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Download DUET Master 2018 DU MA MSc Mathematics Question Paper With Answer Key

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

This post was last modified on 29 January 2020

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DU MA MSc Mathematics

Topic:- DU_J18_MA_MATHS_Topic01

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The complete integral of the partial differential equation xpq + yq² - 1 = 0 where p = dz/dx and q = dz/dy is
1. (z + b)² = 4(ax + y).
2. z + b = 2(ax + y).
3. z + b = 4(ax + y)².
4. z + b = 2(ax + y)².
Correct Answer :-

(z + b)² = 4(ax + y).
Let P be the set of all the polynomials with rational coefficients and S be the set of all sequences of natural numbers. Then which one of the following statements is true?
1. S is countable but P is not.
2. Both the sets P and S are uncountable.
3. Both the sets P and S are countable.
4. P is countable but S is not.
Correct Answer :-

P is countable but S is not.
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For the differential equation

x dy/dx + 6y = 3xy^4/3

consider the following statements:

(i) The given differential equation is a linear equation.
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(ii) The differential equation can be reduced to linear equation by the transformation V = y^-1/3.

(iii) The differential equation can be reduced to linear equation by the transformation V = x^-1/3.

Which of the above statements are true?
Which one of the following statements is not true for Simpson's 1/3 rule to find approximate value of the definite integral I = ∫ f(x)dx?
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1. If y₀ = f(0), y₁ = f(0.5), y₂ = f(1), the approximate value of I is 1/3 [y₀ + 3y₁ + y₂].
2. The approximating function has odd number of points common with the function f(x).
3. Simpson's 1/3 rule improves trapezoidal rule.
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4. The function f(x) is approximated by a parabola.
Correct Answer :-

If y₀ = f(0), y₁ = f(0.5), y₂ = f(1), the approximate value of I is 1/3 [y₀ + 3y₁ + y₂].
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The equation of the tangent plane to the surface z = 2x² - y² at the point (1, 1, 1) is
1. x - y - 2z = 2.
2. 4x - y - 3z = 1.
3. 2x - y - 2z = 1.
4. 4x - 2y - z = 1.
Correct Answer :-

4x - 2y - z = 1.
If {x,y} is an orthonormal set in an inner product space then the value of ||x - y|| + ||x + y || is
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1. 2√2.
2. 2 + √2.
3. √2.
4. 2.
Correct Answer :-
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2√2.
Which one of the following spaces, with the usual metric, is not separable?
1. The space C[a, b] of the set of all real valued continuous functions defined on [a, b].
2. The space l^∞ of all bounded real sequences with supremum metric.
3. The Euclidean space Rⁿ.
4. The space l¹ of all absolutely convergent real sequences.
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Correct Answer :-

The space l^∞ of all bounded real sequences with supremum metric.
Let G be an abelian group of order 2018 and f: G→ G be defined as f(x) = x⁵. Then
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1. f is not injective.
2. f is not surjective.
3. there exists e ≠ x ∈ G such that f(x) = x^-1.
4. f is an automorphism of G.
Correct Answer :-
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f is an automorphism of G.
If f: R → R is a continuous function such that

f(x + y) = f(x) + f(y), for all x, y ∈ R,

then
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1. f is increasing if f (1) ≥ 0 and decreasing if f (1) ≤ 0.
2. f is increasing if f (1) ≤ 0 and decreasing if f (1) ≥ 0.
3. f is a not an increasing function.
4. f is neither an increasing nor a decreasing function.
Correct Answer :-
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f is increasing if f (1) ≥ 0 and decreasing if f (1) ≤ 0.
The central difference operator δ and backward difference operator ∇ are related as
1. δ = √(1 − ∇)^½.
2. δ = (1 + ∇)^½.
3. δ = √(1 - √)^-½.
4. δ = √(1 + ∇)^-½.
Correct Answer :-

δ = √(1 - √)^-½.
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How many continuous real functions f can be defined on R such that (f(x))² = x² for every x∈R?
1. Infinitely many.
2. None.
3. 4.
4. 2.
Correct Answer :-

4.
The greatest common divisor of 11 + 7i and 18-i in the ring of Gaussian integers Z[i] is
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1. 3i.
2. 1.
3. 1+i.
4. 2+i.
Correct Answer :-
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1.
The complete integral of the partial differential equation

∂²z/∂x² - 2 ∂²z/∂x∂y + ∂²z/∂y² = e^x+2y

is
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1. φ₁(y - x) + xφ₂(y + x) + e^x+2y.
2. φ₁(y + x) + xφ₂(y + x) + xe^x+2y.
3. φ₁(y - x) + φ₂(y + x) + e^x+2y.
4. φ₁(y + x) + xφ₂(y + x) + e^x+2y.
Correct Answer :-
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φ₁(y + x) + xφ₂(y + x) + e^x+2y.
If S = {(1, 0, i), (1, 2, 1)} ∈ C³ then S^⊥ is
1. span {(i, -½(i + 1), -1)}.
2. span {(-i, -½(i + 1), 1)}.
3. span {(i, -½(i + 1), 1)}.
4. span {(i, ½(i + 1), -1)}.
Correct Answer :-

span {(i, -½(i + 1), 1)}.

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The improper integral ∫_-∞^∞ 2^-x²dx is
1. convergent and converges to 2.
2. divergent.
3. convergent and converges to 1/ln2.
4. convergent and converges to -ln2.
Correct Answer :-

convergent and converges to 1/ln2.
Let f: R → R be a continuous function which takes irrational values at rational points and rational values at irrational points. Then which one of the following statements is true?
1. f is uniformly continuous on Q.
2. f is uniformly continuous on R.
3. f is uniformly continuous on Q^c.
4. No such function exists.
Correct Answer :-

No such function exists.
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If f: [0,10] → R is defined as

f(x) = { 0, 0 ≤ x < 2, 1, 2 ≤ x ≤ 5 0, 5 < x ≤ 10,

and F(x) = ∫₀^x f(t)dt then
1. F(x) = 3 for x ≤ 5.
2. F'(x) = f(x) for every x.
3. F is not differentiable at x = 2 and x = 5.
4. F is differentiable everywhere on [0, 10].
Correct Answer :-

F is not differentiable at x = 2 and x = 5.
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The Maclaurin series expansion

ln(1 + x) = x - x²/2 + x³/3 ...

is valid
1. only if x ∈ [-1,1].
2. if x > -1.
3. only if x ∈ (-1,1].
4. for every x ∈ R.
Correct Answer :-

only if x ∈ (-1,1].
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If 4x = 2(mod 6) and 3x = 5(mod 8) then one of the value of x is
1. 32
2. 34
3. 26
4. 23
Correct Answer :-

23
If f(x) = lim_n→∞ S_n(x), where
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S_n(x) = x/(x+1)(2x+1) + x/(2x+1)(3x+1) + ... + x/(nx+1)((n+1)x+1)

then the function f is
1. continuous nowhere.
2. continuous everywhere.
3. continuous everywhere except at countably many points.
4. continuous everywhere except at one point.
Correct Answer :-

continuous everywhere except at one point.
The rate of change of f(x, y) = 4y - x² at the point (1, 5) in the direction from (1, 5) to the point (4, 3) is
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1. -6/√5
2. -14/√13
3. -12/√5
4. -19/√13
Correct Answer :-
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-14/√13
Let G = {a₁, a₂,...., a₂₅} be a group of order 25. For b, c ∈ G let

bG = {ba₁, ba₂,..., ba₂₅}, Gc = {a₁c, a₂c,...., a₂₅c}.

Then
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1. bG = Gc only if b = c.
2. bG ⊆ Gc ∀ b, c ∈ G.
3. bG = Gc only if b^-1 = c.
4. bG ⊆ Gc, if b ≠ c.
Correct Answer :-
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If (x_n) is a sequence such that x_n ≥ 0, for every n ∈ N and if lim_n→∞((-1)ⁿx_n) exists then which one of the following statements is true?
1. The sequence (x_n) is a Cauchy sequence.
2. The sequence (x_n) is not a Cauchy sequence.
3. The sequence (x_n) is unbounded.
4. The sequence (x_n) is divergent.
Correct Answer :-

The sequence (x_n) is a Cauchy sequence.
If n > 2, then n⁵ - 5n³ + 4n is divisible by
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1. 80
2. 120
3. 100
4. 125
Correct Answer :-
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120
Let

S = ∩_n=1^∞ [2 - 1/n, 3 + 1/n].

Then S equals
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1. (2, 3].
2. [2, 3].
3. [2, 3).
4. (2, 3).
Correct Answer :-
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[2, 3].
If a_n = n^sin(nπ/2) then
1. lim sup a_n = +∞, lim inf a_n = -1.
2. lim sup a_n = +∞, lim inf a_n = 0.
3. lim sup a_n = +∞, lim inf a_n = -∞.
4. lim sup a_n = 1, lim inf a_n = -1.
Correct Answer :-

lim sup a_n = +∞, lim inf a_n = 0.

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Let f: R²→ R be defined as f (x, y) = |x| + |y|. Then which one of the following statements is true?
1. f is continuous at (0, 0) and f_x(0,0) ≠ f_y(0,0).
2. f is continuous at (0, 0) and f_x(0,0) = f_y(0,0).
3. f is discontinuous at (0, 0) and f_x(0,0) = f_y(0,0).
4. f is continuous at (0, 0) but f_x and f_y does not exist at (0, 0).
Correct Answer :-

f is continuous at (0, 0) but f_x and f_y does not exist at (0, 0).
Let A and B be two subsets of a metric space X. If intA denotes the interior A of then which one of the following statements is not true?
1. A ⊆ B ⇒ intA ⊆ intB.
2. int(A ∪ B) = intA ∪ intB.
3. int(A ∩ B) = intA ∩ intB.
4. int(A ∪ B) ⊇ intA ∪ intB.
Correct Answer :-

int(A ∪ B) = intA ∪ intB.
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Which one of the following statements is false?
1. A subring of a field is a subfield.
2. A subring of the ring of integers Z, is an ideal of Z.
3. A commutative ring with unity is a field if it has no proper ideals.
4. A field has no proper ideals.
Correct Answer :-

A subring of a field is a subfield.
Let σ = (37125)(43216) ∈ S₇, the symmetric group of degree 7. The order of σ is
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1. 7
2. 4
3. 5
4. 2
Correct Answer :-
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4
Let

S = ∩_n=1^∞ [0, 1/n].

Then which one of the following statements is true?
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1. inf S > 0.
2. sup S=1 and inf S = 0.
3. sup S > 0.
4. sup S = inf S = 0.
Correct Answer :-
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sup S = inf S = 0.
The characteristics of the partial differential equation

36 ∂²z/∂x² - 14 x¹² ∂²z/∂x∂y - 8 ∂²z/∂y² = 0

when it is of hyperbolic type are given by
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1. x + 36/y⁶ = C₁, x - 36/y⁶ = C₂.
2. x + 1/y⁶ = C₁, x - 1/y⁶ = C₂.
3. x + 36/y⁷ = C₁, x - 36/y⁷ = C₂.
4. x + 36/y⁷ = C₁, x + 36/y⁷ = C₂.
Correct Answer :-
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x + 1/y⁶ = C₁, x - 1/y⁶ = C₂.
A bound for the error for the trapezoidal rule for the definite integral ∫₀¹ 1/(1+x²) dx is
1. 1/6
2. 2/25
3. 1/15
4. 1/20
Correct Answer :-

1/6

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Exact value of the definite integral ∫₀¹ f(x)dx using Simpson's rule
1. cannot be given for any polynomial.
2. is given when f (x) is a polynomial of degree 4.
3. is given when f (x) is a polynomial of degree 5.
4. is given when f (x) is a polynomial of degree 3.
Correct Answer :-

is given when f (x) is a polynomial of degree 3.
Let p be a prime and let G be a non-abelian p-group. The least value of m such that p^m|(G/Z(G)) is
1. 0
2. 1
3. 3
4. 2
Correct Answer :-

0
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If φ is Euler's Phi function then the value of φ(720) is
1. 248
2. 144
3. 192
4. 72
Correct Answer :-
The total number of arithmetic operations required to find the solution of a system of n linear equations in n unknowns by Gauss elimination method is
1. 2/3 n³ + 1/2 n² - 5/6 n.
2. 1/3 n³ + 1/2 n² - 1/6 n.
3. 2/3 n³ + 3/2 n² - 7/6 n.
4. 1/3 n³ + 3/2 n² - 5/6 n.
Correct Answer :-

2/3 n³ + 3/2 n² - 7/6 n.
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If (x_n) is a sequence defined as

x_n = [(5+n)/2n] for every n ∈ N

where [.] denotes the greatest integer function then lim_n→∞ x_n
1. 1.
2. 1/2
3. does not exist.
4. 0.
Correct Answer :-

0.
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Let R be a ring with characteristic n where n ≥ 2. If M is the ring of 2 × 2 matrices over R then the characteristic of M is
1. 1.
2. 0.
3. n - 1.
4. n.
Correct Answer :-

n.
If A = [2 1; a b] is a matrix with eigen values √6 and -√6, then the values of a and b are respectively,
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1. 2 and -1.
2. 2 and -2.
3. 2 and 1.
4. -2 and 1.
Correct Answer :-
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2 and -2.
The dimension of the vector space of all 6 × 6 real skew-symmetric matrices is
1. 36
2. 21
3. 30
4. 15
Correct Answer :-

15

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Let (x₀, f(x₀)) = (0,1), (x₁, f(x₁)) = (1, a) and (x₂, f (x₂)) = (2,b). If the first order divided differences f[x₀, x₁] = 5 and f[x₁, x₂] = c and the second order divided difference f[x₀, x₁, x₂] = -3/2, then the values of a, b and c are
1. 4, 2, 4.
2. 2, 4, 6.
3. 4, 6, 2.
4. 6, 2, 4.
Correct Answer :-

4, 6, 2.
Let the polynomial f(x) = 3x⁵ + 15x⁴ - 20x³ + 10x + 20 ∈ Z[x], and f₀(x) be the polynomial in Z₃[x] obtained by reducing the coefficients of f(x) modulo 3. Which one of the following statements is true?
1. f(x) is reducible over Q, f₀(x) is reducible over Z₃.
2. f(x) is irreducible over Q, f₀(x) is reducible over Z₃.
3. f(x) is reducible over Q, f₀(x) is irreducible over Z₃.
  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