**GATE-2020**

**MA: Mathematics**

**GA-General Aptitude**

**Q1-Q5 carry one mark each.**

1. Rajiv Gandhi Khel Ratna Award was conferred _____Mary Kom, a six-time world champion in boxing, recently in a ceremony _____ the Rashtrapati Bhawan (the President’s official residence) in New Delhi.

(A) with, at

(B) on, in

(C) on, at

(D) to, at

2. Despite a string of a poor performances, the changes of K. L. Rahul’s selection in the team are ______.

(A) slim

(B) bright

(C) obvious

(D) uncertain

3. Select the word that fits the analogy:

Cover : Uncover :: Associate : _______

(A) Unassociate

(B) Inassociate

(C) Missassociate

(D) Dissociate

4. Hig by floods, he kharif (summer sown) crops in various parts of the county have been affected. Officials believe that the loss in production of the kharif crops can be recovered in the output of the rabi (winter sown) crops so that the country can achieve its food-grain production target of 291 million tons in the crop year 2019-20 (July-June). They are hopeful that good rains in July-August will help the soil retain moisture for a longer period, helping winter sown crops such as wheat and pulses during the November-February period.

Which of the following statements can be inferred from the given passage?

(A) Officials declared that the food-grain production target will be met due to good rains.

(B) Officials want the food-grain production target to be met by the November-February period.

(C) Officials feel that the food-grain production target cannot be met due to floods.

(D) Officials hope that the food-grain production target will be met due to a good rabi produce.

5. The difference between the sum of the first 2n natural numbers and the sum of the first n odd natural numbers is ______.

(A) n^{2} – n

(B) n^{2} + n

(C) 2n^{2} – n

(D) 2n^{2} + n

**Q6-Q10 carry two marks each.**

6. Repo rate is the at which Reserve Bank of India (RBI) lends commercial banks, and reverse repo rate is the rate at which RBI borrows money from commercial banks.

Which of the following statements can be inferred from the above passage?

(A) Decrease in repo rate will increase cost of borrowing and decrease lending by commercial banks.

(B) Increase in repo rate will decrease cost of borrowing and increase lending by commercial banks.

(C) Increase in repo rate will decrease cost of borrowing and decrease lending by commercial banks.

(D) Decrease in repo rate will decrease cost of borrowing and increase lending by commercial banks.

7. P, Q, R, S, T, U, V, and W are seated around a circular table.

(I) S is seated opposite to W.

(II) U is seated at the second place to the right of R.

(III) T is seated at the third place to the left of R.

(IV) V is a neighbor of S.

Which of the following must be true?

(A) P is a neighbor of R.

(B) Q is a neighbor of R.

(C) P is not seated opposite to Q.

(D) R is the left neighbor of S.

8. The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The two cars crossed each other 75 minutes after the car Q started. Both cars were travelling at constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had travelled when the cars crossed each other?

(A) 66.6

(B) 75.2

(C) 88.2

(D) 116.5

9. For a matrix M = [m_{ij}], i. j= 1, 2, 3, 4, the diagonal elements are all zero and m_{ij} = −m_{ij}. The minimum number of elements required to fully specify the matrix is_______.

(A) 0

(B) 6

(C) 12

(D) 16

10. The profit shares of two companies P and Q are shown in the figure. If the two companies have invested a fixed and equal amount every year, then the ratio of the total revenue of company P to the total revenue of company Q, during 2013-2018 is ______.

(A) 15 : 17

(B) 16 : 17

(C) 17 : 15

(D) 17 : 16

**MA: Mathematics**

**Q1 – Q25 carry one mark each.**

1. Suppose that ℑ_{1} and ℑ_{2} are topologies on X induced by metrics d_{1} and d_{2}, respectively, such that ℑ_{1} ⊆ ℑ_{2}. Then which of the following statements is TRUE?

(A) If a sequence converges in (X, d_{2}) then it converges in (X, d_{1})

(B) If a sequence converges in (X, d_{1}) then it converges in (X, d_{2})

(C) Every open ball in (X, d_{1}) is an open ball in (X, d_{2})

(D) The map x → x from (X, d_{1}) to (X, d_{2}) is continuous

2. Let D = [−1, 1] × [−1, 1]. If the function f : D → ℝ is defined by

then

(A) f is continuous at (0, 0)

(B) both the first order partial derivatives of f exist at (0, 0)

(C) ∬_{D} |f(x, y)|^{1/2} dx dy is finite

(D) ∬_{D} |f(x, y)| dx dy is finite

3. The initial value problem y′ = y^{3/5}, y(0) = b has

(A) a unique solution if b = 0

(B) no solution if b = 1

(C) infinitely many solutions if b = 2

(D) a unique solution if b = 1

4. Consider the following statements:

I : log(|z|) is harmonic on ℂ\{0}

II: log(|z|) has a harmonic conjugate on ℂ\{0}

Then

(A) both I and II are true

(B) I is true but II is false

(C) I is false but II is true

(D) both I and II are false

5. Let G and H be defined by

G = ℂ\ {z = x + iy ∈ ℂ : x ≤ 0, y = 0},

H = ℂ\ {z = x + iy ∈ ℂ : x ∈ ℤ, x ≤ 0, y = 0}.

Suppose f: G → ℂ and g : H → ℂ are analytic functions. Consider the following statements:

I : ∫_{γ} f dz is independent of pathys γ in G joining –i and i

II : ∫_{γ} g dz is independent of paths γ in H joining –i and i

Then

(A) both I and II are true

(B) I is true but II is false

(C) I is false but II true

(D) both I and II are false

6. Let f (z) = e^{1/z}, z ∈ ℂ \ {0} and let, for n ∈ ℕ,

If for a subset S of ℂ, denotes the closure of S in ℂ, then

7. Suppose that

Then, with respect to the Eclidean metric on ℝ^{2},

(A) both U and V are disconnected

(B) U is disconnected but V is connected

(C) U is connected but V is disconnected

(D) both U and V are connected

8. If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and P2), respectively, where

(P1) : minimize x_{1} – 2x_{2} subject to –x_{1} +x_{2} = 10, x_{1}, x_{2} ≥ 0,

(P2) : minimize x_{1} – 2x_{2} subject to –x_{1} + x_{2} = 10, x_{1} – x_{2} = 10, x_{1}, x_{2} ≥ 0, then

(A) both (D1) and (D2) are infeasible

(B) (P2) is infeasible and (D2) is feasible

(C) (D1) is infeasible and (D2) is feasible but unbounded

(D) (P1) is feasible but unbounded and (D1) is feasible

9. If (4, 0) and (0, −1/2) are critical points of the function

f(x, y) = 5 – (α + β) x^{2} + βy^{2} + (α + 1) y^{3} + x^{3},

where α, β ∈ ℝ, then

(A) (4, −1/2) is a point of local maxima of f

(B) (4, −1/2) is saddle point of f

(C) α = 4, β = 2

(D) (4, −1/2) is a point of local minima of f

10. Consider the iterative scheme

with initial point x_{0} > 0. Then the sequence {x_{n}}

(A) converges only if x_{0} > 1

(B) converges only if x_{0} < 3

(C) converges for any x_{0}

(D) does not converge for any x_{0}

11. Let C[0, 1] denote the space of all real-valued continuous functions on [0, 1] equipped with the supremum norm || ∙ ||_{∞}. Let T : C[0, 1] → C[0, 1] be the linear operator defined by

Then

(A) || T || = 1

(B) I – T is not invertible

(C) T is surjective

(D) ||I + T|| = 1+ ||T||

12. Suppose that M is a 5 × 5 matrix with real entries and p(x) = det(xI – M). Then

(A) p(0) = det(M)

(B) every eigenvalue of M is real if p(1) + (p)2 = 0 = p(2) + p(3)

(C) M^{−}^{1} is necessarily a polynomial in M of degree 4 if M is invertible

(D) M is not invertible if M^{2} – 2M = 0

13. Let C[0, 1] denote the space of all real-valued continuous functions on [0, 1] equipped with the supremum norm || ∙ ||_{∞}. Let f ∈[0, 1] be such that

|f(x) – f(y)| ≤ M |x – y|, for all x, y ∈ [0, 1] and for some M > 0.

For n ∈ ℕ, let f_{n}(x) = f(x^{1+1/n}). If S = {f_{n} : n ∈ ℕ}, then

(A) the closure of S is compact

(B) S is closed and bounded

(C) S is bounded but not totally bounded

(D) S is compact

14. Let K : ℝ × (0, ∞) → ℝ be a function such that the solution of the initial value problem u(x, 0), = f(x), x ∈ ℝ, t > 0, is given by

for all bounded continuous functions f. Then the value of is ______

15. The number of cyclic subgroups of the quaternion group is _________

16. The number of elements of order 3 in the symmetric group S_{6} is _______

17. Let F be the field with 4096 elements. The number of proper subfields of F is ______

18. If (x_{1}^{*}, x_{2}^{*}) is an optimal solution of the linear programming problem, minimize x_{1} + 2x_{2} subject to

4x_{1} – x_{2} ≥8

2x_{1} + x_{2} ≥ 10

−x_{1} + x_{2} ≤ 7

x_{1}, x_{2} ≥ 0

and (λ_{1}^{*}, λ_{2}^{*}, λ_{3}^{*}) in an optimal solution of its dual problem, then is equal to ______ (correct up to one decimal place)

19. Let a, b, c ∈ ℝ be such that the quadrature rule

is exact for al polynomials of degree less than or equal to 2. Then b is equal to ______ (rounded off to two decimal places)

20. Let f(x) = x^{4} and let p(x) be the interpolating polynomial of f at nodes 1, 2 and 3. Then p(0) is equal to _______

21. For n ≥ 2, define the sequence {x_{n}} by

Then the sequence {x_{n}} converges to ______ (correct up to two decimal places)

22. Let

equipped with the norm and let T be the linear functional on L^{2}[0, 10] given by

Then ||T|| is equal to ______

23. If {x_{13} x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}} is the set of basic variable of a balanced transportation problem seeking to minimize cost of transportation from origins to destinations, where the cost matrix is,

and λ, μ ∈ ℝ, then x_{32} is equal to ______

24. Let ℤ_{225} be the ring of integers modulo 225. If x is the number of prime ideals and y is the number of nontrivial units in ℤ_{225}, then x + y is equal to _______

25. Let u(x, t) be the solution of

where f is a twice continuously differential function. If f(−2) = 4, f(0) = 0, and u(2, 2) = 8, then the value of u(1, 3) is ________

**Q26 – Q55 carry two marks each.**

26. Let be an orthonormal basis for a separable Hilbert space H with the inner product Define

Then

(A) the closure of the span {f_{n} : n ∈ ℕ} equals H

(B) f = 0 if for all n ∈ ℕ

(C) is an orthogonal subset of H

(D) there does not exist nonzero f ∈ H such that

27. Suppose V is a finite dimensional non-zero vector space over ℂ and T : V → V is a linear transformation such that Range (T) = Nullspace (T). Then which of the following statements is FALSE ?

(A) The dimension of V is even

(B) 0 is the only eigenvalue of T

(C) Both 0 and 1 are eigenvalues of T

(D) T^{2} = 0

28. Let P ∈ M_{m}_{×}_{n}(ℝ). Consider the following statements:

I : If XPY = 0 for all X ∈ M_{1}_{×}_{m}(ℝ), then P = 0.

II : If m = n, P is symmetric and P^{2} = 0, then P = 0.

Then

(A) both I and II are true

(B) I is true but II is false

(C) I is false but II is true

(D) both I and II are false

29. For n ∈ ℕ, let T_{n} : (l^{1}, ||∙||_{1}) → (l^{∞}, ||∙||_{∞}) and T : (l^{1}, ||∙||_{∞}) be the bounded linear operators defined by

and

T(x_{1}, x_{2}, …) = (x_{1}, x_{2} …).

Then

(A) ||T_{n}|| does not converge to ||T|| as n → ∞

(B) ||T_{n} – T|| converges to zero as n → ∞

(C) for all x ∈ l^{1}, ||T_{n}(x) – T(x)|| converges to zero as n → ∞

(D) for each non-zero x ∈ l^{1}, there exists a continuous linear functional g on l^{∞} such that g(T_{n}(x)) does not converge to g(T(x)) as n → ∞

30. Let P(ℝ) denote the power set of ℝ, equipped with the metric

where χ_{U} and χ_{V} denote the characteristic functions of the subjects U and V, respectively of ℝ. The set {{m} : m ∈ ℤ} in the metric space (P(ℝ), d) is

(A) bounded but not totally bounded

(B) totally bounded but not compact

(C) compact

(D) not bounded

31. Let f : ℝ → ℝ be defined by

where χ_{(n n+1]} is the characteristic function of the interval (n, n + 1]. for α ∈ ℝ, let S_{α} = {x ∈ ℝ : f(x) > α}. Then

(A) S_{1/2} is open

(B) S_{√}_{5/2} is not measurable

(C) S_{0} is closed

(D) S_{1/}_{√}_{3} is measurable

32. For n ∈ ℕ, let f_{n}, g_{n} : (0, 1) → ℝ be functions defined by f_{n}(x) = x^{n} and g_{n}(x) = x^{n} (1 – x).

Then

(A) {f_{n}} converges uniformly but {g_{n}} does not converge uniformly

(B) {g_{n}} converges uniformly but {f_{n}} does not converge uniformly

(C) both {f_{n}} and {g_{n}} converge uniformly

(D) neither {f_{n}} nor{g_{n}} converge uniformly

33. Let u be a solution of the differential equation y′ + xy = 0 and let ϕ = uψ be a solution of the differential equation y′′ + 2xy′ + (x^{2} + 2) y = 0 satisfying ϕ(0) = 1 and ϕ′(0) = 0. Then ϕ(x) is

34. For n ∈ ℕ ⋃ {0}, let y_{n} be a solution of the differential equation xy′′ + (1 – x)y′ + ny = 0 satisfying y_{n}(0) = 1. For which of the following functions w(x), the integral is equal to zero?

35. Suppose that

are metric spaces with metrics induced by the Euclidean metric of ℝ^{2}. Let B_{X }and B_{Y} be the open unit balls around (0 0) in X and Y, respectively. Consider the following statements:

I : The closure of B_{X} in X is compact.

II : The closure of B_{Y} is compact.

Then

(A) both I and II are true

(B) I is true but II is false

(C) I is false but II is true

(D) both I and II are false

36. If f : ℂ \ {0} → ℂ is a function such that and its restriction to the unit circle is continuous, then

(A) f is continuous but not necessarily analytic

(B) f is analytic but not necessarily a constant function

(C) f is a constant function

(D)

37. For a subset S of a topological space, let Int(S) and denote the interior and closure of S, respectively. Then which of the following statements is TRUE?

(A) If S is open, then

(B) If the boundary of S is empty, then S is open

(C) If the boundary of S is empty, then S is open

(D) If is a proper subset of the boundary of S, then S is open

38. Suppose ℑ_{1}, ℑ_{2}, and ℑ_{3} are the smallest topologies on ℝ containing S_{1}, S_{2} and S_{3}, respectively, where

Then

(A) ℑ_{3} ⊋ ℑ_{1}

(B) ℑ_{3} ⊋ ℑ_{2}

(C) ℑ_{1} = ℑ_{2}

(D) ℑ_{1} ⊋ ℑ_{2}

39. Let Consider the following statements:

I: There exists a lower triangular matrix L such that M = LL^{t}, where L^{t} denotes transpose of L.

II : Gauss-Seidel method of Mx = b (b ∈ ℝ^{3}) converges for any initial choice x_{0} ∈ ℝ^{3}.

Then

(A) I is not true when α > 9/2, β = 3

(B) II is not true when α > 9/2, β =−1

(C) II is not true when α = 4, β = 3/2

(D) I is true when α = 5, β = 3

40. Let I and J be the ideals generated by {5, √10} and {4, √10} in the ring ℤ[√10] = {a+ b√10|a, b ∈ ℤ}, respectively. Then

(A) both I and J are maximal ideals

(B) I is a maximal ideal but J is not a prime ideal

(C) I is not a maximal ideal but J is a prime ideal

(D) neither I nor J is a maximal ideal

41. Suppose V is finite dimensional vector space over ℝ. If W_{1}, W_{2} and W_{3} are subspaces of V, then which of the following statements is TRUE?

(A) If W_{1} + W_{2} + W_{3} = V then span (W_{1} ⋃ W_{2}) ⋃ span(W_{2} ⋃ W_{3}) ⋃ span(W_{3} ⋃ W_{1}) = V

(B) If W_{1} ⋂ W_{2} = {0} and W_{1} ⋂ W_{3} = {0}, then W_{1} ⋂ (W_{2} + W_{3}) = {0}

(C) If W_{1} + W_{2} = W_{1} + W_{3}, then W_{2} = W_{3}

(D) If W_{1} ≠ V, then span(V\W_{1}) = V

42. Let α, β ∈ ℝ, α ≠ The system

x_{1} – 2x_{2} + αx_{3} = 8

x_{1} – x_{2} + x_{4} = β

x_{1}, x_{2}, x_{3}, x_{4} ≥ 0

has NO basic feasible solution if

(A) α < 0, β > 8

(B) α > 0, 0 < β < 8

(C) α > 0, β < 0

(D) α < 0, β < 8

43. Let 0 < p < 1 and let

Then

(A) |∙| defines a norm on X

(B) |f + g|_{p} ≤ |f|_{p} + |g|_{p} for all f, g ∈ X

(C) |f +g|_{p}^{p} ≤ |f|_{p}^{p} + |g|_{p}^{p} for all, f, g ∈ X

(D) if f_{n} converges to f pointwise on ℝ, then

44. Suppose the ϕ_{1} and ϕ_{2} are linearly independent solutions of the differential equation 2x^{2}y′′ – (x + x^{2})y′ + (x^{2} – 2)y = 0, and ϕ_{1}(0) = 0. Then the smallest positive integer n such that is _______

45. Suppose that z ∈ ℂ and γ(t) = e^{2it}, t ∈[0, 2π]. If then the value of α is equal to ________

46. If t ∈ [0, 2] and then β is equal to _______ (correct up to one decimal place)

47. Let where ω is a primitive cube root of unity. Then the degree of extension of K over ℚ is _______

48. Let α ∈ ℝ. If (3, 0, 0, β) is an solution of the linear programming problem minimize x_{1} + x_{2} + x_{3} – αx_{4} subject to

2x_{1} – x_{2} + x_{3} = 6

−x_{1} + x_{2} + x_{4} = 3

x_{1}, x_{2}, x_{3}, x_{4} ≥ 0

then the maximum value of β – α is _________

49. Suppose that T : ℝ^{4} → ℝ[x] is a linear transformation over ℝ satisfying T(−1, 1, 1, 1) = x^{2} + 2x^{4}, T(1, 2, 3, 4) = 1 – x^{2}, T(2, −1, −1, 0) = x^{3} – x^{4}. Then the coefficient of x^{4} in T(−3, 5, 6, 6) is ________

50. Let and let S be the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.If is the unit outward normal to the tetrahedron, then the value of is _______ (rounded off to two decimal places)

51. Let and let S be the surface x^{2} + y^{2} + z = 1, z ≥ If is a unit normal to S and Then α is equal to _____

52. Let G be a non-cyclic group of order 57. Then the number of elements of order 3 in G is _______

53. The coefficient of (x – 1)^{5} in the Taylor expansion about x = 1 of the function is _______ (correct up to two decimal places)

54. Let u(x, y) be the solution of the initial value problem u(x, 0) = 1 + x^{2}. Then the value of u(0, 1) is ______ (rounded off to three decimal places

55. The value of is _______ (rounded off to three decimal places)

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