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Delhi University Entrance Test (DUET) 2020 Previous Year Question Paper With Answer Key

This post was last modified on 27 December 2020

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

Topic:- MATHS MA S2

Let {x_n} and {y_n} be sequences of real numbers such that x_n = y_n for all n = N, where N is some positive integer. Consider the following statements:

(a) lim inf x_n = lim inf y_n
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(b) lim sup x_n = lim sup y_n

Which of the above statements is(are) correct?
1. Neither (a) nor (b)
2. Only (a)
3. Only (b)
4. Both (a) and (b)
Correct Answer :-

• Both (a) and (b)
Which of the sequences {a_n} and {b_n} of real numbers with n^th terms
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a_n = (n²+20n +35) sin n³ / (n² + n + 1)

b_n = 2 cosn - sin n

has(have) convergent subsequences?
1. Neither {a_n} nor {b_n}
2. Only {a_n}
3. Only {b_n}
4. Both {a_n} and {b_n}
Correct Answer :-

• Both {a_n} and {b_n}

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Consider the following series:

(a) ? xⁿ / n!, x ? R, n=1 to 8

(b) ? 1 / (n + sinn), n=1 to 8

(c) ? 1 / (n²vn), n=1 to 8

(d) ? sin n, n=1 to 8
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Which of the above series is (are) convergent?
1. Only (a), (c) and (d)
2. Only (a), (c) and (d)
3. Only (a) and (c)
4. Only (c)
Correct Answer :-

• Only (a) and (c)
The union of infinitely many closed subsets of the real line is
1. uncountable
2. finite
3. always closed
4. need not be closed
Correct Answer :-

• need not be closed
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Consider the series ? a_n, where a_n = (2 + sin(np/2))ⁿrⁿ, r > 0. What are the values of lim inf (a_n+1/a_n) and lim sup (a_n+1/a_n)?
1. r/2 and 2r
2. r/3 and r
3. 2r/3 and 3r/2
4. 0 and 1
Correct Answer :-

• r/2 and 2r
Consider the following series:
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(a) ? 3^-n sin 3ⁿx on R, n=1 to 8

(b) ? n^-2xⁿ on (-2,2), n=1 to 8

(c) ? (1/n) cosnx on R, n=1 to 8

Which of the above series converge uniformly on the indicated domain?
1. Only (a) and (b)
2. Only (b) and (c)
3. Only (a) and (c)
4. All of (a), (b) and (c)
Correct Answer :-

• Only (a) and (c)
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Let {f_n} be a sequence of continuous functions on [a, b] converging uniformly to the function f. Consider the following statements:

(a) f is bounded on [a, b]

If f_n' exists and the sequence {f_n'} converges uniformly to f' on [a, b], f' is the derivative of f.

Which of the following statements is(are) correct?
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1. Only (a) and (b)
2. Only (a) and (c)
3. Only (c)
4. Only (b)
Correct Answer :-
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• Only (a) and (b)
Let G(x) be a real-valued function defined by G(x) = ?₀^4x² cos vt dt. If G' is the derivative of G, then
1. G'(p/2) = -4p
2. G'(p/2) = -4p – 1
3. G'(p/2) = -p
4. G'(p/2) = 0
Correct Answer :-

• G'(p/2) = -4p

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Let f(x) = {(4-x²)/2, |x| < 2; 0, |x| = 2

Consider the following statements:

a. f is not continuous on R

b. f is continuous on R but not differentiable at x = 2, -2

c. f is differentiable on R but f' is not continuous on R
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d. f is differentiable on R and f' is continuous on R

Which of the above statements is(are) correct?
1. Only (a) and (d)
2. Only (b) and (c)
3. Only (c)
4. Only (d)
Correct Answer :-

• Only (d)
Let f(x) be a real-valued function defined on R lie on the interval
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1. (-1, 1)
2. [3, 4]
3. [-2, -1]
4. [-5, -3]
Correct Answer :-
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• [-2, -1]
The Wronskian of cosx, sin x and e^x at x = 0 is
1. 1
2. 2
3. -1
4. -2
Correct Answer :-

• 2

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The solution of the initial value problem y' = 1 + y², y(0) = 1, is:-
1. y = cosec(x + p/4)
2. y = tan(x + p/4)
3. y = sec(x + p/4)
4. y = cot(x + p/4)
Correct Answer :-

• y = tan(x + p/4)
How many solution(s) does the initial value problem y' - 2y = 0, y(0) = 0 have?
1. No solution
2. Unique solution
3. Two solutions
4. Infinitely many solutions
Correct Answer :-

• Infinitely many solutions
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The general solution of the equation y'' + y = x cosx is (c₁ and c₂ are arbitrary constants)
1. c₁ cosx + c₂ sin x - x cosx + sin x ln(sin x)
2. c₁ cosx + c₂ sin x + x cosx + sin x ln(sin x)
3. c₁ cosx + c₂ sin x - x sinx + cos x ln(sin x)
4. c₁ cosx + c₂ sin x + x sin x + cosx ln(sin x)
Correct Answer :-

• c₁ cosx + c₂ sin x + x cosx + sin x ln(sin x)
The particular integral of the differential equation is y'' + y = x³ is
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1. x² + 6x
2. x²-6x
3. x³ + 6x
4. x³-6x
Correct Answer :-
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• x³-6x
The complete integral of the partial differential equation p²z² + q² = 1, where p = ?z/?x, q = ?z/?y is (a, b are arbitrary constants)
1. z + v(z² + a²) + a² ln (z + v(z² + a²) / a) = 0
2. a²z + by + x² = 0
3. z + v(z² + a²) + a² ln (z + v(z² + a²) / a) = 2x + 2ay + 2b
4. z² + y² = x² + 2x + 2ay + 2b
Correct Answer :-

• z + v(z² + a²) + a² ln (z + v(z² + a²) / a) = 2x + 2ay + 2b

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The complete integral of the partial differential equation z = px+qy - sin(pq) where p = ?z/?x, q = ?z/?y is
1. z = ax + by - sm(ab)
2. z = ax + y + sin b
3. z = x + by - sin a
Correct Answer :-
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• z = ax + by - sin(ab)
The partial differential equation y u_xx + 2xy u_xy + x u_yy = u_x + u_y, is
1. Hyperbolic in {(x,y) | 0 < xy < 1}
2. Hyperbolic in {(x,y) | xy > 1}
3. Elliptic in {(x, y) | xy > 1}
4. Elliptic in {(x, y) | xy < 0}
Correct Answer :-

• Hyperbolic in {(x,y) | xy > 1}

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The general solution of the equation ?²z/?y² = x - y is
1. (1/4) x(x - y)² + Ø₁ (x² + y) + Ø₂ (x - y)
2. (1/4) x(x - y)² + Ø₁ (x + y) + Ø₂ (x - y)
3. Ø₁ (x + y) + Ø₂ (x² - y)
4. Ø₁ (x² + y) + Ø₂ (x² - y) - (1/4) x(x + y)
Correct Answer :-

• (1/4) x(x - y)² + Ø₁ (x + y) + Ø₂ (x - y)
The general solution of ?²u/?t² = c² ?²u/?x² with u(0,t) = u(2,t) = 0, u(x, 0) = sin³(px/2) and u_t(x, 0) = 0 is
1. (3/4) sin(px/2) sin(pct/2) - (1/4) sin(3px/2) sin(3pct/2)
2. (3/4) sin(px/2) cos(pct/2) - (1/4) sin(3px/2) cos(3pct/2)
3. (3/4) cos(px/2) sin(pct/2) - (1/4) cos(3px/2) sin(3pct/2)
4. (3/4) sin(px/2) cos(pct/2) - (1/4) cos(3px/2) sin(3pct/2)
Let f: R² ? R be given by f(x) = {(x² + y²) ln(x² + y²), if (x, y) ? (0,0); 0, if (x,y) = (0,0)}
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Then,
1. f_xy and f_yx are continuous at (0, 0), and f_xy(0,0) = f_yx (0,0)
2. f_xy and f_yx are discontinuous at (0, 0), but f_xy (0,0) = f_yx (0,0)
3. f_xy and f_yx are continuous at (0, 0), but f_xy (0,0) ? f_yx (0,0)
4. f_xy and f_yx are discontinuous at (0, 0) and f_xy(0,0) ? f_yx (0,0)
Correct Answer :-

• f_xy and f_yx are discontinuous at (0, 0), but f_xy (0,0) = f_yx (0,0)
The directional derivative of f(x,y,z) = xy² + yz² + zx² defined on R³ along the tangent to the curve x = t, y = t², z = t³ at the point (1, 1, 1) is
1. 18/v14
2. 13/v14
3. 13/v14
4. 18/v14
Correct Answer :-

• 18/v14
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The unique polynomial of degree 2 passing through (1, 1), (3, 27) and (4, 64) obtained by Lagrange interpolation is
1. 8x²-17x + 12
2. 8x²-19x- 12
3. 8x²+14x- 12
4. 8x²-19x + 12
Correct Answer :-

• 8x²-19x + 12
The approximate value of ?₀¹ dx/(1+x)² by Simpson's 1/3-rd rule, using the least number of equal subintervals, is
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1. 0.8512
2. 0.8125
3. 0.7625
4. 0.6702
Correct Answer :-
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• 0.8512
Consider the differential equation, dy/dx = y - x, y(0) = 2. The absolute value of the difference in the solutions obtained by Euler method and Runge-Kutta second order method at y(0.1) using step size 0.1 is
1. 2.205
2. 2.252
3. 0.005
4. 0.055
Correct Answer :-

• 0.005

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The approximate value of (17)^1/3 obtained after two iterations of Newton-Raphson method starting with initial approximation x₀ = 2 is
1. 2.7566
2. 2.5826
3. 2.6713
4. 2.4566
Correct Answer :-

• 2.5826
For an infinite discrete metric space (X, d), which of the following statements is correct?
1. x is compact
2. For every A ? X, A° ? A = X, where A and A° denote respectively the closure and interior of A in X
3. x is connected
4. x is not totally bounded
Correct Answer :-

• y is not totally bounded
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Consider the metric space (l²,d) of square summable sequences with the Euclidean metric. Y = {e₁, e₂, ...} ? l² where e_i is the sequence of 1 at the i – th place and 0 elsewhere. Then,
1. y is not compact and has no limit point
2. y is compact and each e_i is a limit point of y
3. y is not compact and has a limit point
4. y is compact and has no limit point
Correct Answer :-

• y is not compact and has no limit point
Let C[0, 1] be the set of real valued continuous functions on [0, 1] with sup-metric. Let A = {f ? C[0, 1] | f(0) = 0} and B = {f ? C[0, 1] | f(0) > 0} be the subspaces of C[0, 1]. Then,
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1. Both A and B are complete
2. A is complete but B is incomplete
3. A is incomplete but B is complete
4. Neither A nor B is complete
Correct Answer :-
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• A is complete but B is incomplete
Let (R, d) and (R, u) be the metric spaces with the discrete metric space d and usual metric u respectively. Let f: (R, d) ? (R, u) and g: (R, u) ? (R, d) be the functions given by f(x) = g(x) = {x + 1, x = 0; x > 0}

Then,
1. Both f and g are continuous
2. Neither f nor g is continuous
3. f is continuous but g is not
4. g is continuous but f is not
Correct Answer :-

• f is continuous but g is not
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Let Y₁ = {(x,y) ? R² | y = sin(1/x), 0 < x = p} and Y₂ = {(0, y) ? R² | y ? [-2,2]} be subspaces of the metric space (R²) being the Euclidean metric. For any A ? R², A denotes the closure of A
1. Y₁ ? Y₂ is connected
2. Y₁ ? Y₂ is connected
3. Y₁ n Y₂ is disconnected
4. Y₁ n Y₂ is a non-empty bounded subset of R²
Correct Answer :-

• Y₁ ? Y₂ is connected
Let R be the set of all real-valued Riemann integrable functions on and let f be the function given by f(x) = {0 if x = 0; 1/(n+1) if 1/(n+1) < x = 1/n for n ? N}
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Which of the following statements is correct?
1. f is monotonically decreasing on [0, 1] but f ? R[0, 1]
2. f is monotonically decreasing on [0, 1] and f ? R[0, 1]
3. f is discontinuous at infinitely many points in [0, 1] but f ? R[0, 1]
4. f is discontinuous at infinitely many points in [0, 1] and f ? R[0, 1]
Correct Answer :-

• f is discontinuous at infinitely many points in [0, 1] and f ? R[0, 1]
The improper integral ?_-8⁸ dx/(x²+1)
1. Converges to p
2. Converges to p/2
3. Converges to 0
4. Diverges
Correct Answer :-

• Converges to p
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Consider the functions f(x) = (x²-1)/(x-1) and g(x) = |x-1|/(x-1), x ? 1. Then
1. Both f and g have removable discontinuity at x = 1
2. f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1
3. f has a jump discontinuity at x = 1 while g has a removable discontinuity at x = 1
Correct Answer :-

• f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1
What is the length of the interval on which the function f(x) = x³ - 6x² + 15x + 8 is decreasing?
1. 8
2. 6
3. 4
4. 2
Correct Answer :-

• 6
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Let f: [a, b] ? R be a monotonic function. Consider the following statements:

a. The function f obeys the maximum principle

b. The function f is Riemann integrable on [a, b]

Which of the above statement(s) is (are) true?
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1. Only (a)
2. Only (b)
3. Both (a) and (b)
4. Neither (a) nor (b)
Correct Answer :-
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• Both (a) and (b)
Consider the following:

a. ((a,b), (c, d)) = ac - bd, (a, b), (c, d) ? R²

b. (f(x), g(x)) = ?₀¹ f'(x)g(x) dx, where f(x), g(x) are polynomials over R
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Which of the above is(are) an inner product?
1. Neither (a) nor (b)
2. Both (a) and (b)
3. Only (a)
4. Only (b)
Let T = ((0,1), (1,0), (0,0), (0,2)). Then T³ + 4T² + 5T - 2I is equal to
1. 10T +4I
2. 10T-4I
3. -10T +4I
4. -10T-4I
Correct Answer :-

• 10T-4I
Let V be an infinite dimensional vector space over a field F.
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Consider the following statements:

a. Any one-one linear transformation from V to itself is onto

b. Any onto linear transformation from V to itself must be one-one

Which of the above statements is (are) correct?
1. Both (a) and (b)
2. Only (a)
3. Only (b)
4. Neither (a) nor (b)
Correct Answer :-

• Neither (a) nor (b)
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Let P_n(R) be the set of all polynomials over R of degree at most n. Let T: P_n(R) ? P_n+1(R) be given by T(f(x)) = xf(x). Then
1. T is one-one and onto linear transformation
2. T is an onto function but neither a linear transformation nor one-one
3. T is not onto but a one-one linear transformation
4. T is one-one but neither a linear transformation nor onto
Correct Answer :-

• T is not onto but a one-one linear transformation
Let Z is the set of integers. The inverse of a is
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1. a-6
2. a-4
3. 4-a
4. 6-a
Correct Answer :-
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• 4-a
Let G be a group of even order. Suppose that exactly half of G consists of elements of order 2 and the rest forms a subgroup H of G. Which of the following statements is incorrect?
1. H is a normal subgroup of G
2. Order of H is even
3. H is abelian
4. G: H = 2
Correct Answer :-

• Order of H is even

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Let G and K be finite groups such that |G| = 21 and |K| = 49. Suppose G does not have a normal subgroup of order 3. Let be the set of all group homomorphism from G to K. Then the number of elements in is
1. 1
2. 3
3. 5
4. 7
Correct Answer :-

• 1
Let G be a finite group of a ? G has exactly two conjugates. Suppose that C(a) = {x^-1ax | x ? G} and N(a) = {x ? G | ax = xa}.

Which of the following statements is incorrect?
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1. The number of elements in C(a) is a prime number
2. G is a simple group
3. N(a) is a subgroup of G
Correct Answer :-

• G is a simple group
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Let G be a finite group of order 385. Let H, K and L be p-Sylow subgroups of G for p = 5,7 and 11, respectively. Which of the following statements is incorrect?
1. K is a normal subgroup of G
2. L is normal subgroup of G
3. HK is a non-abelian subgroup of G
4. G = HKL
Correct Answer :-

• HK is a non-abelian subgroup of G
The remainder when 2020²⁰²⁰ is divided by 12 is
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1. 0
2. 2
3. 4
4. 8
Correct Answer :-
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• 4
The smallest integer a > 2 such that 2|a, 3|(a+1), 4|(a + 2), 5|(a + 3) and 6|(a + 4) is
1. 14
2. 56
3. 122
4. 62
Correct Answer :-

• 62

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Let R = {(a, b) | a, b ? Z} be a ring and f: R ? Z be given by f((a, b)) = a - b. Which of the following statements is incorrect?
1. f is a ring homomorphism
2. ker f is a prime ideal but not maximal
3. ker f is maximal ideal

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Consider the following statements

a. A polynomial is irreducible over a field F if it has no zeros in F

b. Let f(x) ? Z[x]. If f(x) is reducible over Q, then it is reducible over Z

c. For any prime p, the polynomial x^p-1 + x^p-2 + ... + x +
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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

This post was last modified on 27 December 2020

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