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) n2 – n
(B) n2 + n
(C) 2n2 – n
(D) 2n2 + 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 = [mij], i. j= 1, 2, 3, 4, the diagonal elements are all zero and mij = −mij. 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 ℑ1 and ℑ2 are topologies on X induced by metrics d1 and d2, respectively, such that ℑ1 ⊆ ℑ2. Then which of the following statements is TRUE?
(A) If a sequence converges in (X, d2) then it converges in (X, d1)
(B) If a sequence converges in (X, d1) then it converges in (X, d2)
(C) Every open ball in (X, d1) is an open ball in (X, d2)
(D) The map x → x from (X, d1) to (X, d2) 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′ = y3/5, 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) = e1/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 ℝ2,
(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 x1 – 2x2 subject to –x1 +x2 = 10, x1, x2 ≥ 0,
(P2) : minimize x1 – 2x2 subject to –x1 + x2 = 10, x1 – x2 = 10, x1, x2 ≥ 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) x2 + βy2 + (α + 1) y3 + x3,
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 x0 > 0. Then the sequence {xn}
(A) converges only if x0 > 1
(B) converges only if x0 < 3
(C) converges for any x0
(D) does not converge for any x0
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 M2 – 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 fn(x) = f(x1+1/n). If S = {fn : 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 S6 is _______
17. Let F be the field with 4096 elements. The number of proper subfields of F is ______
18. If (x1*, x2*) is an optimal solution of the linear programming problem, minimize x1 + 2x2 subject to
4x1 – x2 ≥8
2x1 + x2 ≥ 10
−x1 + x2 ≤ 7
x1, x2 ≥ 0
and (λ1*, λ2*, λ3*) 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) = x4 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 {xn} by
Then the sequence {xn} converges to ______ (correct up to two decimal places)
22. Let
equipped with the norm and let T be the linear functional on L2[0, 10] given by
Then ||T|| is equal to ______
23. If {x13 x22, x23 = 10, x31, x32, x34} 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 x32 is equal to ______
24. Let ℤ225 be the ring of integers modulo 225. If x is the number of prime ideals and y is the number of nontrivial units in ℤ225, 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 {fn : 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) T2 = 0
28. Let P ∈ Mm×n(ℝ). Consider the following statements:
I : If XPY = 0 for all X ∈ M1×m(ℝ), then P = 0.
II : If m = n, P is symmetric and P2 = 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 Tn : (l1, ||∙||1) → (l∞, ||∙||∞) and T : (l1, ||∙||∞) be the bounded linear operators defined by
and
T(x1, x2, …) = (x1, x2 …).
Then
(A) ||Tn|| does not converge to ||T|| as n → ∞
(B) ||Tn – T|| converges to zero as n → ∞
(C) for all x ∈ l1, ||Tn(x) – T(x)|| converges to zero as n → ∞
(D) for each non-zero x ∈ l1, there exists a continuous linear functional g on l∞ such that g(Tn(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 χU and χV 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 Sα = {x ∈ ℝ : f(x) > α}. Then
(A) S1/2 is open
(B) S√5/2 is not measurable
(C) S0 is closed
(D) S1/√3 is measurable
32. For n ∈ ℕ, let fn, gn : (0, 1) → ℝ be functions defined by fn(x) = xn and gn(x) = xn (1 – x).
Then
(A) {fn} converges uniformly but {gn} does not converge uniformly
(B) {gn} converges uniformly but {fn} does not converge uniformly
(C) both {fn} and {gn} converge uniformly
(D) neither {fn} nor{gn} 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′ + (x2 + 2) y = 0 satisfying ϕ(0) = 1 and ϕ′(0) = 0. Then ϕ(x) is
34. For n ∈ ℕ ⋃ {0}, let yn be a solution of the differential equation xy′′ + (1 – x)y′ + ny = 0 satisfying yn(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 ℝ2. Let BX and BY be the open unit balls around (0 0) in X and Y, respectively. Consider the following statements:
I : The closure of BX in X is compact.
II : The closure of BY 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 ℑ1, ℑ2, and ℑ3 are the smallest topologies on ℝ containing S1, S2 and S3, respectively, where
Then
(A) ℑ3 ⊋ ℑ1
(B) ℑ3 ⊋ ℑ2
(C) ℑ1 = ℑ2
(D) ℑ1 ⊋ ℑ2
39. Let Consider the following statements:
I: There exists a lower triangular matrix L such that M = LLt, where Lt denotes transpose of L.
II : Gauss-Seidel method of Mx = b (b ∈ ℝ3) converges for any initial choice x0 ∈ ℝ3.
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 W1, W2 and W3 are subspaces of V, then which of the following statements is TRUE?
(A) If W1 + W2 + W3 = V then span (W1 ⋃ W2) ⋃ span(W2 ⋃ W3) ⋃ span(W3 ⋃ W1) = V
(B) If W1 ⋂ W2 = {0} and W1 ⋂ W3 = {0}, then W1 ⋂ (W2 + W3) = {0}
(C) If W1 + W2 = W1 + W3, then W2 = W3
(D) If W1 ≠ V, then span(V\W1) = V
42. Let α, β ∈ ℝ, α ≠ The system
x1 – 2x2 + αx3 = 8
x1 – x2 + x4 = β
x1, x2, x3, x4 ≥ 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|pp ≤ |f|pp + |g|pp for all, f, g ∈ X
(D) if fn converges to f pointwise on ℝ, then
44. Suppose the ϕ1 and ϕ2 are linearly independent solutions of the differential equation 2x2y′′ – (x + x2)y′ + (x2 – 2)y = 0, and ϕ1(0) = 0. Then the smallest positive integer n such that is _______
45. Suppose that z ∈ ℂ and γ(t) = e2it, 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 x1 + x2 + x3 – αx4 subject to
2x1 – x2 + x3 = 6
−x1 + x2 + x4 = 3
x1, x2, x3, x4 ≥ 0
then the maximum value of β – α is _________
49. Suppose that T : ℝ4 → ℝ[x] is a linear transformation over ℝ satisfying T(−1, 1, 1, 1) = x2 + 2x4, T(1, 2, 3, 4) = 1 – x2, T(2, −1, −1, 0) = x3 – x4. Then the coefficient of x4 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 x2 + y2 + 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 + x2. 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)