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Download DUET Master 2018 DU MPhil Phd in Mathematics Question Paper With Answer Key

Download DUET (Delhi University Entrance Test conducted by the NTA) 2018 DU MPhil Phd in Mathematics Question Paper With Solution Key

This post was last modified on 29 January 2020

Tags: DUET Delhi University Entrance Test NTA Last 10 Years 2011-2021 Question Papers Answer Key DU MPhil PhD Mathematics 2018 duet-master-2018 frdA290120A1938177

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Topic:- DU_J18_MPHIL_MATHS_Topic01

The mathematician who was awarded Abel's prize for a proof of Fermat's Last Theorem is [Question ID = 19249]
1. Andrew Wiles. [Option ID = 46987]
2. Johan F. Nash. [Option ID = 46988]
3. S. R. Srinivasa Varadhan. [Option ID = 46989]
4. Lennart Carleson. [Option ID = 46990]
Correct Answer :-

Andrew Wiles. [Option ID = 46987]
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Founder of Indian Mathematical Society(IMS) was [Question ID = 19252]
1. Asutosh Mukherjee. [Option ID = 47000]
2. S. Narayana Aiyer. [Option ID = 47001]
3. M.T. Narayaniyengar. [Option ID = 47002]
4. V. Ramaswamy Aiyer. [Option ID = 46999]
Correct Answer :-

V. Ramaswamy Aiyer. [Option ID = 46999]
Let R be a commutative ring with identity. If R is an Artinian domain, then the total number of prime ideals in R is [Question ID = 19280]
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1. 1 [Option ID = 47111]
2. infinite. [Option ID = 47114]
3. 3 [Option ID = 47113]
4. 2 [Option ID = 47112]
Correct Answer :-
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1 [Option ID = 47111]
Riemann hypothesis is associated with the function [Question ID = 19250]
1. \(f(s) = \int_0^\infty t^{s-1}e^{-t} dt\). [Option ID = 46991]
2. \(f(x, y) = \int_0^1 t^{x-1}(1 - t)^{y-1} dt\). [Option ID = 46992]
3. Hermite polynomial \(f(s) = \sum_{n=1}^\infty \frac{1}{n^s}, s \in \mathbb{C}\) [Option ID = 46993]
4. [Option ID = 46994]
Correct Answer :-

Hermite polynomial \(f(s) = \sum_{n=1}^\infty \frac{1}{n^s}, s \in \mathbb{C}\) [Option ID = 46993]

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For the stream function of a two dimensional motion, which of the following is not true [Question ID = 19297]
1. Stream function is constant along a stream line. [Option ID = 47181]
2. Stream function is harmonic. [Option ID = 47180]
3. Stream function exists for steady motion of compressible fluid. [Option ID = 47179]
4. Stream function has dimension \(L^2T^{-2}\). [Option ID = 47182]
Correct Answer :-

Stream function has dimension \(L^2T^{-2}\). [Option ID = 47182]
The famous Indian mathematician Srinivas Ramanujan passed away in the year [Question ID = 19248]
1. 1920 [Option ID = 46984]
2. 1922 [Option ID = 46985]
3. 1921 [Option ID = 46983]
4. 1919 [Option ID = 46986]
Correct Answer :-

1920 [Option ID = 46984]
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Let F be a finite field with 9 elements. How many elements are there in F?
1. 1 [Option ID = 47142]
2. 4 [Option ID = 47140]
3. 8 [Option ID = 47139]
4. 2 [Option ID = 47141]
Correct Answer :-

4 [Option ID = 47140]
For a viscous compressible fluid Consider the following statements:
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(I) Stress matrix is symmetric.

(II) Kinematic coefficient of viscosity is dependent on the mass.

(III) Rate of dilatation is \(\nabla \cdot \vec{q}\).

Then
1. all of I, II and III are true. [Option ID = 47163]
2. only I and III are true. [Option ID = 47164]
3. only I and II are true. [Option ID = 47165]
4. only II and III are true. [Option ID = 47166]
Correct Answer :-

only I and III are true. [Option ID = 47164]
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Let \(f: R \rightarrow R'\) be a ring homomorphism. Assume that 1 and 1' are multiplicative identities of the rings R and R' respectively. Then f(1) = 1' if

I f is onto.

II f is one-one.

III R is a domain.
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IV R' is a domain.

The correct options are
1. III and IV only. [Option ID = 47096]
2. II and III only [Option ID = 47098]
3. I and IV only. [Option ID = 47097]
4. I and II only. [Option ID = 47095]
Correct Answer :-

I and IV only. [Option ID = 47097]
For a solid stationary sphere of radius a placed in an incompressible fluid of uniform stream with velocity -Ui:
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(I) velocity potential \(\phi(r, \theta) = U \cos \theta (r + \frac{a^3}{2r^2})\).

(II) there exist two stagnation points (a, 0), (\(\alpha\), \(\pi\)).

(III) stagnation pressure \(p_\infty + \frac{1}{2} \rho U^2\), \(p_\infty\) is a pressure at \(\infty\).

(IV) velocity at any point of surface of sphere is (0, U sin \(\theta\), 0).

Then
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1. only I, II, IV are true. [Option ID = 47175]
2. only I, III, IV are true. [Option ID = 47177]
3. only I, II, III are true. [Option ID = 47176]
4. only II, III, IV are true. [Option ID = 47178]
Correct Answer :-
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only I, II, III are true. [Option ID = 47176]
Let \(R = \{a + ib : a, b \in \mathbb{Z}\}\). Then R is a Euclidean domain with
1. exactly two units. [Option ID = 47099]
2. exactly eight units. [Option ID = 47101]
3. exactly four units. [Option ID = 47100]
4. infinitely many units. [Option ID = 47102]
Correct Answer :-

exactly four units. [Option ID = 47100]

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Consider the sequence of Lebesgue measurable functions \((f_n)\) on \(\mathbb{R}\)

\(f_n(x) = \begin{cases} 5, & x \geq 2^n \\ 0, & x < 2^n \end{cases}\)

Then \(\lim_{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) dx\)
1. does not exist [Option ID = 47046]
2. equals 0. [Option ID = 47043]
3. equals 5. [Option ID = 47044]
4. equals \(\infty\). [Option ID = 47045]
Correct Answer :-

equals \(\infty\). [Option ID = 47045]

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Let \(f(x) = \sin x + \cos x\) on \([0, \pi]\). Then \(||f||_\infty\) is equal to
1. 1 [Option ID = 47067]
2. \(\sqrt{2}\) [Option ID = 47068]
3. \(2\sqrt{2}\) [Option ID = 47070]
4. \(1/\sqrt{2}\) [Option ID = 47069]
Correct Answer :-

\(\sqrt{2}\) [Option ID = 47068]
Let f be a continuous function on a finite interval [a, b]. Then

\(\lim_{t \rightarrow \infty} \int_a^b f(x) \sin tx dx\)
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1. equals 0 [Option ID = 47033]
2. equals \(\sup_{x \in [a, b]} f(x)\) [Option ID = 47034]
3. does not exist [Option ID = 47032]
4. equals \(\int_a^b f(x) dx\). [Option ID = 47031]
Correct Answer :-
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equals 0 [Option ID = 47033]
Let (X, d) be a metric space and A ⊂ X, B ⊂ X. Consider the following statements:

I If x ∉ A then d(x, A) > 0.

II If A ∩ B = ∅, then d(A, B) ≥ 0.
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III If A is closed and x ∉ A then d(x, A) > 0.

IV If A and B are closed and A ∩ B = ∅ then d(A, B) ≥ 0.

Then,
1. all statements are correct. [Option ID = 47030]
2. only III is correct. [Option ID = 47028]
3. only II, III, IV are correct. [Option ID = 47027]
4. only III and IV are correct. [Option ID = 47029]
The set \(A = \{x \in \mathbb{Q} | -\sqrt{7} \leq x \leq \sqrt{7}\}\) in the subspace \(\mathbb{Q}\) of the real line \(\mathbb{R}\) is
1. neither open nor closed [Option ID = 47078]
2. open but not closed [Option ID = 47075]
3. both open and closed [Option ID = 47077]
4. closed but not open [Option ID = 47076]
Correct Answer :-

both open and closed [Option ID = 47077]
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A Lipschitz's constant associated with the function \(f(x, y) = y^{2/3}\) on \(\mathbb{R}: |x| \leq 1, |y| \leq 1\)
1. does not exist. [Option ID = 47146]
2. equals 1/2. [Option ID = 47145]
3. equals 0. [Option ID = 47143]
4. equals 1. [Option ID = 47144]
Correct Answer :-

does not exist. [Option ID = 47146]
Let \(I = \int_C y dx + (x + 2y) dy\), where \(C = C_1 + C_2\), \(C_1\) being the line joining (0, 1) to (1, 1) and \(C_2\) is the line joining (1, 1) to (1, 0). The value of I is
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1. 2 [Option ID = 47017]
2. -1 [Option ID = 47018]
3. 1 [Option ID = 47015]
4. 0 [Option ID = 47016]
Correct Answer :-
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1 [Option ID = 47015]
Let \(F(x) = \int_0^x \frac{\sin t}{t} dt\), \(0 < x < \infty\). The local maximum value is at the point
1. \(x = \pi/2\) [Option ID = 47013]
2. \(x = 4\pi\) [Option ID = 47014]
3. \(x = \pi\) [Option ID = 47011]
4. \(x = 2\pi\). [Option ID = 47012]
Correct Answer :-

\(x = \pi\) [Option ID = 47011]

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The general integral of the partial differential equation yzp + xzq = xy, where \(p = \frac{\partial z}{\partial x}\), \(q = \frac{\partial z}{\partial y}\) (G being an arbitrary function) is
1. \(z^2 = x^2 - G(x^2 + y^2)\). [Option ID = 47150]
2. \(2z^2 = y^2 + G(x^2 + y^2)\). [Option ID = 47147]
3. \(z^2 = y^2 + G(x^2 - y^2)\). [Option ID = 47149]
4. \(z^2 = x - G(x^2 - y^2)\). [Option ID = 47148]
Correct Answer :-

\(z^2 = y^2 + G(x^2 - y^2)\). [Option ID = 47149]
Let \(f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}\) Then
1. For any \(\delta > 0\), f is not monotonic on [0, \(\delta\)) [Option ID = 47020]
2. f has a local extremum at x = 0 [Option ID = 47021]
3. For any \(\delta > 0\), f is convex on [0, \(\delta\)) [Option ID = 47022]
4. f' is continuous at x = 0 [Option ID = 47019]
Correct Answer :-

For any \(\delta > 0\), f is not monotonic on [0, \(\delta\)) [Option ID = 47020]
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Let \(F = \mathbb{Q}(\sqrt{2}, \sqrt{3})\). Then F is minimal splitting field of the polynomial \((x^2 - 2)(x^2 - 3)\) over \(\mathbb{Q}\). The field F is not the minimal splitting field of which of the following polynomials over \(\mathbb{Q}\)
1. \(x^4 - 10x^2 + 1\). [Option ID = 47135]
2. \(x^4 - x^2 + 6\). [Option ID = 47137]
3. \(x^4 + x^2 + 1\). [Option ID = 47136]
4. \(x^4 + x^2 + 25\). [Option ID = 47138]
An elementary solution of the partial differential equation \(\frac{\partial u}{\partial x^2} + \frac{\partial u}{\partial y^2} = 0\) is of the form (\(\vec{r} = xi + yj\), \(\vec{r'} = x'i + y'j\))
1. \(u = \log |\vec{r}\vec{r'}|\). [Option ID = 47154]
2. \(u = \log \frac{1}{|\vec{r} + \vec{r'}|}\) [Option ID = 47151]
3. \(u = \log \frac{1}{|\vec{r}\vec{r'}|}\) [Option ID = 47153]
4. \(u = \log \frac{1}{|\vec{r} - \vec{r'}|}\) [Option ID = 47152]
Correct Answer :-

\(u = \log \frac{1}{|\vec{r} - \vec{r'}|}\) [Option ID = 47152]

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Let \(E = \{x \in (0, \sqrt{2}] : x \text{ is a rational number}\} \cup \{y \in [2, 3] : y \text{ is an irrational number}\}\) Then the Lebesgue measure of E is
1. 1 [Option ID = 47048]
2. \(\sqrt{2}\) [Option ID = 47049]
3. 1/2 [Option ID = 47050]
4. \(\sqrt{2} + 1\) [Option ID = 47047]
Correct Answer :-

1 [Option ID = 47048]
Let H be a Sylow p-subgroup and K be a p-subgroup of a finite group G. Which of the following is incorrect is incorrect (H char G means H is characteristic in G)
1. K < G ⇒ K ⊂ H. [Option ID = 47119]
2. K < G ⇒ K char H. [Option ID = 47121]
3. K ⊂ H if K < G. [Option ID = 47120]
4. K < G ⇒ H ∩ K < H [Option ID = 47122]
Correct Answer :-

K < G ⇒ H ∩ K < H [Option ID = 47122]
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A two dimensional motion with complex potential \(w = U(z + \frac{a^2}{z}) + ik \log \frac{z}{a}\) has

(I) stream lines as circle |z| = a.

(II) circulation zero about circle |z| = a.

(III) has two stagnation points in general.
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(IV) velocity at infinity equal to (-U).

Then
1. only I, II, IV are true. [Option ID = 47172]
2. only I, III, IV are true. [Option ID = 47173]
3. only I, II, III are true. [Option ID = 47171]
4. only II, III, IV are true. [Option ID = 47174]
Correct Answer :-

only I, III, IV are true. [Option ID = 47173]
Let G be an abelian group of order 15. Define a map \(\phi: G \rightarrow G\) by \(\phi(g) = g^8\) for all \(g \in G\). Consider the statements:
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I \(\phi\) is a homomorphism.

II \(\phi\) is one-to-one.

III \(\phi\) is onto.

Then
1. only I and III are true. [Option ID = 47117]
2. only I and II are true. [Option ID = 47116]
3. only I is true. [Option ID = 47115]
4. all statements are true. [Option ID = 47118]
Correct Answer :-

all statements are true. [Option ID = 47118]
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Let \(\xi\) be a primitive \(n^{th}\) root of unity where n ≡ 2 (mod 4). Then \([\mathbb{Q}(\xi) : \mathbb{Q}(\xi^2)]\) is

(Here [V : F] denotes the dimension of the vector space V over F)
1. 1 [Option ID = 47131]
2. 2 [Option ID = 47132]
3. \(\phi(n)\) [Option ID = 47133]
4. \(\phi(n)/2\) [Option ID = 47134]
Correct Answer :-

1 [Option ID = 47131]

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The closed topologist's sine curve \(\{(x, \sin \frac{1}{x}) | x \in (0, 1]\}\) as subspace of real line \(\mathbb{R}\) is
1. a path connected space [Option ID = 47081]
2. connected but not locally connected [Option ID = 47079]
3. a locally path connected space [Option ID = 47082]
4. locally connected but not connected [Option ID = 47080]
Correct Answer :-

connected but not locally connected [Option ID = 47079]
Let R(T) and N(T) denote the range space and null space of the linear transformation \(T: P_2(\mathbb{R}) \rightarrow M_{2 \times 2}(\mathbb{R})\) which is given by

\(T(f) = \begin{pmatrix} f(1) - f(2) & 0 \\ 0 & f(0) \end{pmatrix}\)
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Then
1. dim(R(T)) = 2 and dim(N(T)) = 1 [Option ID = 47094]
2. dim(R(T)) = 0 and dim(N(T)) = 2 [Option ID = 47093]
3. dim(R(T)) = 2 and dim(N(T)) = 0 [Option ID = 47091]
4. dim(R(T)) = 1 and dim(N(T)) = 1 [Option ID = 47092]
Correct Answer :-

dim(R(T)) = 2 and dim(N(T)) = 1 [Option ID = 47094]
The bilinear transformation on \(\mathbb{C}\) which maps z = 0, -i, -1 into w = i, 1, 0 is
1. \(\frac{i z + 1}{z - 1}\) [Option ID = 47053]
2. \(\frac{z + 1}{z - 1}\) [Option ID = 47052]
3. \(\frac{i z + 1}{z - 1}\) [Option ID = 47051]
4. \(\frac{i z - 1}{z + 1}\) [Option ID = 47054]
Correct Answer :-

\(\frac{i z + 1}{z - 1}\) [Option ID = 47053]
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Let \(A, B \in M_n(\mathbb{C})\). Consider the following statements

I If A, B and A + B are invertible, then \(A^{-1} + B^{-1}\) is invertible.

II If A, B and A + B are invertible, then \(A^{-1} - B^{-1}\) is invertible.

III If AB is nilpotent, then BA is nilpotent.
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IV Characteristic polynomials of AB and BA are equal if A is invertible.

Then
1. only I, III, and IV are true [Option ID = 47089]
2. all the statements are true.. [Option ID = 47090]
3. only III is true [Option ID = 47088]
4. only I and II are true [Option ID = 47087]
Correct Answer :-

only I, III, and IV are true [Option ID = 47089]
For the boundary value problem: L(y) = y" = 0, y'(0) = 0, y'(1) = 0, the Green's function is
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1. \(G(x, \xi) = \begin{cases} \xi, & x \leq \xi \\ x, & x > \xi \end{cases}\) [Option ID = 47156]
2. \(G(x, \xi) = \begin{cases} -x, & x \leq \xi \\ -\xi, & x > \xi \end{cases}\) [Option ID = 47157]
3. \(G(x, \xi) = \begin{cases} x, & x \leq \xi \\ -\xi, & x > \xi \end{cases}\) [Option ID = 47158]
4. \(G(x, \xi) = \begin{cases} \xi, & x \leq \xi \\ x, & x > \xi \end{cases}\) [Option ID = 47155]
Correct Answer :-
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\(G(x, \xi) = \begin{cases} \xi, & x \leq \xi \\ x, & x > \xi \end{cases}\) [Option ID = 47155]
Let \(E = \{x \in [0, \pi) : \sin 4x < 0\}\). Then Lebesgue measure of E is
1. \(\pi/2\) [Option ID = 47040]
2. \(\pi/4\) [Option ID = 47039]
3. \(3\pi/4\) [Option ID = 47041]
4. \(\pi/3\) [Option ID = 47042]
Correct Answer :-

\(\pi/2\) [Option ID = 47040]

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Let \(x_1, x_2, ..., x_n\) be non-zero real numbers. With \(x_{ij} = x_i x_j\), let X be the n × n matrix \((x_{ij})\). Then
1. the matrix X is positive definite if \((x_1, x_2, ..., x_n)\) is a non-zero vector [Option ID = 47084]
2. the matrix X is positive semi definite for all \((x_1, x_2, ..., x_n)\) [Option ID = 47085]
3. for all \((x_1, x_2, ..., x_n)\), zero is an eigenvalue of X. [Option ID = 47086]
4. it is possible to chose \(x_1, x_2, ..., x_n\) so as to make the matrix X non singular [Option ID = 47083]
Let \(A = \{f : \mathbb{R} \rightarrow \mathbb{R} | f \text{ is continuous on } \mathbb{Q} \text{ and discontinuous } \mathbb{Q'}\}\), where \(\mathbb{Q}\) is the set of all rational numbers and \(\mathbb{Q'}\) is the set of all irrational numbers. Let \(\mu\) be a counting measure on A. Then
1. \(\mu(A) = \sum_{q \in \mathbb{Q}} \frac{1}{2^q}\) [Option ID = 47026]
2. \(\mu(A)\) is infinite [Option ID = 47023]
3. \(\mu(A) = 0\) [Option ID = 47024]
4. \(\mu(A) = 2\) [Option ID = 47025]
Correct Answer :-

\(\mu(A) = 0\) [Option ID = 47024]
Let \(R = \mathbb{Z}_2 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5\). Then
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This download link is referred from the post: DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA

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This post was last modified on 29 January 2020