GATE 2020 Mathematics Previous Year Paper

GATE 2020 Mathematics Previous Year Paper

GA – General Aptitude

Q1 – Q5 carry one mark each.

Q.No. 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,

(B) at on,

(D) to, at

Q.No. 2 Despite a string of poor performances, the chances of K. L. Rahul’s selection in the team are

(A) slim

(B) bright

(D) uncertain

Q.No. 3 Select the word that fits the analogy:

Cover : Uncover :: Associate :

(A) Unassociate

(B) Inassociate

(D) Dissociate

Q.No. 4 Hit by floods, the kharif (summer sown) crops in various parts of the country 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.

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

Q.No. 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

(B) n² + n

(D) 2n² +n

Q6 – Q10 carry two marks each.

Q.No. 6 Repo rate is the rate 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 the cost of borrowing and increase lending byC commercial banks.

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

Q.No. 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 neighbour of S.

Which of the following must be true?

(A) P is a neighbour of R.

(B) Q is a neighbour of R.

(D) R is the left neighbour of S.

Q.No. 8 The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car 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 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

(D) 116.5

Q. No. 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

(D) 16

Q.No. 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 Company P Company Q

(A) 15:17

(B) 16:17

(D) 17:16

MA: Mathematics

Q1 – Q25 carry one mark each.

Q.No. 1 Suppose that I, and I2 are topologies on X induced by metrics dį and dz, respectively, such that I, S 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, d) to (X, dz) is continuous

Q.No.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)∫∫₀, |f(x,y)½ dx dy is finite

(D)∫∫_D, |f(x,y)| dx dy is finite

Q.No. 3 The initial value problem

has

(A) a unique solution if b = 0

(B) no solution if b = 1

(D) a unique solution if b = 1

Q.No.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) I and II are true

(B) I is true but II is false

(D) both I and II are false

Q.No. 5 Let G and H be defined by

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

I. ∫_Y f dz is independent of paths y in G joining – i and i

II. ∫_Y g dz is independent of paths y 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 is true

(D)both I and II are false

Q.No. 6 Let f (z) = e^1/z, Z ∈ C\{0} and let, for n ∈ N,

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

Q.No. 7 Suppose that

Then, with respect to the Euclidean metric on R2,

(A) both U and V are disconnected

(B) U is disconnected but V is connected

(D) both U and V are connected

Q.No. 8 If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where

(Pl): 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

Q.No. 9 If (4,0) and (0, -½) are critical points of the function

where a, β E R, then

(A) (4, – ½)point of local maxima of f

(B) (4, – ½) is a saddle point of f

(D) (4, – ½) point of local minima of f

Q.No. 10 Consider the iterative scheme

with initial point x > 0. Then the sequence {X_n}

(A)converges only if X, > 1

(B)converges only if Xo <3

(C)converges for any Xo

(D)does not converge for any Xo

Q.No. 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

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

Q.No. 12 Suppose that M is a 5 x 5 matrix with real entries and p(x) = det(xl – M). Then

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

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

(D) M is not invertible if M2 – 2M = 0

Q.No. 13 Let C[0, 1] denote the space of all real-valued continuous functions on [0, 1] equipped with the supremum norm ||-||_∞ … Let f ∈ C[0, 1] be such that

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

(D) S is compact

Q.No.14 Let K: R × (0, ∞) → R be a function such that the solution of the initial value problem is, given by

for all bounded continuous functions f. Then the value of ∫_R K(x, t) is _________.

Q.No. 15 The number of cyclic subgroups of the quaternion group is____________

Q.No. 16 The number of elements of order 3 in the symmetric group So is ________

Q.No. 17 Let F be the field with 4096 elements. The number of proper subfields of F is________

Q.No. 18 If (x1, x2*) is an optimal solution of the linear programming problem, and (λ1,λ2,λ3) is an optimal solution of its dual problem, then 22-1x* + =12; is equal to____________(correct up to one decimal place)

Q.No. 19 Let a,b,c E R be such that the quadrature ruleis exact for all polynomials of degree less than or equal to 2. Then b is equal to _________(rounded off to two decimal places)

Q.No.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 ____________

Q.No. 21 For n ≥ 2, define the sequence {xn} by Then the sequence {X_n} converges to___________(correct up to two decimal places).

Q.No. 22 Let L₂ [0,10] = {f:[0, 10] — R : f is Lebesgue measurable and 010f2dx>} equipped with the norm ||f|| = 010f2dx12 and let T be the linear functional on L² [0, 10) given by

Then ||T || is equal to _______________

Q.No. 23 If {X13, X22, X23 = 10, X31, X32, X34} is the set of basic variables 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 ______________

Q.No. 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 ____________

Q.No. 25 Let u(x,t) be the solution of

where f is a twice continuously differentiable 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.

Q.No. 26 Let be an orthonormal basis for a separable Hilbert space H with the

inner product (:, ). Define

Then

Q.No. 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

(D) T² = 0

Q.No. 28 Let PE M_m_×_n(R). Consider the following statements:

I: If XPY = 0 for all X € Mixm(R) and Ye Mnx1(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

(D) both I and II are false

Q.No. 29 For n EN, let Tn: (11, II.1) → (1o, 11•||..) and T: (11, 11:||) (1o, |I||..) be the bounded linear operators defined by

Then

(A) || Tn || does not converge to ||T|| as n →

(B) || Tn – T|| converges to zero as n 700

(C)for all x € 11, || Tn(x) – T(x) || converges to zero as n →

(D) for each non-zero x € 11, there exists a continuous linear functional g on lo such that g (Tn (x)) does not converge to g(T(x)) as n

Q.No. 30 Let P(R) denote the power set of R, equipped with the metric

where Xu and Xy denote the characteristic functions of the subsets U and V, respectively, of R. The set { {m}: m e Z} in the metric space (P(R), d) is bounded but not totally bounded totally bounded but not compact compact not bounded

Q.No. 31 Let f:R → R be defined by

where X(n,n+1) is the characteristic function of the interval (n,n + 1]. For a ER, let Sa = {x E R : f(x) > a}. Then

Q.No. 32 For n E N, let frIn: (0,1) → R be functions defined by

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

(D) neither {f_n} nor {g_n} converge uniformly

Q.No. 33 Let u be a solution of the differential equation y’ + xy = 0 and let = uy be a solution of the differential equation y” + 2xy’ + (x2 + 2)y = 0 satisfying (0) = 1 and ‘(0) = 0. Then (x) is

Q.No. 34 For n E NU{0}, let yn be a solution of the differential equation

satisfying yn (0) = 1. For which of the following functions w(x), the integral

Is equal to zero ?

Q.No. 35 Suppose that

are metric spaces with metrics induced by the Euclidean metric of R². 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 Bx in X is compact.

II : The closure of By in Y is compact.

Then

(A) both I and II are true

(B) I is true but II is false

(D) both I and II are false

Q.No. 36 If f:C\{0} → C is a function such that f(2) = f (z/|z|)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)Z0f(z) exists

Q.No. 37 For a subset S of a topological space, let Int(s) and S denote the interior and closure of S, respectively. Then which of the following statements is TRUE?

(A) If S is open, then S = Int(5)

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

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

Q.No.38 Suppose I1, I2 and I3 are the smallest topologies on R containing S1, S2 and S3, respectively, where

Then

Q.No. 39 Let Consider the following statements:

I: There exists a lower triangular matrix L such that M = LL”, where Lt denotes transpose of L.

II: Gauss-Seidel method for Mx = b (be R3) converges for any initial choice X, E R3

Then

(A) I is not true when a >}, B = 3

(B) II is not true when a >

(D) I is true when a = 5, B = 3

Q.No. 40 Let I and J be the ideals generated by {5, √10} and {4, 10} in the ring Z[110] = {a + b/10 | a, b € Z}, respectively. Then

(A) both / and J are maximal ideals

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

(D) (neither / nor ) is a maximal ideal

Q.No. 41 Suppose V is a finite dimensional vector space over R. If W,W2 and W3 are subspaces of V, then which of the following statements is TRUE?

(A) If W. + W2+W3 = V then span(W. U W2) U span(W2 U W3) U span(W3 U W) = V

(B) If W W2 = {0} and W, W3 = {0}, then W, n (W2 + W3) = {0}

(D) If W#V, then span(V \W) = V

Q.No.42 Let a, ER, a 0. The system

has NO basic feasible solution if

Q.No. 43 Let 0 <p < 1 and let

For f eX, define

Then

(A)|·|_p defines a norm on X

(B) If + glp < If lp + Iglp for all f.g EX

(C)If+gl% = |f1 + gl% for all f, g ex

(D) if fn converges to f pointwise on R, then limfnly = Iflg.

Q.No. 44 Suppose that 01 and $2 are linearly independent solutions of the differential equation

and Ø₁(0) = 0. Then the smallest positive integer n such that

is______________.

Q.No. 45 Suppose that . If

then the value of a is equal to _______________.

Q.No. 46 If y(t) = ½e³^it, t e [0, 2] and

then ß is equal to______________(correct up to one decimal place)

Q.No. 47 Let where is a primitive cube root of unity. Then the degree of extension of K over Q is_________________.

Q.No. 48 Let a E R. If (3,0,0,B) is an optimal solution of the linear programming problem

then the maximum value of ß-a is __________.

Q.No. 49 Suppose that T: R4 → R[x] is a linear transformation over R satisfying

Then the coefficient of x4 in T(-3,5,6,6) is ______________.

Q.No. 50 Let F(x, y, z) = (2x – 2y cos x) î + (2y – y2 sin x) s + 4z k 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 ñ is the unit outward normal to the tetrahedron, then the value of

is__________________(rounded off to two decimal places)

Q.No. 51 Let F = (x + 2y)ez î + (yez + x2) ĵ + y2z ħ and let S be the surface x2 + y2 + z = 1, 2 > 0. If ñ is a unit normal to S and

Then a is equal to ______________.

Q.No. 52 Let G be a non-cyclic group of order 57. Then the number of elements of order 3 in G is ________________.

Q.No.53 The coefficient of (x – 1)5 in the Taylor expansion about x = 1 of the function

is _________________ (correct up to two decimal places)

Q.No. 54 Let u(x,y) be the solution of the initial value problem

Then the value of u(0,1) is ______________ (rounded off to three decimal places)

Q.No. 55 The value of

is ________________ (rounded off to three decimal places)

Answer Key

Q.No.	1	2	3	4	5	6	7	8	9	10
Ans.	C	B	D	D	B	D	C	B	B	B
Q.No.	1	2	3	4	5	6	7	8	9	10
Ans.	A	C	D	B	B	A OR D	C	A	B	C
Q.No.	11	12	13	14	15	16	17	18	19	20
Ans.	D	C	A	1 TO 1	5 TO 5	80 TO 80	5 TO 5	5.5 TO 5.5	1.70 TO 1.80	36 TO 36
Q.No.	21	22	23	24	25	26	27	28	29	30
Ans.	0.25 TO 0.25	3 TO 3	5 TO 5	121 TO 121	10 TO 10	A	C	A	C	A
Q.No.	31	32	33	34	35	36	37	38	39	40
Ans.	D	B	B	B	C	A	B	C	D	B
Q.No.	41	42	43	44	45	46	47	48	49	50
Ans.	D	D	C	3 TO 3	56 TO 56	0.5 TO 0.5	4 TO 4	7 TO 7	5 TO 5	1.30 TO 1.40
Q.No.	51	52	53	54	55
Ans.	2 TO 2	38 TO 38	0.04 TO 0.04	1.61 TO 1.625	2.710 TO 2.725