = 0.

C) closed in R2, d(A,B) = 0, but not disjoint.

D) disjoint, d(A,B) = 0, but not closed in R2.

(15) Let V be a Lebesgue non-measurable subset of R and p ∈ R be fixed. Define a

function fp : R→ R by

fp(x) =

{2, x− p ∈ V ;

0, otherwise.

Then fp is

A) Lebesgue integrable.

B) not Lebesgue measurable only for p = 0.

C) Lebesgue measurable only for p = 0.

D) not Lebesgue measurable.

(16) Let zi ∈ C (i = 1, 2, 3, 4) with z1 6= 0. Which one of the following identities about

cross ratio is false?

A) (z1, z2, z3, z4) = (z1 + z4, z2 + z4, z3 + z4, 2z4).

B) (z1, z2, z3, z4) = (z21 , z1z2, z1z3, z1z4).

C) (z1, z2, z3, z4) = (z21 , z

22 , z

23 , z

24).

[ 4 ]



D) (z1, z2, z3, z4) =(

1, z2z1, z3z1, z4z1

).

(17) Which of the following is not a reflexive Banach space?

A) For 1 < p <∞, lp = {(xn) :∑|xn|p <∞} with ‖.‖p.

B) for 1 ≤ p ≤ ∞, lnp = (Rn, ‖.‖p).C) C0, the space of sequences converging to zero with sup norm.

D) C{1, 2, . . . , n}, space of real valued continuous functions on {1, 2, . . . , n} with

maximum norm.

(18) Which of the following is not true?

A) If X is uncountable and d is an arbitrary metric on X, then (X, d) is separable.

B) Product of two separable metric spaces is separable.

C) ([a, b], d) (d is usual metric on [a, b]) is separable.

D) For a Lebesgue measurable subset E of R and 1 ≤ p ≤ ∞, Lp(E) is separable.

(19) Which of the following is not true?

A) R2 \Q2 is path connected.

B) For (x, y) ∈ R2, the subspace R2 \ {(x, y)} is connected and is homeomorphic

to R.

C) {(x, y) ∈ R2 : x = 0, −1 ≤ y ≤ 1} ∪ {(x, y) : y = sin 1x, 0 < x ≤ 1} is

connected.

D) C = {(x, y) ∈ R2 : x2 + y2 = 1} is connected.

(20) Let f(x) = sinx, then which of the following is not true?

A)

∫ ∞0

f(x)

xdx exists. B)

∫ ∞0

|f(x)|x

dx = π/2.

C)

∫ ∞0

f(x)

xdx does not exist in R. D)

∫∞0e−x

2dx =

√π/2.

(21) Let H and K be normal solvable subgroups of a group G. Then, HK is

A) solvable but not normal in G.

B) not solvable but normal in G.

C) neither solvable nor normal in G.

D) solvable and normal in G.

(22) The alternating group A∞ on infinitely many symbols

A) has a proper subgroup of finite index.

B) has no proper subgroup of finite index.

C) not a simple group.

[ 5 ]



D) none of A), B), C).

(23) Let G be a group of order 26. The number of subgroups of order 5 is

A) 0. B) 5. C) 6. D) 2.

(24) Which one of the following statements is true?

A) An infinite abelian group has a composition series.

B) a finite abelian group has no composition series.

C) a finite abelian group has a composition series.

D) every group has a composition series.

(25) Suppose ED stands for Euclidean domain, PID and UID stands for principal ideal

domain and unique factorization domain respectively. Then which of the following

statements is true?

A) PID =⇒ UFD =⇒ ED .

B) UFD =⇒ PID =⇒ ED.

C) ED =⇒ PID =⇒ UFD.

D) ED =⇒ UFD =⇒ PID.

(26) Let R be a commutative ring with unity satisfying descending chain condition

(d.c.c.) on its ideals. Consider the following statements.

1. R satisfies ascending chain condition (a.c.c.) on its ideals.

2. Every prime ideal in R is maximal.

Which of the following is correct?

A) Statement 1 is correct and statement 2 is not correct.

B) Statement 2 is correct and statement 1 is not correct.

C) Both the statements 1 and 2 are correct.

D) Both the statements 1 and 2 are false.

(27) Let Zp(α) be an extension of Zp obtained by adjoining α to Zp, where α is a root

of a degree two, irreducible polynomial over Zp. Then

A) |Zp(α)| = p3.

B) |Zp(α)| = p2.

C) |Zp(α)| =∞.

D) Zp(α) is a finite field but cannot say about its number of elements.

(28) Let F ⊆ E be a field extension. Let α ∈ E be a root of an irreducible polynomial

f(x) over F of multiplicity three. Let β be any other root of f(x) in E. Then the

multiplicity of β is

[ 6 ]



A) two. B) greater than three.

C) three. D) one.

(29) The total number of subfields of a field with 729 elements is

A) one. B) two. C) three. D) four.

(30) Let α be a positive constructible number. Then which one of the following is a

constructible number?

A) (α)1/3. B) (α)1/10. C) (α)1/14. D) (α)1/16.

(31) Let K = Q(√

2, i) be the field generated over Q by√

2 and i. Then the dimension

of Q(√

2, i), as a Q-vector space is equal to

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

(32) Let A =

0 1 1

1 0 1

1 1 0

and B =

−1 1 0

0 −1 1

0 0 2

. Then

A) A and B are diagonalizable.

B) A is diagonalizable but B is not.

C) B is diagonalizable but A is not.

D) neither A nor B is diagonalizable.

(33) Let T : R2 → R3 be the linear transformation defined by

T

(x1

x2

)=

x2

x1 + x2

x1 − x2

and let B = {(1, 2)t, (3, 1)t}, B′ = {(1, 0, 0)t, (1, 1, 0)t, (1, 1, 1)t} be ordered

bases for R2 and R3 respectively. Then the matrix of T relative to B and B′ is

A)

−1 −3

4 2

−1 2

. B)

1 3

2 4

1 −2

.

C)

−1 3

−4 2

−1 2

. D)

1 3

−2 −4

1 −2

.

(34) Let B = {(1, 1, 0), (−1, 0, 1), (0, 1,−1)} be a basis of R3 and its dual basis B∗ =

{f1, f2, f3}, where fi, i = 1, 2, 3 are linear functionals defined on R3. Then for

x = (x1, x2, x3)

[ 7 ]



A) 2f1(x) = x1 + x2 + x3; 2f2(x) = x1 − x2 + x3; 2f3(x) = −x1 + x2 − x3.

B) 2f1(x) = x1 − x2 + x3; 2f2(x) = −x1 − x2 + x3; 2f3(x) = −x1 + x2 − x3.

C) 2f1(x) = x1 + x2 + x3; 2f2(x) = −x1 − x2 + x3; 2f3(x) = −x1 + x2 + x3.

D) 2f1(x) = x1 + x2 + x3; 2f2(x) = −x1 + x2 + x3; 2f3(x) = −x1 + x2 − x3.

(35) Let A =

1 1 0 0

−1 −1 0 0

−2 −2 2 1

1 1 −1 0

. Then its minimal polynomial is

A) x2(x− 2)2. B) x2(x− 1)2.

C) (x− 1)2(x− 2)2. D) x(x− 1)3.

(36) The equation uxx + xuyy = 0 is

A) elliptic for x > 0, y ∈ R and hyperbolic for x < 0, y ∈ R.

B) elliptic for all (x, y) ∈ R2.

C) hyperbolic for all (x, y) ∈ R2.

D) hyperbolic for x > 0, y ∈ R and elliptic for x < 0, y ∈ R.

(37) Let G(x, ξ) be the Green’s function for the linear second order ordinary differential

equation L[y] = f(x), where L =d

dx

(p(x)

d

dx

)+ q(x) with q, f ∈ C[a, b], p ∈

C ′[a, b] and p(x) 6= 0, x ∈ [a, b]. Then which of the following is not true?

A) at the given point x = ξ ∈ [a, b], the first derivative has jump discontinuity

given by ddxG(x, ξ)

∣∣ξ+x=ξ−

= −1p(ξ)

.

B) G(x, ξ) = G(ξ, x).

C) G(x, ξ) and its first and second order derivatives are continuous for all x 6= ξ

in a ≤ x, ξ ≤ b.

D) for fixed ξ, G(x, ξ) is the solution of the associated homogeneous problem L[y] =

0, except at point x = ξ.

(38) The initial boundary value problem

ut = uxx, 0 < x < 1, t > 0;

u(0, t) = u(1, t) = 0; u(x, 0) = x(1− x), 0 ≤ x ≤ 1

has solution u(x, t) =∑∞

n=1 an exp (−n2πt) sin (nπx), where an is equal to

A) 8/(n3π3) if n is odd and 0 if n is even.

B) 4/(n3π3) if n is odd and 0 if n is even.

C) 8/(n2π2) if n is odd and 0 if n is even.

D) 8/(n3π3) if n is even and 0 if n is odd.

[ 8 ]



(39) Consider the following partial differential equation

z2(p2 + q2 + 1) = r2

where p = ∂z∂x

and q = ∂z∂y

. Which of the following statements is true about the

above equation?

A) (x− a)2 + (y − b)2 + z2 = r2 is a general integral.

B) it is a semi-linear PDE of order 2 and degree 1.

C) 4x2 + y2 − 4xy + 5z2 − 5r2 = 0 is a particular solution.

D) A), B), C) are true.

(40) The vertical displacement u(x, t) of an infinitely long elastic string is goverened

by the initial value problem

utt = 4uxx, −∞ < x <∞, t > 0

u(x, 0) = −x, ut(x, 0) = 0.

The value of u(x, t) at x = 2, t = 2 is

A) −2. B) −4. C) 2. D) 4.

(41) The equation of the surface satisfying 4yzp + q + 2y = 0, p = ∂z∂x, q = ∂z

∂yand

passing through y2 + z2 = 1, x+ z = 2 is given by

A) x2 + z + z2 + y2 = 3. B) x+ z2 + z + y2 = 3.

C) y + x+ z + z2 = 3. D) x2 + z + z2 + y = 3.

(42) An inviscid incompressible fluid of density ρ moves steadily with velocity ~q =

(kx,−ky, 0), where k is constant, under no extremal force. The pressure p(x, y, z)

in the fluid motion when p(0, 0, 0) = p0, is

A) p0 − ρk2(y2 − x2). B) p0 − ρk2(y2 + x2).

C) p0 − ρk2(y2+x2)2

. D) p0 − ρk2(y2−x2)2

.

(43) A sphere is moving with constant velocity U in a liquid which is otherwise at rest.

The velocity potential φ(r, θ) for the flow is

A) 12Ua3r−2 cos θ. B) 1

2Ua3r cos θ.

C) 12Ua2r−2 cos θ. D) none of these.

(44) In two dimensional motion, the vorticity vector is

A) perpendicular to the plane of flow.

B) parallel to the plane of flow.

C) oblique to the plane of flow.

[ 9 ]



D) may be parallel or perpendicular to the plane of flow.

(45) The singular solution of p3 − 4xyp+ 8y2 = 0, where p = dydx

is

A) 27y − 4x2 = 0. B) 27y − 4x3 = 0.

C) 27y + 4x3 = 0. D) 27y + 4x2 = 0.

(46) The arrangement of sources and sinks for w = log(z − a2

z

)are

A) a source of unit strength at (a, 0).

B) two sinks of unit strength at (±a2, 0).

C) a source of unit strength at origin and two sinks of unit strength at (a, 0) and

(−a, 0) .

D) a source of unit strength at (a, 0) and a sink at (0,−a) of unit strength.

(47) By the method of variation of parameters, the particular solution of the equation

x2y′′ + xy′ − y = x2ex is

A) ex + 1x. B) ex − ex

x.

C) ex + ex

x. D) ex − 1

x.

(48) The solution of uxx − 4uxy + 4uyy = 0 is

A) u = f(y + 2x) + g(y + 2x). B) u = f(y − 2x) + g(y + 2x).

C) u = f(y + 2x) + xg(y + 2x). D) u = f(y + 2x)− xg(y + 2x).

(49) A complete integral of f = xzxzy + yz2x − 1 = 0 obtained by Charpit’s method is

A) (z + b)2 = 4(ax− y). B) (z + b)2 = 4(ax+ y).

C) (z − b)2 = 4(ax+ y). D) (z + b)2 = 4ax+ y2.

(50) The tensor form of Navier Stokes equations with xi as space coordinates, ui as

velocity components, Fi as components of body forces, ∆ as rate of dilation, p as

pressure, ρ as density and ν as coefficient of kinematic viscosity, is given by

A) duidt

= Fi − 1ρ∂p∂xi

+ ν∇2ui + 23ν ∂∆∂xi

.

B) duidt

= Fi + 1ρ∂p∂xi

+ ν∇2ui + 23ν ∂∆∂xi

.

C) duidt


+ ν∇2ui + 13ν ∂∆∂xi

.

D) duidt


+ ν∇2ui + 23ν∆.

[ 10 ]


(1) -

(2) D

(3) C

(4) D

(5) A

(6) B

(7) C

(8) D

(9) A

(10) D

(11) C

(12) B

(13) A

(14) B

(15) D

(16) C

(17) C

(18) -

(19) B

(20) C

(21) D

(22) B

(23) A

(24) C

(25) C

(26) C

(27) B

(28) C

(29) D

(30) D

(31) A

(32) B

(33) A

(34) D

(35) B

(36) A

(37) C

(38) A

(39) C

(40) A

(41) B

(42) C

1

[ 11 ]


2 M.Phil./Ph.D Admission Entrance Test — 2016

(43) A

(44) A

(45) B

(46) C

(47) B

(48) C

(49) B

(50) C

[ 12 ]


[ 13 ]


[ 14 ]


[ 15 ]


[ 16 ]


[ 17 ]


[ 18 ]


[ 19 ]


[ 20 ]


[ 21 ]


[ 22 ]


[ 23 ]


[ 24 ]


[ 25 ]


[ 26 ]


[ 27 ]


[ 28 ]


[ 29 ]


[ 30 ]


[ 31 ]


[ 32 ]


[ 33 ]


[ 34 ]


[ 35 ]


[ 36 ]


DEPARTMENT OF MATHEMATICS Answer key for M.Phil/Ph.D Entrance Test (SET – A)

Question No. Answer

1. D

2. A

3. C

4. D

5. D

6. A

7. C

8. B

9. C

10. D

11. B

12. A

13. B

14. A

15. B

16. B

17. D

18. C

19. C

20. C

21. A

22. D

23. C

24. B

[ 37 ]


25. C

26. C

27. A

28. C

29. B

30. B

31. D

32. A

33. A

34. D

35. D

36. C

37. B

38. B

39. C

40. D

41. B

42. B

43. C

44. A

45. C

46. B

47. C

48. D

49. B

50. A

[ 38 ]


1)

2)

3)

4)

DU MPhil Phd in Mathema�cs

Topic:- DU_J19_MPHIL_MATHS

Which of the following journals is published by Indian Mathematical Society

[Question ID = 13918]

1. Indian Journal of Pure and Applied Mathematics. [Option ID = 25669]2. Indian Journal of Mathematics. [Option ID = 25671]3. Ramanujan Journal of Mathematics. [Option ID = 25670]4. The Mathematics Students . [Option ID = 25672]

Correct Answer :-Indian Journal of Pure and Applied Mathematics. [Option ID = 25669]

Name a Fellow of Royal Society who expired in 2019 [Question ID = 13917]

1. M. S. Ragunathan. [Option ID = 25665]2. Manjul Bhargava. [Option ID = 25666]3. Michael Atiyah. [Option ID = 25667]4. S. R. Srinivasa Varadhan. [Option ID = 25668]

Correct Answer :-M. S. Ragunathan. [Option ID = 25665]

Which of the following statements is true? [Question ID = 13973]

1.

[Option ID = 25890]2.

[Option ID = 25891]3.

[Option ID = 25889]

4. [Option ID = 25892]

Correct Answer :-

[Option ID = 25889]

Which of the following statements is true for topological spaces? [Question ID = 13927]

1. Every second countable space is separable. [Option ID = 25706]

[ 39 ]


5)

6)

7)

2. Every separable space is second countable. [Option ID = 25705]3. Every first countable space is second countable. [Option ID = 25708]4. Every first countable space is separable. [Option ID = 25707]

Correct Answer :-Every separable space is second countable. [Option ID = 25705]

Which of the following statements is not true? [Question ID = 13997]

1.

[Option ID = 25987]2.

[Option ID = 25986]3.

[Option ID = 25985]4.

[Option ID = 25988]

Correct Answer :-

[Option ID = 25985]

Which one of the following fellowship is based on merit in M.A/M.Sc. of the University [Question ID= 13920]

1. NBHM-JRF. [Option ID = 25679]2. INSPIRE-JRF [Option ID = 25677]3. UGC-JRF. [Option ID = 25680]4. CSIR-JRF [Option ID = 25678]

Correct Answer :-INSPIRE-JRF [Option ID = 25677]

The Abel prize 2019 was awarded to [Question ID = 13919]

1. Lennert Carleson. [Option ID = 25673]2. Mikhail Gromov. [Option ID = 25676]3. Karen Keskulla Uhlenbeck. [Option ID = 25674]4. Peter Lax. [Option ID = 25675]

Correct Answer :-Lennert Carleson. [Option ID = 25673]

[ 40 ]


8)

9)

10)


1. [Option ID = 25922]

2. [Option ID = 25921]

3. [Option ID = 25924]

4. [Option ID = 25923]

Correct Answer :-

[Option ID = 25921]


1. [Option ID = 25869]

2. [Option ID = 25871]

3. [Option ID = 25870]

4. [Option ID = 25872]

Correct Answer :-

[Option ID = 25869]


1. [Option ID = 25951]

2. [Option ID = 25949]

3. [Option ID = 25952]

4. [Option ID = 25950]

Correct Answer :-

[ 41 ]


11)

12)

13)

[Option ID = 25949]


1. cosec(y/x) = c/x. [Option ID = 26036]2. cosec(y/x) = cx. [Option ID = 26035]3. sec(y/x) = cx. [Option ID = 26033]4. sec(y/x) = c/x. [Option ID = 26034]

Correct Answer :-sec(y/x) = cx. [Option ID = 26033]


1. [Option ID = 26029]

2. [Option ID = 26032]

3. [Option ID = 26031]

4. [Option ID = 26030]

Correct Answer :-

[Option ID = 26029]


1. [Option ID = 25916]

2. [Option ID = 25913]

3. [Option ID = 25915]

4. [Option ID = 25914]

[ 42 ]


14)

15)

16)

17)

Correct Answer :-

[Option ID = 25913]


1. 2x + y + z = 0. [Option ID = 25979]2. x + 2y + z = 0. [Option ID = 25978]3. x + y + z = 0. [Option ID = 25980]4. x + y + 2z = 0. [Option ID = 25977]

Correct Answer :-x + y + 2z = 0. [Option ID = 25977]


1. [Option ID = 26027]

2. [Option ID = 26028]

3. [Option ID = 26026]

4. [Option ID = 26025]

Correct Answer :-

[Option ID = 26025]


1. 1/15 [Option ID = 25876]2. 2 [Option ID = 25875]3. 1/10 [Option ID = 25874]4. 1/5 [Option ID = 25873]

Correct Answer :-1/5 [Option ID = 25873]

[ 43 ]


18)

19)


1. lines parallel to z-axis. [Option ID = 25975]2. lines perpendicular to z-axis. [Option ID = 25976]3. planes perpendicular to xz- plane. [Option ID = 25973]4. planes parallel to yz- plane. [Option ID = 25974]

Correct Answer :-planes perpendicular to xz- plane. [Option ID = 25973]


1. Neither (I) nor (II) is correct. [Option ID = 26004]2. Both (I) and (II) are correct. [Option ID = 26003]3. Only (I) is correct. [Option ID = 26001]4. Only (II) is correct. [Option ID = 26002]

Correct Answer :-Only (I) is correct. [Option ID = 26001]


1. [Option ID = 25877]

2. [Option ID = 25879]

3. [Option ID = 25878]

4. [Option ID = 25880]

Correct Answer :-

[ 44 ]


20)

21)

22)

[Option ID = 25877]


1. 2π. [Option ID = 25935]2. 0 [Option ID = 25933]3. π/2. [Option ID = 25934]4. π. [Option ID = 25936]

Correct Answer :-0 [Option ID = 25933]


1. [Option ID = 25895]

2. [Option ID = 25893]

3. [OptionID = 25896]

4. [Option ID =25894]

Correct Answer :-

[Option ID = 25893]


1. [Option ID = 25899]

2. [Option ID = 25897]

3. [Option ID = 25900]

4. [Option ID = 25898]

Correct Answer :-

[Option ID = 25897]

[ 45 ]


23)

24)

25)

26)


1. 0 < α < 1, β = 1 [Option ID = 25942]

2. α = 0, β = ∞. [Option ID = 25943]

3. α = β = 0. [Option ID = 25941]

4. α = 0, β = 2. [Option ID = 25944]

Correct Answer :-

α = β = 0. [Option ID = 25941]


1. [−1, 1]. [Option ID = 25938]2. [0, 1]. [Option ID = 25937]3. {0}. [Option ID = 25939]4. [−1, 0]. [Option ID = 25940]

Correct Answer :-[0, 1]. [Option ID = 25937]


1. All (I), (II) and (III) are true. [Option ID = 26007]2. None of (I), (II)and (III) is true. [Option ID = 26008]3. Only (I) is true. [Option ID = 26005]4. Only (I) and (II) are true. [Option ID = 26006]

Correct Answer :-Only (I) is true. [Option ID = 26005]

[ 46 ]


27)

28)

29)


1. 2 [Option ID = 25963]2. 1 [Option ID = 25964]3. 3 [Option ID = 25962]4. infinity. [Option ID = 25961]

Correct Answer :-infinity. [Option ID = 25961]


1. (I) is true but (II) is false. [Option ID = 25925]2. (I) is false but (II) is true. [Option ID = 25926]3. Neither (I) nor (II) is true. [Option ID = 25927]4. Both (I) and (II) are true. [Option ID = 25928]

Correct Answer :-(I) is true but (II) is false. [Option ID = 25925]


1. [Option ID = 25911]

2. [Option ID = 25912]

3. [Option ID = 25909]

4. [Option ID = 25910]

Correct Answer :-

[Option ID = 25909]


[ 47 ]


30)

31)

32)

1. 2 [Option ID = 25989]2. 5 [Option ID = 25991]3. 8 [Option ID = 25992]4. 4 [Option ID = 25990]



1. [Option ID = 25931]

2. [Option ID = 25932]

3. [Option ID = 25929]

4. [Option ID = 25930]

Correct Answer :-

[Option ID = 25929]


1. [Option ID = 26018]

2. [Option ID = 26019]

3. [Option ID = 26020]

4. [Option ID = 26017]

Correct Answer :-

[Option ID = 26017]


[ 48 ]


33)

34)

35)

1. [Option ID = 25902]2.

[Option ID = 25903]

3. [Option ID = 25904]4.

[Option ID = 25901]

Correct Answer :-

[Option ID = 25901]


1. dimension of kernel of D is 2. [Option ID = 25969]2. dimension of kernel of D is 4. [Option ID = 25970]3. range of D = V . [Option ID = 25972]4. range of D is a finite dimensional space [Option ID = 25971]

Correct Answer :-dimension of kernel of D is 2. [Option ID = 25969]


1. [Option ID = 25998]

2. [Option ID = 26000]

3. [Option ID = 25997]

4. [Option ID = 25999]

Correct Answer :-

[Option ID = 25997]

[ 49 ]


36)

37)


1. [Option ID = 26023]

2. [Option ID = 26022]

3. [Option ID = 26024]

4. [Option ID = 26021]

Correct Answer :-

[Option ID = 26021]


1. continuous, one-one but not onto. [Option ID = 25886]2. not a continuous map. [Option ID = 25885]3. a continuous bijection but not an open map. [Option ID = 25887]4. a homeomorphism. [Option ID = 25888]

Correct Answer :-not a continuous map. [Option ID = 25885]


1. [Option ID = 25959]

2. [OptionID = 25958]

3. [Option ID =25957]

4.

[Option ID = 25960]

Correct Answer :-

[ 50 ]


38)

39)

40)

[Option ID =25957]


1. [Option ID = 25967]

2. [Option ID = 25966]

3. [Option ID = 25965]

4. [Option ID = 25968]

Correct Answer :-

[Option ID = 25965]


1. [Option ID = 26015]

2. [Option ID = 26014]

3. [Option ID = 26013]

4. [Option ID = 26016]

Correct Answer :-

[Option ID = 26013]


[ 51 ]


41)

42)

43)

1. [Option ID = 25918]

2. [Option ID = 25917]

3. [Option ID = 25919]

4. [Option ID = 25920]

Correct Answer :-

[Option ID = 25917]


1. [Option ID = 25882]

2. [Option ID = 25884]

3. [Option ID = 25881]

4. [Option ID = 25883]

Correct Answer :-

[Option ID = 25881]


1. G is a cyclic group under composition. [Option ID = 25994]2. G is a non-cyclic group under composition. [Option ID = 25995]3. G does not form a group under composition. [Option ID = 25993]4. G is a non-abelian group under composition. [Option ID = 25996]

Correct Answer :-G does not form a group under composition. [Option ID = 25993]

[ 52 ]


44)

45)

46)


1. [Option ID = 25954]

2. [Option ID = 25956]

3. [Option ID = 25953]

4. [Option ID = 25955]

Correct Answer :-

[Option ID = 25953]


1. Only (III) and (IV) are true. [Option ID = 26012]2. Only (I) and (II) are true. [Option ID = 26009]3. Only (II) and (III) are true. [Option ID = 26011]4. Only (I) and (IV) are true. [Option ID = 26010]

Correct Answer :-Only (I) and (II) are true. [Option ID = 26009]


1. 0 [Option ID = 25866]2. 1/3 [Option ID = 25867]3. 1/2 [Option ID = 25868]4. 1 [Option ID = 25865]



[ 53 ]


47)

48)

49)

1. [Option ID = 25945]

2. [Option ID = 25947]

3. [Option ID = 25948]

4. [Option ID = 25946]

Correct Answer :-

[Option ID = 25945]

Which of the following statements is true? [Question ID = 13977]

1.

[Option ID = 25906]

2. [Option ID =25907]

3. [Option ID = 25905]

4. [Option ID =25908]

Correct Answer :-

[Option ID = 25905]

Which one of the following statements is not true? [Question ID = 13966]

1.

[Option ID = 25861]

2. [Option ID= 25864]

3. [Option ID =25863]

4. [OptionID = 25862]

Correct Answer :-

[Option ID = 25861]

Present President of the Ramanujan Mathematical Society is [Question ID = 13916]

1. V. Kumar Murty. [Option ID = 25664]

[ 54 ]


50)

2. Dinesh Singh [Option ID = 25661]3. S. Ponnusamy [Option ID = 25662]4. R. Balakrishnan. [Option ID = 25663]

Correct Answer :-Dinesh Singh [Option ID = 25661]

The characteristic and the minimal polynomial are same for the matrix


1. [Option ID = 25983]

2. [Option ID = 25982]

3. [Option ID = 25981]4. All of the above matrices [Option ID = 25984]

Correct Answer :-

[Option ID = 25981]

[ 55 ]


phd mathematics entrance · m.phil./ph.d. (mathematics) entrance examination 2016-17 max time: 2...

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