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

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

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 n B = Ø, 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 n B = Ø 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 n K < H [Option ID = 47122]
Correct Answer :-

K < G ? H n 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 post was last modified on 29 January 2020

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