DUET 2019 MPhil PhD in Mathematics Previous Queston Papers

Delhi University Entrance Test (DUET) 2019 MPhil PhD in Mathematics Previous Queston Papers

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DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA

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DU MPhil Phd in Mathematics

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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]

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Which of the following statements is true? [Question ID = 13973]
1. Every topological space having Bolzano-Weierstrass property is a compact space. [Option ID = 25890]
2. If {Xn} is a convergent sequence in a topological space X with a limit x then Y = {x}?{xn: n = 1,2,... } is a compact subset of X. [Option ID = 25891]
3. The projection map p: X×Y ? Y defined by p(x, y) = y is a closed map for all topological spaces X, Y. [Option ID = 25889]
4. Every topological space is a first countable space. [Option ID = 25892]
Correct Answer :-
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The projection map p: X×Y ? Y defined by p(x, y) = y is a closed map for all topological spaces X, Y. [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 = 25705]
2. Every separable space is second countable. [Option ID = 25706]
3. Every first countable space is second countable. [Option ID = 25708]
4. Every first countable space is separable. [Option ID = 25707]
Correct Answer :-

Every second countable space is separable [Option ID = 25705]
Which of the following statements is not true? [Question ID = 13997]
1. If H and K are normal subgroups of G, then the subgroup generated by H?K is also a normal subgroup of G. [Option ID = 25987]
2. Let G be a finite group and H a subgroup of order n. If H is the only subgroup of order n, then H is normal in G. [Option ID = 25986]
3. The set of all permutations s of Sn (n = 3) such that s(n) = n is a normal subgroup of Sn. [Option ID = 25985]
4. For groups G and H and f: G? H a group homomorphism. If H is abelian and N is a subgroup of G containing kerf then N is a normal subgroup of G. [Option ID = 25988]
Correct Answer :-

The set of all permutations s of Sn (n = 3) such that s(n) = n is a normal subgroup of Sn. [Option ID = 25985]
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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]

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Let X be a normed space over C and f a non-zero linear functional on X. Then [Question ID = 13981]
1. f is surjective and a closed map. [Option ID = 25922]
2. f is surjective and open. [Option ID = 25921]
3. f is continuous and bijective. [Option ID = 25924]
4. f is open and continuous. [Option ID = 25923]
Correct Answer :-
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f is surjective and open. [Option ID = 25921]
Let f: R?R be defined as f(x) =
```
 { x² sin(1/x), x?0 0, x=0 
```
Then which of the following statements is not true? [Question ID = 13968]
1. f is bounded above on (a, 8). [Option ID = 25869]
2. f' is not continuous at 0. [Option ID = 25871]
3. f is infinitely differentiable at every non zero x ? R. [Option ID = 25870]
4. f is neither convex nor concave on (0, d). [Option ID = 25872]
Correct Answer :-

f is bounded above on (a, 8). [Option ID = 25869]
The principal part of the Laurent series of f(z) = 1/(z(z-1)(z-3)) in the annulus {z: 0 < |z| <1} is [Question ID = 13988]
1. 1/z² [Option ID = 25951]
2. 1/(2z) [Option ID = 25949]
3. 1/(3z²) [Option ID = 25952]
4. 1/(3z²) [Option ID = 25950]
Correct Answer :-

1/(2z) [Option ID = 25949]
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The general solution of the differential equation dy/dx = y/x + cot(y/x) , where c is a constant, is [Question ID = 14009]
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]
Velocity potential for the uniform stream flow with velocity q = -Ui, where U is constant and i is the unit vector in x-direction, past a stationary sphere of radius a and centre at origin, for r= a is [Question ID = 14008]
1. U cos ? (r + a²/r²) [Option ID = 26029]
2. U cos ? (r² + a²/r) [Option ID = 26032]
3. U cos ? (r² + a/r²) [Option ID = 26031]
4. U cos ? (r + a²/r) [Option ID = 26030]
Correct Answer :-

U cos ? (r + a²/r²). [Option ID = 26029]

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Let X = P[a, b] be the linear space of all polynomials on [a, b]. Then which of the following statements is not true? [Question ID = 13979]
1. X is dense in C[a, b] with ||. ||p-norm, 1 = p=8. [Option ID = 25916]
2. X is a Banach space with ||. ||p- norm, 1 = p < 8. [Option ID = 25913]
3. X has a denumerable basis. [Option ID = 25915]
4. X is incomplete with ||. ||8-norm. [Option ID = 25914]
Correct Answer :-
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X is a Banach space with ||. ||p- norm, 1 = p=8. [Option ID = 25913]
Let W = {(x,x,x): x ? R} be a subspace of the inner product space R³ over R. The orthogonal complement of W in R³ is the plane [Question ID = 13995]
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]
The integral surface of the partial differential equation x²p+y²q=z², p = dz/dx, q = dz/dy which passes through the hyperbola xy = x + y, z = 1 is [Question ID = 14007]
1. x^-1 + y^-1 + z^-1=3. [Option ID = 26027]
2. x^-1 + y^-2 + z^-1=3. [Option ID = 26028]
3. x^-2 + y^-1 + z^-1=3. [Option ID = 26026]
4. x^-1 + y^-1 + z^-2=3. [Option ID = 26025]
Correct Answer :-

x^-1 + y^-1 + z^-2=3. [Option ID = 26025]
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The value of ?_C x²dx + (xy + y²)dy, where C is the boundary of the region R bounded by y = x and y = x² and is oriented in positive direction is [Question ID = 13969]
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]
Let W = {(x, y, 0): x,y ? R} be a subspace of R³. The cosets of W in R³ are [Question ID = 13994]
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]

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Let R be a ring with unity. An element a of R is called nilpotent if aⁿ = 0 for some positive integer n. An element a of R is called unipotent if and only if 1 - a is nilpotent. Consider the following statements: (I) In a commutative ring with unity, product of two unipotent elements is in- vertible. (II) In a ring with unity, every unipotent element is invertible. Then [Question ID = 14001]
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 :-
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Only (I) is correct. [Option ID = 26001]
Which of the following statements is not true? [Question ID = 13970]
1. gn(x) = 1/(n(1+x²)) ?0, n?8 uniformly on R. [Option ID = 25877]
2. fn(x) = (sin nx)/(x² + nx) converges uniformly on R. [Option ID = 25879]
3. hn(x) = xⁿ/n converges uniformly on R. [Option ID = 25878]
4. Un(x) = xⁿ/n converges uniformly on [0, 1]. [Option ID = 25880]
Correct Answer :-

gn(x) = 1/(n(1+x²)) ?0, n?8 uniformly on R. [Option ID = 25877]
The value of the integral ?_C dz/(z²+4) where C is the anticlockwise circle |z|= 2 is [Question ID = 13984]
1. 2p. [Option ID = 25935]
2. 0 [Option ID = 25933]
3. p/2. [Option ID = 25934]
4. p. [Option ID = 25936]
Correct Answer :-

0 [Option ID = 25933]
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Which of the following statements is true for the product ?_a? X_a with product topology of a family {X_a}_a? of topological spaces? [Question ID = 13974]
1. If each X_a is metrizable then ?_a? X_a is metrizable. [Option ID = 25895]
2. If each X_a is normal then ?_a? X_a is normal. [Option ID = 25893]
3. If each X_a is completely regular then ?_a? X_a is completely regular. [Option ID = 25896]
4. If each X_a is locally connected then ?_a? X_a is locally connected. [Option ID = 25894]
Correct Answer :-

If each X_a is normal then ?_a? X_a is normal. [Option ID = 25893]
Consider R with usual metric and a continuous map f: R ? R then [Question ID = 13975]
1. f(A) is bounded for every bounded subset A of R. [Option ID = 25899]
2. f is bounded. [Option ID = 25897]
3. f^-1(A) is compact for all compact subset A of R. [Option ID = 25900]
4. Image of f is an open subset of R. [Option ID = 25898]
Correct Answer :-

f is bounded. [Option ID = 25897]

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Define a sequence of functions fn(x) =
```
 { 1, if x ? [-n -2, -n) 0, otherwise. 
```
Let a = ?⁸_-8 lim_n?8 fn(x)dx and ß = lim_n?8?⁸_-8 fn(x)dx. Then [Question ID = 13986]
1. 0 < a < 1, ß = 1 [Option ID = 25942]
2. a = 0, ß = 8. [Option ID = 25943]
3. a = ß = 0. [Option ID = 25941]
4. a = 0, ß = 2. [Option ID = 25944]
Correct Answer :-
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a = ß = 0. [Option ID = 25941]
Suppose f is an entire function with f(0) = 0 and u be the real part of f such that |u(x, y)| = 1 for all (x, y) ? R². Then the range of u is [Question ID = 13985]
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]
For the minimal splitting field F of a polynomial f(x) of degree n over a field K. Consider the following statements: (I) F over K is a normal extension. (II) n|[F: K]. (III) F over K is a separable extension. Then [Question ID = 14002]
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]
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Let V = {x+ay: a, x, y ? R} be a normed space [Question ID = 13991]
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]
Let X = C² with ||. ||1 norm and Xo = {(X1, X2) ? X: X2 = 0}. Define g: Xo? C by g(x) = x1, x = (x1, 0). Consider the following statements: (I) Every f? X' (dual space of X) is of the form f(x1, x2) = ax1 + bx2 for some a, b ? C. (II) Hahn-Banach extensions of g are precisely of the form f(x) = x1 + bx2, x = (X1, X2) ? X, |b| = 1, b ? C. Then [Question ID = 13982]
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]

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Which of the following statements is not true for a subset A of a metric space X, whose closure is A? [Question ID = 13978]
1. If X is totally bounded then A is totally bounded. [Option ID = 25911]
2. A is connected if and only if A is connected. [Option ID = 25912]
3. A is bounded if and only if A is bounded. [Option ID = 25909]
4. A is totally bounded if and only if A is totally bounded. [Option ID = 25910]
Correct Answer :-
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A is bounded if and only if A is bounded. [Option ID = 25909]
How many pairs of elements are there that generate D₈ = (a, b|a² = b⁴ = 1, ab = ba^-1) [Question ID = 13998]
1. 2 [Option ID = 25989]
2. 5 [Option ID = 25991]
3. 8 [Option ID = 25992]
4. 4 [Option ID = 25990]
Correct Answer :-

2 [Option ID = 25989]
For each n ? N, define xn ? C[0, 1] by xn(t) =
```
 { n²t, 0<t=1/n 1/t, 1/n<t=1 
```
where C[0, 1] is endowed with sup-norm. Then which of the following is not true: [Question ID = 13983]
1. The sequence {xn}_n?N is uniformly bounded on [0, 1]. [Option ID = 25931]
2. The set {xn(t): n ? N} is bounded for each t ? [0, 1]. [Option ID = 25929]
3. Each xn is uniformly continuous on [0, 1]. [Option ID = 25932]
4. ||xn||₈= n for all n. [Option ID = 25930]
Correct Answer :-

The set {xn(t): n ? N} is bounded for each t ? [0, 1]. [Option ID = 25929]
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The eigenvalues of the boundary value problem y" + y' + (1 + x)y = 0, y(0) = 0, y(1) = 0 are [Question ID = 14005]
1. -1/4 + n²p², n ? N. [Option ID = 26018]
2. -1/2 + n²p², n ? N. [Option ID = 26019]
3. -1 + n²p², n ? N. [Option ID = 26020]
4. 3/4 + n²p², n ? N. [Option ID = 26017]
Correct Answer :-

3/4 + n²p², n ? N. [Option ID = 26017]
Let (X, d) be a complete metric space. Then which of the following statements holds true? [Question ID - 13976]
1. If X is compact [Option ID = 25902]
2. If {F_n} is a decreasing sequence of non-empty closed subsets of X then n⁸₁Fn is non-empty. [Option ID = 25903]
3. Every open subspace of X is complete. [Option ID = 25904]
4. If X is union of a sequence of its subsets then the closure of at least one set in the sequence must have non-empty interior. [Option ID = 25901]
Correct Answer :-

If X is union of a sequence of its subsets then the closure of at least one set in the sequence must have non-empty interior. [Option ID = 25901]

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Let V be the set of all polynomials over IR. A linear transformation D: V ? V is defined by D(f(x)) = d³/dx³ (f(x)). Then [Question ID = 13993]
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 :-
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dimension of kernel of D is 2. [Option ID = 25969]
If G = Z₆ ? Z₂₀ ? Z₇₂, then G is isomorphic to [Question ID = 14000]
1. Z₈ ? Z₉ ? Z₄₀. [Option ID = 25998]
2. Z₂ ? Z₁₂ ? Z₃₆₀. [Option ID = 26000]
3. Z₅ ? Z₂₇ ? Z₆₄ [Option ID = 25997]
4. Z₆ ? Z₃₂ ? Z₄₅ [Option ID = 25999]
Correct Answer :-

Z₅ ? Z₂₇ ? Z₆₄ [Option ID = 25997]
The general solution of the partial differential equation ?²z/?x?y + ?z/?x - ?z/?y - z = xy is [Question ID = 14006]
1. e^xf1(y) + e^-xf2(x) +xy+y-x-1. [Option ID = 26023]
2. e^xf1(y) + e^-xf2(x) - xy - y + x + 1. [Option ID = 26022]
3. e^xf1(y) + e^xf2(x) + xy + y - x - 1. [Option ID = 26024]
4. e^-xf1(y) + e^xf2(x) – xy - y + x + 1. [Option ID = 26021]
Correct Answer :-

e^-xf1(y) + e^xf2(x) - xy - y + x + 1. [Option ID = 26021]
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The function f: [0, 2p] ? S¹ defined by f(t) = e^it, where S¹ is the unit circle, is [Question ID = 13972]
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]
Define f on C by f(z) =
```
 { z²/|z|², z? 0 0, z = 0. 
```
Let u and v denote the real and imaginary parts of f. Then at the origin [Question ID = 13990]
1. u, v do not satisfy the Cauchy Riemann equations but f is differentiable. [Option ID = 25959]
2. u, v satisfy the Cauchy Riemann equations but f is not differentiable [Option ID = 25958]
3. f is differentiable and u, v satisfy the Cauchy Riemann equations. [Option ID = 25957]
4. f is not differentiable and u, v do not satisfy the Cauchy Riemann equations. [Option ID = 25960]
Correct Answer :-

f is differentiable and u, v satisfy the Cauchy Riemann equations. [Option ID = 25957]

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Let V be the set of all polynomials over IR. Define W = {xⁿ f(x) : f(x) ? V}, n? N is fixed. Then which of the following statements is not true? [Question ID = 13992]
1. V is infinite dimensional over IR. [Option ID = 25967]
2. The quotient space V/W is finite dimensional. [Option ID = 25966]
3. W is not a subspace of V. [Option ID = 25965]
4. V has linearly independent set of m vectors for every m ? N. [Option ID = 25968]
Correct Answer :-
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W is not a subspace of V. [Option ID = 25965]
Navier Stokes equation of motion for steady viscous incompressible fluid flow in absence of body force is (where q, p, ?, ? and ? are velocity, pressure, density, vorticity, and kinematic coefficient of viscosity respectively) [Question ID = 14004]
1. ?(½|q|² + p/?) + q × ? = ?²q. [Option ID = 26015]
2. ?(½|q|² + p/?) - q × ? = ?²q. [Option ID = 26014]
3. ?(½|q|² + p/?) + q × ? = ?²q. [Option ID = 26013]
4. ?(½|q|² + p/?) - q × ? = -?²q. [Option ID = 26016]
Correct Answer :-

?(½|q|² + p/?) + q × ? = ?²q. [Option ID = 26013]
Let X = C₀₀ (the space of all real sequences having only finitely many non-zero terms) with ||.||₈-norm. Define P: X ? X by P(x)(2j-1) = x(2j - 1) + jx(2j) P(x)(2j) = 0 for x ? X, j? N. Then which of the following statements is not true? [Question ID = 13980]
1. P is a bounded linear map. [Option ID = 25918]
2. P is linear and P² = P. [Option ID = 25917]
3. Range(P) is a closed subspace of X. [Option ID = 25919]
4. P is a continuous map. [Option ID = 25920]
Correct Answer :-

P is linear and P² = P. [Option ID = 25917]
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The value of ?_C 2x ds, where C consists of the arc C₁ of the parabola y (0, 0) to (1, 1) followed by the line segment from (1, 1) to (0, 0) is [Question ID = 13971]
1. (5v5-1)/6 +2v2. [Option ID = 25882]
2. (5v5-4)/3 + 2v2. [Option ID = 25884]
3. (5v5-1)/6 + v2. [Option ID = 25881]
4. (3v5-1)/5 + v2. [Option ID = 25883]
Correct Answer :-

(5v5-1)/6 + v2. [Option ID = 25881]
For each integer n, define f_n(x) = x + n, x ? R and let G = {f_n: n?Z}. Then [Question ID = 13

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This post was last modified on 19 June 2020

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