GATE Exam 2019 Mathematics (MA) Question Paper With Answer Key

GATE 2019 General Aptitude (GA) Set-8

Q. 1 – Q. 5 carry one mark each.

1. The fishermen, ________ the flood victims owed their lives, were rewarded by the government.

(A) whom

(B) to which

(C) to whom

(D) that

Answer: (C)

2.  Some students were not involved in the strike.

If the above statement is true, which of the following conclusions is/are logically necessary?

1. Some who were involved in the strike were students.

2. No student was involved in the strike.

3. At least one student was involved in the strike.

4. Some who were not involved in the strike were students.

(A) 1 and 2

(B) 3

(C) 4

(D) 2 and 3

Answer: (C)

3. The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is .

(A) 17.1

(B) 21.0

(C) 33.1

(D) 72.8

Answer: (C)

4. Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following the directions given below:

1. No two odd or even numbers are next to each other.

2. The second number from the left is exactly half of the left-most number.

3. The middle number is exactly twice the right-most number.

Which is the second number from the right?

(A) 2

(B) 4

(C) 7

(D) 10

Answer: (C)

5. Until Iran came along, India had never been in kabaddi.

(A) defeated

(B) defeating

(C) defeat

(D) defeatist

Answer: (A)

Q. 6 – Q. 10 carry two marks each.

6.  Since the last one year, after a 125 basis point reduction in repo rate by the Reserve Bank of India, banking institutions have been making a demand to reduce interest rates on small saving schemes. Finally, the government announced yesterday a reduction in interest rates on small saving schemes to bring them on par with fixed deposit interest rates.

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

(A) Whenever the Reserve Bank of India reduces the repo rate, the interest rates on small saving schemes are also reduced

(B) Interest rates on small saving schemes are always maintained on par with fixed deposit interest rates

(C) The government sometimes takes into consideration the demands of banking institutions before reducing the interest rates on small saving schemes

(D) A reduction in interest rates on small saving schemes follow only after a reduction in repo rate by the Reserve Bank of India

Answer: (C)

7. In a country of 1400 million population, 70% own mobile phones. Among the mobile phone owners, only 294 million access the Internet. Among these Internet users, only half buy goods from e-commerce portals. What is the percentage of these buyers in the country?

(A) 10.50

(B) 14.70

(C) 15.00

(D) 50.00

Answer: (A)

8. The nomenclature of Hindustani music has changed over the centuries. Since the medieval period dhrupad styles were identified as baanis. Terms like gayaki and baaj were used to refer to vocal and instrumental styles, respectively. With the institutionalization of music education the term gharana became acceptable. Gharana originally referred to hereditary musicians from a particular lineage, including disciples and grand disciples.

Which one of the following pairings is NOT correct?

(A) dhrupad, baani

(B) gayaki, vocal

(C) baaj, institution

(D) gharana, lineage

Answer: (C)

9. Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is AM.

(A) 9

(B) 10

(C) 11

(D) 11.30

Answer: (B)

10.  “I read somewhere that in ancient times the prestige of a kingdom depended upon the number of taxes that it was able to levy on its people. It was very much like the prestige of a head-hunter in his own community.”

Based on the paragraph above, the prestige of a head-hunter depended upon

(A) the prestige of the kingdom

(B) the prestige of the heads

(C) the number of taxes he could levy

(D) the number of heads he could gather

Answer: (D)

MATHEMATICS

MATHEMATICS

Q.1 – Q.25 carry one mark each.

1. For a balanced transportation problem with three sources and three destinations where costs, availabilities and demands are all finite and positive, which one of the following statements is FALSE?

(A) The transportation problem does not have unbounded solution

(B) The number of non-basic variables of the transportation problem is 4

(C) The dual variables of the transportation problem are unrestricted in sign

(D) The transportation problem has at most 5 basic feasible solutions

Answer: (D)

2. Let f : [a, b]→ ℝ (the set of all real numbers) be any function which is twice differentiable in (a, b) with only one root a in (a, b). Let f’(x) and f’’(x) denote the first and second order derivatives of f(x) with respect to x If a is a simple root and is computed by the Newton-Raphson method, then method converges if

Answer: (A)

3. Let f : ℂ → ℂ (the set of all complex numbers) be defined by

f(x + iy) =x3 + 3xy2 + i(y3 + 3x23y), i = √−i

Let f’(z) denote the derivative of f with respect to z.

Then which one of the following statements is TRUE?

(A) f’(1+ i) exists and |f’(1+i)| = 3√5

(B) f is analytic at the origin

(C) f is not differentiable at i

(D)f is differentiable at 1

Answer: (D)

4. The partial differential equation

is

(A)parabolic in the region x2 + y2 > 2

(B) hyperbolic in the region x2 + y2 > 2

(C) elliptic in the region 0 < x2 + y2 < 2

(D) hyperbolic in the region x < x2 + y2 < 2

Answer: (D)

5. If

then which one of the following statements is TRUE?

Answer: (C)

6. Let Γ = {(x, y, z) ∈ ℝ : −1 < x < 1, −1 < y < 1, −1 < z < 1} and ϕ : Γ →ℝ be a function whose all second order partial derivatives exist and are continuous. If ϕ satisfies the Laplace equation ∇2ϕ= 0 for all (x, y, z) ∈Γ, then which one of the following statements is TRUE in Γ?

(ℝ is the set of all real numbers, and ℝ3 = {(x, y, z) : x, y, z ∈ ℝ})

Answer: (C)

7. Let X = {(x1, x2 ….) : x1 ∈ ℝ and only finitely many xi’s are non-zero} and d : X ×X → ℝ be a metric on X defined by

(ℝ is the set of all real number and ℕ is the set of all natural numbers)

Consider the following statements:

P : (X, d) is a complete metric space.

Q : The set {x ∈ d(0, x) ≤ 1} is compact, where 0 is the zero element of X.

Which of the above statements is/are TRUE?

(A) Both P and Q

(B) P only

(C) Q only

(D) Neither P nor Q

Answer: (D)

8. Consider the following statements:

(I) The set ℚ × ℤ is uncountable.

(II) The set {f : f is the function from ℕ to {0, 1}} is uncountable.

(III) The set {√p : p is a prime number} is uncountable.

(IV) For any infinite set, there exists a bijection from the set of one of its proper subsets.

(ℚ is the set of all rational numbers, ℤ is the set of all integers and ℕ is the set of all natural numbers)

Which of the above statements are TRUE?

(A) I and IV only

(B) II and IV only

(C) II and III only

(D) I, II and IV only

Answer: (B)

9. Let f : ℝ2 → ℝ be defined by

f(x, y ) = x6 – 2x2 – x4y + 2y2.

(ℝ is the set of all real number and ℝ2 = {(x, y) : x, y ∈ ℝ})

Which one of the following statements is TRUE?

(A) f has a local maximum at origin

(B) f has a local minimum at origin

(C) f has saddle point at origin

(D) The origin is not a critical point of f

Answer: (C)

10. Let be any sequence of real numbers such that  If the radius of convergence of  then which one of the following statements is necessarily TRUE?

Answer: (A)

11. Let T1 be the co-countable topology on ℝ(the set of real numbers) and T2 be the co-finite topology on ℝ.

Consider the following statements:

(I) In (ℝ, T1), the sequence  converges to 0.

(II) In (ℝ, T2) the sequence  converges to 0.

(III) In (ℝ, T1), there is no sequence of rational numbers which converges to √3.

(IV) In (ℝ, T2), there is no sequence of rational numbers which converges to √3.

Which of the above statements are TRUE?

(A) I and II only

(B) II and III only

(C) III and IV only

(D I and IV only

Answer: (B)

12. Let X and Y be normed linear spaces, and let T : X → Y be any bijective linear map with closed graph. Then which one of the following statements is TRUE?

(A) The graph of T is equal to X × Y

(B) T−1 is continuous

(C) The graph of T−1 is closed

(D)  T is continuous

Answer: (C)

13. Let g : ℝ2 → ℝ2 be a function defined by g(x, y) = (ex cosy, ex siny) and (a, b) = g(1, π/3).

(ℝ is the set of all real numbers and ℝ2= ({(x, y): x, y ∈ ℝ})

Which one of the following statements is TRUE?

(A) g is injective

(B) If h is the continuous inverse of g, defined in some neighbourhood of (a, b) ∈ ℝ2, such that h(a, b)= (1, π/3), then the Jacobian of h at (a, b) is e2

(C) If h is the continuous inverse of g, defined in some neighbourhood of (a, b) ∈ ℝ2, such that h(a, b) = (1, π/3), then the Jacobian of h at (a, b) is e−2

(D) g is sujective

Answer: (C)

14. Let

(the set of all natural numbers).

Answer: (0 to 0)

15. If the differential equation

         

is solved using the Euler’s method with step-size h = 0.1, then y(1.2) is equal to______(round off to 2 places of decimal)

Answer: (2.40 to 2.50)

16. Let f be any polynomial function of degrees at most 2 over ℝ (the set of all real numbers). If the constants a and b are such that

             

then 4a + 3b is equal to _______(round off to 2 places of decimal).

Answer: (-7.5 to -7.5)

17. Let L denote the value of the line integral ∮C (3x- 4x2y)dx + (4xy2 + 2y)dy, where C, a circle of radius 2 with centre at origin of the xy-place, is traversed once in the anti-clockwise direction. Then L/π is equal to ______.

 

Answer: (31.90 to 32.10)

18. The temperature T : ℝ3\{(0, 0, 0)} → ℝ at any point P(x, y, z) is inversely proportional to the square of the distance of P from the origin. If the value of the temperature T at the point R(0, 0, 1) is √3, then the rate of change of T at the point Q(1, 1, 2) in the direction of  is equal to ______(round off to 2 places of decimal).

(ℝ is the set of all real numbers, ℝ3 = {(x, y, z) : x, y, z ∈ ℝ} and ℝ3\{(0, 0, 0)} denotes ℝ3 excluding the origin)

Answer: (0.21 to 0.23)

19. Let f be a continuous function defined on [0, 2] such that f(x) ≥ 0 for all x ∈ [0, 2]. If the area bounded by y = f(x), x = 0, y = 0 and x = b is  , where b ∈ (0, 2], then f(1) is equal to _____(round off to 1 place of decimal).

Answer: (0.5 to 0.5)

20. If the characteristic polynomial and minimal polynomial of a square matrix A are (λ – 1) (λ + 1)4 (λ −2)5 and (λ – 1) (λ + 1) (λ – 2), respectively, then the rank of the matrix A + I is ____, where I is the identity matrix of appropriate order.

 

Answer: (6 to 6)

21. Let ω be a primitive complex cube root of unity and i = √−1. Then the degree of the field extension ℚ(i, √3, ω) over ℚ (the field of rational numbers) is ______.

 

Answer: (4 to 4)

22. Let

          

Then the greatest integer has less than or equal to |α| is ______.

 

Answer: (2 to 2)

23. Consider the system:

3x1 + x2 + 2x3 – x4=0

x1 + x2 + x3 – 2x4 = 3,

x1, x2, x3, x4 ≥ 0.

If x1 = 1, x2 = b, x3 = 0, x4 = c is a basic feasible solution of the above system (where a, b and c are real constants), then a + b + c is equal to ______.

Answer: (7 to 7)

24. Let f : ℂ → ℂ be a function defined by f(z) = z6 – 5z4 + 10. Then the number of zeros of f in {z ∈ ℂ : |z| < 2} is _____.

(ℂ is the set of all complex numbers)

 

Answer: (4 to 4)

25. Let

           

be a normed linear space with the norm

         

Let g : ℓ2 → ℂ be the bounded linear functional defined by

             

Then (sup{|g(x)| : ||x||2 ≤ 1})2 is equal to ______ (round off to 3 places of decimal).

(ℂ is the set of all complex numbers).

Answer: (0.125 to 0.125)

Q.26 – Q.55 Carry two marks each.

26. For the linear programming problem (LPP):

Maximize Z = 2x1 + 4x2

subject to – x1 + 2x2 ≤ 4,

                       3x1 + βx2 ≤ 6,

                       x1, x2 ≥ 0, β ∈ ℝ,

(ℝ is the set of all real numbers)

consider the following statements:

(I) The LPP always has a finite optimal value for any β ≥ 0.

(II) The dual of the LPP may be infeasible for some β ≥ 0.

(III) If for some β, the point (1, 2) is feasible to the dual of the LPP, then Z ≤ 16, for any feasible solution (x1, x2) of the LPP.

(IV) If for some β, x1 and x2 are the basic variables in the optimal table of the LPP with x1 = ½, then the optimal value of dual of the LPP is 10.

Then which of the above statements are TRUE?

(A) I and III only

(B) I, III and IV only

(C) III and IV only

(D) II and IV only

Answer: (B)

27. Let f : ℝ2 → ℝ2 be defined by

           

Consider the following statements:

(I) The partial derivatives ∂f/∂x, ∂f/∂y exist at (0, 0) but are unbounded in any neighbourhood of (0, 0).

(II) f is continuous but not differentiable at (0, 0).

(III) f is not continuous at (0, 0).

(IV) f is differentiable at (0, 0).

(ℝ is the set of all real numbers and ℝ2 = {(x, y): x, y ∈ ℝ})

Which of the above statements is/are TRUE?

(A)  I and II only

(B) I and IV only

(C) IV only

(D) III only

Answer: (B)

28. Let  be an infinite matrix cover ℂ (the set of all complex numbers) such that (i) for each i ∈ ℕ (the set of all natural numbers), the ith row( ki, 1, ki, 2…..) of K is in ℓ and (ii) for every  is summable for all i ∈ ℕ, and (y1, y2 …) ∈ ℓ1, where 

Let the set of all rows of K be denoted by E. Consider the following statements:

P: E is bounded set in ℓ

Q: E is dense set in ℓ.

Which of the above statements is/are TRUE?

(A) Both P and Q

(B) P only

(C) Q only

(D) Neither P nor Q

Answer: (B)

29. Consider the following heat conduction problem for a finite rod

with the boundary conditions u(0, t) = −t2, u(π, t) = −πet – t2, t > 0 and the initial condition u(x, 0) = sin x –  sin3 x – x, 0 ≤ x ≤ π. If v(x, t) = u(x, t) + xet+ t2, then which one of the following is CORRECT?

Answer: (A)

30. Let f : ℂ → ℂ be non-zero and analytic at all points in ℤ.

If F(z) = π f(z) cot(πz) for z ∈ℂ\ℤ, then the residue of F at n ∈ ℤ is ______.

(ℂ is the set of all complex numbers, ℤ is the set of all integers and ℂ\ℤ denotes the set of all complex numbers excluding integers)

(A) π f(n)

(B) f(n)

(C) f(n)/π

(D) (df/dz)z = N

Answer: (B)

31. Let the general integral of the partial differential equation

be given by F(u v) = 0, where F: ℝ2 → ℝ is the continuously differentiable function.

(ℝ is the set of all real numbers and ℝ2 = {(x, y): x, y ∈ ℝ})

Then which one of the following is TRUE?

(A) u = x2 + y2 + z, v = xz + y

(B) u = x2 + y2 – z, v = xz – y

(C) u = x2 – y2 + z, v = yz + x

(D) u = x2 + y2 – z, v = yz – x

Answer: (A)

32. Consider the following statements:

(I) If ℚ denotes the additive group of rational numbers and f: ℚ → ℚ is a non-trivial homomorphism, then f is an isomorphism.

(II) Any quotient group of a cyclic group is cyclic.

(III) If every subgroup of a group G is a normal subgroup, then G is abelian.

(IV) Every group of order 33 is cyclic.

Which of the above statements are TRUE?

(A) II and IV only

(B) II and III only

(C) I, II and IV only

(D) I, III and IV only               

Answer: (C)

33. A solution of the Dirichlet problem

2u(r, θ) = 0, 0<r<1, −π≤θ≤π

u(1, θ) =|θ|, −π≤θ≤π,

is given by

Answer: (C)

34. Consider the subspace Y = {(x, x) : x ∈ ℂ} of the normed linear space (ℂ2, || ||).

If ϕ is a bounded linear functional on Y, defined by ϕ(x, x) = x, then which one of the following sets is equal to

{Ψ(1, 0): Ψ is a norm preserving extension of ϕ to(ℂ2, || ||)}.

(ℂ is the set of all complex numbers, ℂ2 = {(x, y) : x, y ∈ ℂ} and ||(x1, x2)|| = sup{|x1|, |x2|})

(A) {1}

(B) [1/2, 3/2]

(C) [1, ∞)

(D)[0, 1]

Answer: (D)

35. Consider the following statements:

(I) The ring ℤ[√−1] is a unique factorization domain.

(II) The ring ℤ[√−5] is a principal ideal domain.

(III) In the polynomial ring ℤ2[x], the ideal generated by x3 + x + 1 is a maximal ideal.

(IV) In the polynomial ring ℤ3[x], the ideal generated by x6 + 1 is a prime ideal.

(ℤ denotes the set of all integers, ℤn denotes the set of all integers modulo, n, for any positive integer n)

Which of the above statements are TRUE?

(A) I, II and III only

(B) I and III only

(C) I, II and IV only

(D) II and III only

Answer: (B)

36. Let M be a 3 × 3 real symmetric matrix with eigenvalues 0, 2 and a with the respective eigenvectors u = (4, b, c)T, v =(−1, 2, 0)T and w = (1, 1, 1)T.

Consider the following statements.

(I) a + b – c = 10.

(II) The vector x = (0, 3/2, ½)T satisfies Mx = v + w.

(III) For any d ∈ span{u, v, w}, Mx= d has a solution.

(IV) The trace of the matrix M2 + 2M is 8.

(yT denotes the transpose of the vector y)

Which of the above statements are TRUE?

(A) I, II and III only

(B) I and II only

(C) II and IV only

(D) III and IV only

Answer: (B)

37. Consider the region

 

in the complex plane. The transformation x + iy → ex+iy maps the region Ω onto the region S ⊂ ℂ (the set of all complex numbers). Then the area of the region S is equal to

(A) π/3(e4 − e−2)

(B) π/4(e4 + e−2)

(C) 2π/3(e4 − e−2)

(D) π/6(e4 − e−2)

Answer: (A)

38. Consider the sequence  of functions, where  is the derivative of gn(x) with respect to x.

(ℝ is the set of all real numbers, ℕ is the set of all natural numbers.)

Answer: (D)

39. Consider the boundary value problem (BVP)

with the boundary conditions y(0) = 0, y(π) = k (k is a non-zero real number).

Then which one of the following statements is TRUE?

(A) For α = 1, the BVP has infinitely many solutions

(B) For α =1, the BVP has a unique solution

(C) For α = −1, k < 0, the BVP has a solution y(x) such that y(x) > 0 for all x ∈ (0, π)

(D)For α = −1, k = 0, the BVP has a solution y(x) such that y(x) > 0 for all x ∈(0, π

Answer: (D)

40. Consider the ordered square I02, the set [0, 1] × [0, 1] with the dictionary order topology. Let the general element of I02 be denoted by x × y, where x, y ∈[0, 1]. Then the closure of the subset  is

(A) S ⋃ ((a, b] ×{0}) ⋃([a, b) × {1})

(B) S ⋃ ([a, b] ×{0}) ⋃((a, ]) × {1})

(C) S ⋃ ((a, b) ×{0}) ⋃((a, b) × {1})

(D) S ⋃ ((a, b] ×{0})

Answer: (A)

41. Let P2 be the vector space of all polynomials of degree at most 2 over ℝ (the set of real numbers). Let a linear transformation T : P2 → P2 be defined by

T(a + bx + cx2) = (a + b) + (b – c)x + (a + c)x2.

Consider the following statements:

(I) The null space of T is {α(−1 + x +x2) : α ∈ ℝ}.

(II) The range space of T is spanned by the set {1 + x2, 1 + x}.

(III) T(T(1 + x) = 1 + x2.

(IV) If M is the matrix representation of T with respect of the standard basis {1, x, x2} of P2 then the trace of the matrix M is 3.

Which of the above statements are TRUE?

(A) I and II only

(B) I, III and IV only

(C) I, II and IV only

(D) II and IV only

Answer: (C)

42. Let T1 and T2 be two topologies defined on ℕ (The set of all natural numbers), where T1 is the topology generated byB = {{2n – 1, 2n} : n ∈ ℕ} and T2 is the discrete topology on ℕ.

Consider the statements:

(I) In (ℕ, T1), every infinite subset has a limit point.

(II) The function f : (ℕ, T1) → (ℕ, T2) define by

                                                   

is a continuous function.

Which of the above statements is/are TRUE?

(A) Both I and II

(B) I only

(C) II only

(D) Neither I nor II

Answer: (A)

43. Let 1 ≤ p < q < ∞. Consider the following statements:

(I) ℓp ⊂ ℓq

(II) Lp[0, 1] ⊂ Lq[0,1],

(ℝ is the set of all real numbers

Which of the above statements is/are TRUE?

(A) Both I and II

(B) I only

(C) II only

(D)  Neither I nor II

Answer: (B)

44. Consider the differential equation

                           

If Y(s) is the Laplace transform of y(t), then the value of Y(1) s _____(round off to 2 places of decimal).

(Here, t he inverse trigonometric functions assume principal values only)

 

Answer: (0.76 to 0.83)

45. Let R be the region in the xy-place bounded by the curves y = x2, y = 4x2, xy = 1 and xy = 5.

Then the value of integral  is equal to _______

 

Answer: (12 to 12)

46. Let V be the vector space of all 3 × 3 matrices with complex entries over the real field. If

                   

then the dimension of W1 + W2 is equal to _____.

 

Answer: (17 to 17)

47. The number of elements of order 15 in the additive group ℤ60 × ℤ50 is ______

(ℤn denotes the group of integers modulo n, under the operation of addition modulo n, for any positive integer n)

Answer: (48 to 48)

48. Consider the following cost matrix of assigning four jobs to f our persons:

Then the minimum cost of the assignment problem subject to the constraint that job J4 is assigned to person P2, is _____.

Answer: (27 to 27)

49. Let y : [−1, 1] → ℝ with y(1) = 1 satisfy the Legendre differential equation

         

Then the value of  is equal to _______(round off to 2 places of decimal).

Answer: (0.25 to 0.30)

50. Let ℤ125 be the ring of integers modulo 125 under the operations of addition modulo 125 and multiplication modulo 125. If m is the number of maximal ideals of ℤ125 and n is the number of non-units of ℤ125, then m + n is equal to ______.

Answer: (26 to 26)

51. The maximum value of the error term of the composite Trapezoidal rule when it is used to evaluate the definite integral

                 

with  12 sub-intervals of equal length, is equal to ______ (round off to 3 places of decimal).

Answer: (0.022 to 0.028)

52. By the Simplex method, the optimal table of the linear programming problem:

             

where α, β, γ are real, constants, is

Then the value of α + β + γ is ______

Answer: (14.0 to 16.0 OR 17.0 to 18.5)

53. Consider the inner product space P2 of all polynomials of degree at most 2 over the field of real numbers with the inner product 

Let {f0, f1, f2) be an orthogonal set in P2, where f0 = 1, f1 = t +c1, f2 = t2 + c2f1 +c3 and c1, c2, c3 are real constants. Then the value of 2c1 + c2+ 3c3 is equal to _____.

Answer: (-3 to -3)

54. Consider the system of linear differential equations

         

with the initial conditions x1(0) = 0, x2(0) = 1.

The loge(x2(2) – x1(2)) is equal to

 

Answer: (1.95 to 2.05)

55. Consider the differential equation

             

The sum of the roots of the indicial equation of the Frobenius series solution for the above differential equation in a neighbourhood of x = 0 is equal to ____.

Answer: (10 to 10)

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