**GATE-2022**

**MA: Mathematics**

**General Aptitude**

1. As you grow older, an injury to your _________ may take longer to _________.

(A) heel / heel

(B) heal / heel

(C) heal / heal

(D) heel / heal

2. In a 500 m race, P and Q have speeds in the ratio of 3 ∶ Q starts the race when P has already covered 140 m.

What is the distance between P and Q (in m) when P wins the race?

(A) 20

(B) 40

(C) 60

(D) 140

3. Three bells P, Q, and R are rung periodically in a school. P is rung every 20 minutes; Q is rung every 30 minutes and R is rung every 50 minutes.

If all the three bells are rung at 12:00 PM, when will the three bells ring together again the next time?

(A) 5:00 PM

(B) 5:30 PM

(C) 6:00 PM

(D) 6:30 PM

4. Given below are two statements and four conclusions drawn based on the statements.

Statement 1: Some bottles are cups.

Statement 2: All cups are knives.

Conclusion I: Some bottles are knives.

Conclusion II: Some knives are cups.

Conclusion III: All cups are bottles.

Conclusion IV: All knives are cups.

Which one of the following options can be logically inferred?

(A) Only conclusion I and conclusion II are correct

(B) Only conclusion II and conclusion III are correct

(C) Only conclusion II and conclusion IV are correct

(D) Only conclusion III and conclusion IV are correct

5. The figure below shows the front and rear view of a disc, which is shaded with identical patterns. The disc is flipped once with respect to any one of the fixed axes 1-1, 2-2 or 3-3 chosen uniformly at random.

What is the probability that the disc DOES NOT retain the same front and rearviews after the flipping operation?

(A) 0

(B) 1/3

(C) 2/3

(D) 1

6. Altruism is the human concern for the well being of others. Altruism has been shown to be motivated more by social bonding, familiarity and identification of belongingness to a group. The notion that altruism may be attributed to empathy or guilt has now been rejected.

Which one of the following is the CORRECT logical inference based on the information in the above passage?

(A) Humans engage in altruism due to guilt but not empathy

(B) Humans engage in altruism due to empathy but not guilt

(C) Humans engage in altruism due to group identification but not empathy

(D) Humans engage in altruism due to empathy but not familiarity

7. There are two identical dice with a single letter on each of the faces. The following six letters: Q, R, S, T, U, and V, one on each of the faces. Any of the six outcomes are equally likely.

The two dice are thrown once independently at random.

What is the probability that the outcomes on the dice were composed only of any combination of the following possible outcomes: Q, U and V?

(A) 1/4

(B) 3/4

(C) 1/6

(D) 5/36

8. The price of an item is 10% cheaper in an online store S compared to the price at another online store M. Store S charges Rs. 150 for delivery. There are no delivery charges for orders from the store M. A person bought the item from the store S and saved Rs. 100.

What is the price of the item at the online store S (in Rs.) if there are no other charges than what is described above?

(A) 2500

(B) 2250

(C) 1750

(D) 1500

9. The letters P, Q, R, S, T and U are to be placed one per vertex on a regular convex hexagon, but not necessarily in the same order.

Consider the following statements:

• The line segment joining R and S is longer than the line segment joining P and Q.

• The line segment joining R and S is perpendicular to the line segment joining P and Q.

• The line segment joining R and U is parallel to the line segment joining T and Q.

Based on the above statements, which one of the following options is CORRECT?

(A) The line segment joining R and T is parallel to the line segment joining Q and S

(B) The line segment joining T and Q is parallel to the line joining P and U

(C) The line segment joining R and P is perpendicular to the line segment joining Uand Q

(D) The line segment joining Q and S is perpendicular to the line segment joining Rand P

10.

An ant is at the bottom-left corner of a grid (point P) as shown above. It aims to move to the top-right corner of the grid. The ant moves only along the lines marked in the grid such that the current distance to the top-right corner strictly decreases.

Which one of the following is a part of a possible trajectory of the ant during the movement?

**Mathematics**

**Q.1 – Q.5 Carry ONE mark each.**

11. Suppose that the characteristic equation of M ∈ ℂ^{3}^{×}^{3} is

λ^{3} + αλ^{2} + βλ – 1 = 0,

where α, β ∈ ℂ with α + β ≠ 0.

Which of the following statements is TRUE?

(A) M(I – βM) = M^{−}^{1}(M + αI)

(B) M(I + βM) = M^{−}^{1}(M – αI)

(C) M^{−}^{1}(M^{−}^{1 }+ βI) = M – αI

(D) M^{−}^{1}(M^{−}^{1} – βI) = M + αI

12. Consider

P: Let M ∈ ℝ^{m}^{×}^{n} with m > n ≥ 2. If rank (M) = n, then the system of linear equations M_{x} = 0 has x = 0 as the only solution.

Q: Let E ∈ ℝ^{n}^{×}^{n}, n ≥ 2 be a non-zero matrix such that E^{3} = 0. Then I + E^{2} is a singular matrix.

Which of the following statements is TRUE?

(A) Both P and Q are TRUE

(B) Both P and Q are FALSE

(C) P is TRUE and Q is FALSE

(D) P is FALSE and Q is TRUE

13. Consider the real function of two real variables given by

u(x, y) = e^{2x}[sin 3x cos 2y cosh 3y − cos 3x sin 2y sinh 3y].

Let v(x, y) be the harmonic conjugate of u(x, y) such that v(0, 0) = 2. Let z = x + iy and f(z) = u(x, y) + iv(x, y), then the value of 4 + 2if (iπ) is

(A) e^{3}^{π} + e^{−}^{3}^{π}

(B) e^{3}^{π} – e^{−}^{3}^{π}

(C) −e^{3}^{π} + e^{−}^{3}^{π}

(D) −e^{3}^{π} – e^{−}^{3}^{π}

14. The value of the integral

where C is the circle of radius 2 centred at the origin taken in the anti-clockwise direction is

(A) −2πi

(B) 2π

(C) 0

(D) 2πi

15. Let X be a real normed linear space. Let X_{0} = {x ∈ X : ∥x∥ = 1}. If X_{0} contains two distinct points x and y and the line segment joining them, then, which of the following statements is TRUE?

(A) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly independent

(B) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly dependent

(C) ∥x + y∥^{2} = ∥x∥^{2} + ∥y∥^{2} and x, y are linearly independent

(D) ∥x + y∥ = 2∥x∥∥y∥ and x, y are linearly dependent

16. Let {e_{k} : k ∈ ℕ} be an orthonormal basis for a Hilbert space H.

Define f_{k} = e_{k} + e_{k+1}, k ∈ ℕ and

Then

(A) 0

(B) j^{2}

(C) 4j^{2}

(D) 1

17. Consider ℝ^{2} with the usual metric. Let A = {(x, y) ∈ ℝ^{2} : x^{2} + y^{2} ≤ 1} and B = {(x, y) ∈ ℝ^{2} : (x−2)^{2 }+ y^{2} ≤ 1}. Let M = A ⋃ B and N = interior(A) ⋃ interior(B). Then, which of the following statements is TRUE?

(A) M and N are connected

(B) Neither M nor N is connected

(C) M is connected and N is not connected

(D) M is not connected and N is connected

18. The real sequence generated by the iterative scheme

(A) converges to √2, for all x_{0} ∈ ℝ\{0}

(B) converges to √2, whenever

(C) converges to √2, whenever x_{0} ∈ (−1, 1)\{0}

(D) diverges for any x_{0} ≠ 0

19. The initial value problem where y_{0} is a real constant, has

(A) a unique solution

(B) exactly two solutions

(C) infinitely many solutions

(D) no solution

20. If eigenfunctions corresponding to distinct eigenvalues λ of the Sturm-Liouville problem

are orthogonal with respect to the weight function w(x), then w(x) is

(A) e^{−}^{3x}

(B) e^{−}^{2x}

(C) e^{2x}

(D) e^{3x}

21. The steady state solution for the heat equation

with the initial condition u(x, 0) = 0, 0 < x < 2 and the boundary conditions u(0, t) = 1 and u(2, t) = 3, t > 0, at x = 1 is

(A) 1

(B) 2

(C) 3

(D) 4

22. Consider ([0, 1], T_{1}), where T_{1} is the subspace topology induced by the Euclidean topology on ℝ, and let T_{2} be any topology on [0, 1]. Consider the following statements:

P : If T_{1} is a proper subset of T_{2}, then ([0, 1], T_{2}) is not compact.

Q : If T_{2} is a proper subset of T_{1}, then ([0, 1], T_{2}) is not Hausdorff.

Then

(A) P is TRUE and Q is FALSE

(B) Both P and Q are TRUE

(C) Both P and Q are FALSE

(D) P is FALSE and Q is TRUE

23. Let p : ([0, 1], T_{1}) → ({0, 1}, T_{2}) be the quotient map, arising from the characteristic function on [1/2, 1], where T_{1} is the subspace topology induced by the Euclidean topology on ℝ. Which of the following statements is TRUE?

(A) p is an open map but not a closed map

(B) p is a closed map but not an open map

(C) p is a closed map as well as an open map

(D) p is neither an open map nor a closed map

24. Set X_{n} = ℝ for each n ∈ ℕ. Define Endow Y with the product topology, where the topology on each X_{n} is the Euclidean topology. Consider the set

∆ = {(x, x, x, …) | x ∈ ℝ}

with the subspace topology induced from Y . Which of the following statements is TRUE?

(A) Δ is open in Y

(B) Δ is locally compact

(C) Δ is dense in Y

(D) Δ is disconnected

25. Consider the linear system of equations Ax = b with

Which of the following statements are TRUE?

(A) The Jacobi iterative matrix is

(B) The Jacobi iterative method converges for any initial vector

(C) The Gauss-Seidel iterative method converges for any initial vector

(D) The spectral radius of the Jacobi iterative matrix is less than 1

26. The number of non-isomorphic abelian groups of order 2^{2}.3^{3}.5^{4} is ______.

27. The number of subgroups of a cyclic group of order 12 is _____.

28. The radius of convergence of the series

is ______ (round off to TWO decimal places).

29. The number of zeros of the polynomial 2z^{7} – 7z^{5} + 2z^{3} – z + 1

in the unit disc {z ∈ ℂ : |z| < 1} is _______.

30. If P(x) is a polynomial of degree 5 and

where x_{0}, x_{1}, · · · , x_{6} are distinct points in the interval [2, 3], then the value of α^{2} − α + 1 is ______.

31. The maximum value of f(x, y) = 49 − x^{2} − y^{2} on the line x + 3y = 10 is _____.

32. If the function x ≠ 0, y ≠ 0 attains its local minimum value at the point (a, b), then the value of a^{3} + b^{3} is _________ (round off to TWO decimal places).

33. If the ordinary differential equation

has a solution of the form where a_{n}’s are constants and a_{0} ≠ 0, then the value of r^{2} + 1 is ________.

34. The Bessel functions J_{α}(x), x > 0, α ∈ ℝ satisfy Then, the value of (πJ_{3/2}(π))^{2} is ______.

35. The partial differential equation

is transformed to

using ξ = y − 2x and η = 7y − 2x.

Then, the value of is _______.

**Q.36 – Q.65 Carry TWO marks each.**

36. Let ℝ[X] denote the ring of polynomials in X with real coefficients. Then, the quotient ring ℝ[X]/(X^{4} + 4) is

(A) a field

(B) an integral domain, but not a field

(C) not an integral domain, but has 0 as the only nilpotent element

(D) a ring which contains non-zero nilpotent elements

37. Consider the following conditions on two proper non-zero ideals J_{1} and J_{2} of a non-zero commutative ring R.

P: For any r_{1}, r_{2} ∈ R, there exists a unique r ∈ R such that r − r_{1} ∈ J_{1}

and r − r_{2} ∈ J_{2}.

Q: J_{1} + J_{2} = R

Then, which of the following statements is TRUE?

(A) P implies Q but Q does not imply P

(B) Q implies P but P does not imply Q

(C) P implies Q and Q implies P

(D) P does not imply Q and Q does not imply P

38. Let f : [−π, π] → ℝ be a continuous function such that for some δ satisfying 0 < δ < π. Define P_{n,δ}(x) = (1+cos x−cos δ)^{n}, for n = 1, 2, 3, · · · . Then, which of the following statements is TRUE?

39. P : Suppose that converges at x = −3 and diverges at x = 6. Then converges.

Q: The interval of convergence of the series is [−4, 4].

Which of the following statements is TRUE?

(A) P is true and Q is true

(B) P is false and Q is false

(C) P is true and Q is false

(D) P is false and Q is true

40. Let x ∈ [0, 1], n = 1, 2, 3, · · · .

Then, which of the following statements is TRUE?

(A) {f_{n}} is not equicontinuous on [0, 1]

(B) {f_{n}} is uniformly convergent on [0, 1]

(C) {f_{n}} is equicontinuous on [0, 1]

(D) {f_{n}} is uniformly bounded and has a subsequence converging uniformly on [0, 1]

41. Let (ℚ, d) be the metric space with d(x, y) = |x − y|. Let E = {p ∈ ℚ : 2 < p^{2} < 3}. Then, the set E is

(A) closed but not compact

(B) not closed but compact

(C) compact

(D) neither closed nor compact

42. Let T : L^{2}[−1, 1] → L^{2}[−1, 1] be defined by where almost everywhere. If M is the kernel of I − T, then the distance between the function ϕ(t) = e^{t} and M is

43. Let X, Y and Z be Banach spaces. Suppose that T : X → Y is linear and S : Y → Z is linear, bounded and injective. In addition, if S ∘ T : X → Z is bounded, then, which of the following statements is TRUE?

(A) T is surjective

(B) T is bounded but not continuous

(C) T is bounded

(D) T is not bounded

44. The first derivative of a function f ∈ C^{∞}(−3, 3) is approximated by an interpolating polynomial of degree 2, using the data

(−1, f(−1)), (0, f(0)) and (2, f(2)).

It is found that

Then, the value of 1/αβ is

(A) 3

(B) 6

(C) 9

(D) 12

45. The work done by the force where are unit vectors in directions, respectively, along the upper half of the circle x^{2} + y^{2} = 1 from (1, 0) to (−1, 0) in the xy-plane is

(A) −π

(B) –π/2

(C) π/2

(D) π

46. Let u(x, t) be the solution of the wave equation

0 < x < π, t > 0,

with the initial conditions

u(x, 0) = sin x + sin 2x + sin 3x,

and the boundary conditions u(0, t) = u(π, t) = 0, t ≥ 0. Then, the value of is

(A) −1/2

(B) 0

(C) 1/2

(D) 1

47. Let T : ℝ^{2} → ℝ^{2} be a linear transformation defined by

T((1, 2)) = (1, 0) and T((2, 1)) = (1, 1).

For p, q ∈ R, let T^{−1}((p, q)) = (x, y).

Which of the following statements is TRUE?

(A) x = p − q; y = 2p − q

(B) x = p + q; y = 2p − q

(C) x = p + q; y = 2p + q

(D) x = p − q; y = 2p + q

48. Let y = (α,−1)^{T} , α ∈ ℝ be a feasible solution for the dual problem of the linear programming problem

Maximize: 5x_{1} + 12x_{2}

subject to: x_{1} + 2x_{2} + x_{3} ≤ 10

2x_{1} − x_{2} + 3x_{3} = 8

x_{1}, x_{2}, x_{3} ≥ 0.

Which of the following statements is TRUE?

(A) α < 3

(B) 3 ≤ α < 5.5

(C) 5.5 ≤ α < 7

(D) α ≥ 7

49. Let K denote the subset of ℂ consisting of elements algebraic over ℚ. Then, which of the following statements are TRUE?

(A) No element of ℂ \K is algebraic over ℚ

(B) K is an algebraically closed field

(C) For any bijective ring homomorphism f : ℂ → C, we have f(K) = K

(D) There is no bijection between K and ℚ

50. Let T be a M¨obius transformation such that T(0) = α, T(α) = 0 and T(∞) = −α, where α = (−1 + i)/√2. Let L denote the straight line passing through the origin with slope −1, and let C denote the circle of unit radius centred at the origin. Then, which of the following statements are TRUE?

(A) T maps L to a straight line

(B) T maps L to a circle

(C) T^{−1} maps C to a straight line

(D) T^{−1} maps C to a circle

51. Let a > 0. Define D_{a} : L^{2}(ℝ) → L^{2}(ℝ) by almost everywhere, for f ∈ L^{2}(ℝ). Then, which of the following statements are TRUE?

(A) D_{a} is a linear isometry

(B) D_{a} is a bijection

(C) Da ∘ D_{b} = D_{a+b}, b > 0

(D) D_{a} is bounded from below

52. Let {ϕ_{0}, ϕ_{1}, ϕ_{2}, · · · } be an orthonormal set in L^{2}[−1, 1] such that ϕ_{n} = C_{n}P_{n}, where C_{n} is a constant and Pn is the Legendre polynomial of degree n, for each n ∈ ℕ ⋃{0}. Then, which of the following statements are TRUE?

(A) ϕ_{6}(1) = 1

(B) ϕ_{7}(−1) = 1

(C)

(D)

53. Let X = (ℝ, T), where T is the smallest topology on ℝ in which all the singleton sets are closed. Then, which of the following statements are TRUE?

(A) [0, 1) is compact in X

(B) X is not first countable

(C) X is second countable

(D) X is first countable

54. Consider (ℤ, T), where T is the topology generated by sets of the form

A_{m, n} = {m + nk | k ∈ ℤ},

for m, n ∈ ℤ and n ≠ 0. Then, which of the following statements are TRUE?

(A) (ℤ, T) is connected

(B) Each A_{m, n} is a closed subset of (ℤ, T)

(C) (ℤ, T) is Hausdorff

(D) (ℤ, T) is metrizable

55. Let A ∈ ℝ^{m}^{×}^{n}, c ∈ ℝ^{n} and b ∈ ℝ^{m}. Consider the linear programming primal problem

Minimize: c^{T}x

subject to : Ax = b

x ≥ 0.

Let x^{0} and y^{0} be feasible solutions of the primal and its dual, respectively. Which of the following statements are TRUE?

(A) c^{T} x^{0} ≥ b^{T} y^{0}

(B) c^{T} x_{0} = bT y^{0}

(C) If c^{T} x^{0} = b^{T} y^{0}, then x^{0} is optimal for the primal

(D) If c^{T} x^{0} = b^{T} y^{0}, then y0 is optimal for the dual

56. Consider ℝ^{3} as a vector space with the usual operations of vector addition and scalar multiplication. Let x ∈ ℝ^{3} be denoted by x = (x_{1}, x_{2}, x_{3}). Define subspaces W_{1} and W_{2} by

W_{1} := {x ∈ ℝ^{3} : x_{1} + 2x_{2} − x_{3} = 0}

and

W_{2} := {x ∈ ℝ^{3} : 2x_{1} + 3x_{3} = 0}.

Let dim(U) denote the dimension of the subspace U.

Which of the following statements are TRUE?

(A) dim(W_{1}) = dim(W_{2})

(B) dim(W_{1}) + dim(W_{2}) − dim(ℝ^{3}) = 1

(C) dim(W_{1} +W_{2}) = 2

(D) dim(W_{1} ⋂ W_{2}) = 1

57. Three companies C_{1}, C_{2} and C_{3} submit bids for three jobs J_{1}, J_{2} and J_{3}. The costs involved per unit are given in the table below:

Then, the cost of the optimal assignment is ________.

58. The initial value problem is solved by using the following second order Runge-Kutta method:

K_{1} = hf(x_{i}, y_{i})

K_{2} = hf(x_{i} + αh, y_{i} + βK_{1})

where h is the uniform step length between the points x_{0}, x_{1}, · · · , x_{n} and y_{i} = y(x_{i}). The value of the product αβ is _____ (round off to TWO decimal places).

59. The surface area of the paraboloid z = x^{2} + y^{2} between the planes z = 0 and z = 1 is ______ (round off to ONE decimal place).

60. The rate of change of f(x, y, z) = x + x cos z − y sin z + y at P_{0} in the direction from P_{0}(2,−1, 0) to P_{1}(0, 1, 2) is ______.

61. If the Laplace equation

with the boundary conditions

and

has a solution, then the constant α is ______.

62. Let u(x, y) be the solution of the first order partial differential equation _____

satisfying u(2, y) = y − 4, y ∈ ℝ. Then, the value of u(1, 2) is _____.

63. The optimal value for the linear programming problem

Maximize: 6x_{1} + 5x_{2}

subject to: 3x_{1} + 2x_{2} ≤ 12

−x_{1} + x_{2} ≤ 1

x_{1}, x_{2} ≥ 0

is _______.

64. A certain product is manufactured by plants P_{1}, P_{2} and P_{3} whose capacities are 15, 25 and 10 units, respectively. The product is shipped to markets M_{1}, M_{2}, M_{3} and M_{4}, whose requirements are 10, 10, 10 and 20, respectively. The transportation costs per unit are given in the table below.

Then the cost corresponding to the starting basic solution by the Northwest-corner method is _______.

65. Let M be a 3 × 3 real matrix such that M^{2} = 2M + 3I. If the determinant of M is −9, then the trace of M equals _______.

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