**GATE-2023**

**MA: Mathematics**

**General Aptitude**

**Q.1 – Q.5 Carry ONE mark Each**

1. The village was nestled in a green spot, _______ the ocean and the hills.

(A) through

(B) in

(C) at

(D) between

2. Disagree : Protest : : Agree : _______

(By word meaning)

(A) Refuse

(B) Pretext

(C) Recommend

(D) Refute

3. A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

(A) 125

(B) 720

(C) 60

(D) 80

4. Which one among the following statements must be TRUE about the mean and the median of the scores of all candidates appearing for GATE 2023?

(A) The median is at least as large as the mean.

(B) The mean is at least as large as the median.

(C) At most half the candidates have a score that is larger than the median.

(D) At most half the candidates have a score that is larger than the mean.

5. In the given diagram, ovals are marked at different heights (h) of a hill. Which one of the following options P, Q, R, and S depicts the top view of the hill?

(A) P

(B) Q

(C) R

(D) S

**Q.6 – Q.10 Carry TWO marks Each**

6. Residency is a famous housing complex with many well-established individuals among its residents. A recent survey conducted among the residents of the complex revealed that all of those residents who are well established in their respective fields happen to be academicians. The survey also revealed that most of these academicians are authors of some best-selling books.

Based only on the information provided above, which one of the following statements can be logically inferred with certainty?

(A) Some residents of the complex who are well established in their fields are also authors of some best-selling books.

(B) All academicians residing in the complex are well established in their fields.

(C) Some authors of best-selling books are residents of the complex who are well established in their fields.

(D) Some academicians residing in the complex are well established in their fields.

7. Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:

(1) At any stage, Ankita can move either one or two stairs up.

(2) At any stage, Ankita cannot move to a lower step.

Let F(N) denote the number of possible ways in which Ankita can reach the Nth stair. For example, F(1) = 1, F(2) = 2, F(3) = 3.

The value of F(5) is _______.

(A) 8

(B) 7

(C) 6

(D) 5

8. The information contained in DNA is used to synthesize proteins that are necessary for the functioning of life. DNA is composed of four nucleotides: Adenine (A), Thymine (T), Cytosine (C), and Guanine (G). The information contained in DNA can then be thought of as a sequence of these four nucleotides: A, T, C, and G. DNA has coding and non-coding regions. Coding regions—where the sequence of these nucleotides are read in groups of three to produce individual amino acids—constitute only about 2% of human DNA. For example, the triplet of nucleotides CCG codes for the amino acid glycine, while the triplet GGA codes for the amino acid proline. Multiple amino acids are then assembled to form a protein.

Based only on the information provided above, which of the following statements can be logically inferred with certainty?

(i) The majority of human DNA has no role in the synthesis of proteins.

(ii) The function of about 98% of human DNA is not understood.

(A) only (i)

(B) only (ii)

(C) both (i) and (ii)

(D) neither (i) nor (ii)

9. Which one of the given figures P, Q, R and S represents the graph of the following function?

f(x) = ||x + 2| – |x – 1||

(A) P

(B) Q

(C) R

(D) S

10. An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

(A) P

(B) Q

(C) R

(D) S

**MA: Mathematics**

**Q.11 – Q.35 Carry ONE mark Each**

11. Let f , g : ℝ^{2} → ℝ be defined by

and g(x, y) = 4x^{4} – 5x^{2}y + y^{2}

for all (x, y) ∈ ℝ^{2}.

Consider the following statements:

P: f has a saddle point at (0, 0).

Q: g has a saddle point at (0, 0).

Then

(A) both P and Q are TRUE

(B) P is FALSE but Q is TRUE

(C) P is TRUE but Q is FALSE

(D) both P and Q are FALSE

12. Let ℝ^{3} be a topological space with the usual topology and ℚ denote the set of rational numbers. Define the subspaces X, Y, Z and W of ℝ3 as follows:

X = {(x, y, 𝑧) ∈ ℝ^{ 3}∶ |x| + |y| + |z| ∈ ℚ}

Y = {(x, y, z) ∈ ℝ^{ 3}∶ xyz = 1}

Z = {(x, y, z) ∈ ℝ^{3} ∶ x^{2 }+ y^{2} + z^{2 }= 1}

W ={(x, y, z) ∈ ℝ^{3} ∶ xyz = 0 }

Which of the following statements is correct?

(A) X is homeomorphic to Y

(B) Z is homeomorphic to W

(C) Y is homeomorphic to W

(D) X is NOT homeomorphic to W

13. Let P(x) = 1 + e^{2}^{π}^{ix} + 2 e^{3}^{π}^{ix}, x ∈ ℝ, i = √− Then

is equal to

(A) 0

(B) 1

(C) 3

(D) 4

14. Let T : ℝ^{3} → ℝ^{3} be a linear transformation satisfying

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

Then

(A) T is one-one but T is NOT onto

(B) T is one-one and onto

(C) T is NEITHER one-one NOR onto

(D) T is NOT one-one but T is onto

15. Let 𝔻 = {z ∈ ℂ : |z| < 1} and f: 𝔻 → ℂ be defined by

Consider the following statements:

P: f is three zeros (counting multiplicity) in 𝔻.

Q : f has one zero in

Then

(A) P is TRUE but Q is FALSE

(B) P is FALSE but Q is TRUE

(C) both P and Q are TRUE

(D) both P and Q are FALSE

16. Let 𝒩 ⊆ ℝ be a non-measurable set with respect to the Lebesgue measure on ℝ.

Consider the following statements:

P: If M = {x ∈ : x is irrational}. Then M is Lebesgue measurable.

Q: The boundary of 𝒩 has positive Lebesgue outer measure.

Then

(A) both P and Q are TRUE

(B) P is FALSE and Q is TRUE

(C) P is TRUE and Q is FALSE

(D) both P and Q are FALSE

17. For k ∈ ℕ, let E_{k} be a measurable subset of [0, 1] with Lebesgue measure 1/k^{2}.

Define

and

Consider the following statements:

P: Lebesuge measure of E is equal to zero.

Q: Lebesgue measure of F is equal to zero.

Then

(A) both P and Q are TRUE

(B) both P and Q are FALSE

(C) P is TRUE but Q is FALSE

(D) Q is TRUE but P is FALSE

18. Consider ℝ^{2} with the usual Euclidean metric. Let

Consider the following statements:

P: X is a connected subset of ℝ^{2}.

Q: Y is connected subset of ℝ^{2}.

Then

(A) both P and Q are TRUE

(B) P is FALSE and Q is TRUE

(C) P is TRUE and Q is FALSE

(D) both P and Q are FALSE

19. Let

Consider the following statements:

P: M^{8} + M^{12} is diagonalizable.

Q: M^{7} + M^{9} is diagonalizable.

Which of the following statements is correct?

(A) P is TRUE and Q is FALSE

(B) P is FALSE and Q is TRUE

(C) Both P and Q are FALSE

(D) Both P and Q are TRUE

20. Let C[0, 1] = {f : [0, 1] → ℝ : f is continuous}.

Consider the metric space (C[0, 1], d_{∞}), where

d_{∞}(f, g) = sup{|f(x) – g(x)| : x ∈ [0, 1]} for f, g ∈ C [0, 1].

Let f_{0}(x) = 0 for all x ∈ [0, 1] and

X = {f ∈ (C[0, 1], d_{∞}): d_{∞}(f_{0}, f) ≥ 1/2}.

Let f_{1}, f_{2} ∈ C[0, 1] be defined by f_{1}(x) = x and f_{2}(x) = 1 – x for all x ∈ [0, 1].

Consider the following statements:

P: f_{1} is in the interior of X.

Q: f_{2} is in the interior of X.

Which of the following statements is correct?

(A) P is TRUE and Q is FALSE

(B) P is FALSE and Q is TRUE

(C) Both P and Q are FASLE

(D) Both P and Q are TRUE

21. Consider the metrics ρ_{1} and ρ_{2} on ℝ, defined by

Consider the following statements:

P: The function f : (X ∪ Y, ρ_{1}) → (ℝ, ρ_{1}) is uniformly continuous.

Q: The function f : (X ∪ Y, ρ_{2}) → (ℝ, ρ_{1}) is uniformly continuous.

Then

(A) P is TRUE and Q is FALSE

(B) P is FALSE and Q is TRUE

(C) both P and Q are FALSE

(D) both P and Q are TRUE

22. Let T : ℝ^{4} → ℝ^{4} be a linear transformation and the null space of T be the subspace of ℝ^{4} given by

{(x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ^{4} : 4x_{1} + 3x_{2} + 2x_{3} + x_{4} = 0}.

If Rank(T – 3I) = 3, where I is the identity map of ℝ^{4}, then the minimal polynomial of T is

(A) x(x – 3)

(B) x(x – 3)^{3}

(C) x^{3}(x – 3)

(D) x^{2}(x – 3)^{2}

23. Let C[0, 1] denote the set of all real valued continuous functions defined on [0, 1] and ||f||_{∞} = sup{|f(x)| : x ∈ [0, 1]} for all f ∈ C[0, 1]. Let X = {f ∈ C[0, 1] : f(0) = f(1) = 0}.

Define F : (C[0, 1], ||∙||_{∞}) → ℝ by

Denote S_{X} = {f ∈ X : ||f||_{∞} = 1}.

Then the set {f ∈ X : F(f) = ||F||} ⋂ S_{X} has

(A) NO element

(B) exactly one element

(C) exactly two elements

(D) an infinite number of elements

24. Let X and Y be two topological spaces. A continuous map f ∶ X → Y is said to be proper if f^{−1}(K) is compact in X for every compact subset K of Y, where f^{−1}(K) is defined by f^{−1}(K) = {x ∈ X ∶ f(x) ∈ K} .

Consider ℝ with the usual topology. If ℝ ∖{0} has the subspace topology induced from ℝ and ℝ × ℝ has the product topology, then which of the following maps is proper?

(A) f : ℝ \ {0} → ℝ defined by f(x) = x

(B) f : ℝ × ℝ → ℝ × ℝ defined by f(x, y) = (x + y, y)

(C) f : ℝ × ℝ → ℝ defined by f(x, y) = x

(D) f : ℝ × ℝ → ℝ defined by f(x, y ) = x^{2} – y^{2}

25. Consider the following Linear Programming Problem P:

Minimize 3x_{1} + 4x_{2}

Subject to x_{1} − x_{2} ≤ 1,

x_{1} + x_{2} ≥ 3,

x_{1} ≥ 0, x_{2} ≥ 0.

The optimal value of the problem P is _________.

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

for some positive real number c.

Let the domain of dependence of the solution 𝑢 at the point P(3, 2) be the line segment on the x-axis with end points Q and R.

If the area of the triangle PQR is 8 square units, then the value of c^{2} is _________.

27. Let

for all z in some neighbourhood of 0 in ℂ.

Then the value of a_{6} + a_{5} is equal to _______.

28. Let p(x) = x^{3} – 2x + 2. If q(x) is the interpolating polynomial of degree less than or equal to 4 for the data

then the value of at x = 0 is ________.

29. For a fixed c ∈ ℝ, let

If the value of obtained by using the Trapezoidal rule is equal to α, then the value of c is ________ (rounded off to 2 decimal places.)

30. If for some α ∈ ℝ,

then the value of α equals _______.

31. Let S be the portion of the plane z= 2x + 2y − 100 which lies inside the cylinder x^{2} + 𝑦^{2} = 1. If the surface area of S is απ, then the value of α is equal to ___________.

32. Let L^{2}[−1, 1] = {f : [−1, 1] → ℝ : f is Lebesgue measurable and the norm

Let F : (L^{2}[−1, 1], ||∙||_{2}) → ℝ be defined by

for all f ∈ L^{2}[−1, 1].

If ||F|| denotes the norm of the linear functional F, then 5||F||^{2} is equal to ________.

33. Let y(t) be the solution of the initial value problem

If α = y(π/2), then the value of is _______ (rounded off to 2 decimal places).

34. Consider ℝ^{4} with the inner product < x, y > = for x = (x_{1}, x_{2}, x_{3}, x_{4}) and y = (y_{1}, y_{2}, y_{3}, y_{4}).

Let M = {(x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ^{4} : x_{1} = x_{3}} and M^{⊥} denote the orthogonal complement of M. The dimension of M^{⊥} is equal to ________.

35. Let and If 6M^{−}^{1} = M^{2} – 6M + αI for some α ∈ ℝ, then the value of α is equal to _________.

**Q.36 – Q.65 Carry TWO marks Each**

36. Let GL_{2}(ℂ) denote the group of 2 × 2 invertible complex matrices with usual matrix multiplication. For S, T ∈ GL_{2}(ℂ), < S, T > denotes the subgroup generated by S and T. Let and G_{1}, G_{2}, G_{3} be three subgroups of GL_{2}(ℂ) given by

Let Z(G_{i}) denote the center of G_{i} and i = 1, 2, 3.

Which of the following statements is correct?

(A) G_{1} is isomorphic to G_{3}

(B) Z(G_{1}) is isomorphic to Z(G_{2})

(C)

(D) Z(G_{2}) is isomorphic to Z(G_{3})

37. Let ℓ^{2} = {(x_{1}, x_{2}, x_{3}, …) : x_{n} ∈ ℝ for all n ∈ ℕ and

For a sequence (x_{1}, x_{2}, x_{3}, …) ∈ ℓ^{2}, define ||(x_{1}, x_{2}, x_{3}, …||_{2} = Let S : (ℓ^{2}, ||∙||_{2}) → T : (ℓ^{2}, ||∙||_{2}) and T : (ℓ^{2}, ||∙||_{2}) → (ℓ^{2}, ||∙||_{2}) be defined by

S(x_{1}, x_{2}, x_{3}, …) = (y_{1}, y_{2}, y_{3}, …), where

T(x_{1}, x_{2}, x_{3}, …) = (y_{1}, y_{2}, y_{3}, …), where

Then

(A) S is a compact linear map and T is NOT a compact linear map

(B) S is NOT a compact linear map and T is a compact linear map

(C) both S and T are compact linear maps

(D) NEITHER S NOR T is a compact linear map

38. Let

c_{00} = {(x_{1}, x_{2}, x_{3}…) : x_{i} ∈ ℝ, i ∈ ℕ, x_{i} ≠ 0 only for finitely many indices i}.

For (x_{1}, x_{2}, x_{3}, …) ∈ c_{00}, let||(x_{1}, x_{2}, x_{3}, …)||_{∞} = sup{|x_{i}| : i ∈ ℕ}.

Define F, G : (c_{00}, ||∙||_{∞}) → (c_{00}, ||∙||_{∞}) by

Then

(A) F is continuous but G is NOT continuous

(B) F is NOT continuous but G is continuous

(C) both F and G are continuous

(D) NEITHER F NOR G is continuous

39. Consider the Cauchy problem

u = f(t) on the initial curve Γ = (t, t); t > 0.

Consider the following statements:

P: If f(t) = 2t + 1, then there exists a unique solution to the Cauchy problem in a neighbourhood of Γ.

Q : If f(t) = 2t – 1, then there exist infinitely many solutions to the Cauchy problem in a neighbourhood of Γ.

Then

(A) both P and Q are TRUE

(B) P is FALSE and Q is TRUE

(C) P is TRUE and Q is FALSE

(D) both P and Q are FALSE

40. Consider the linear system Mx = b, where Suppose M = LU, where L and U are lower triangular and upper triangular square matrices, respectively. Consider the following statements:

P : If each element of the main diagonal of L is 1, then trace (U) = 3.

Q : For any choice of the initial vector x^{(0)}, the Jacobi iterates x^{(k)}, k = 1, 2, 3 … converge to the unique solution of the linear system Mx = b.

Then

(A) both P and Q are TRUE

(B) P is FALSE and Q is TRUE

(C) P is TRUE and Q is FALSE

(D) both P and Q are FALSE

41. Let ϕ and ψ be two linearly independent solutions of the ordinary differential equation

yʹʹ + (2 – cos x) y = 0, x ∈ ℝ.

Let α, β ∈ ℝ be such that α < β, ϕ(α) = ϕ(β) = 0 and ϕ(x) = 0 for all x ∈ (α, β).

Consider the following statements:

P : ϕʹ(α) ϕʹ(β) > 0.

Q: ϕ(x) ψ (x) ≠ 0 for all x ∈ (α, β).

Then

(A) P is TRUE and Q is FALSE

(B) P is FALSE and Q is TRUE

(C) both P and Q are FALSE

(D) both P and Q are TRUE

42. Let 𝔻 = {z ∈ ℂ : |z| < 1} and f : 𝔻 → ℂ be an analytic function given by the power series where a_{0} = a_{1} = 1 and a_{n} = 1/2^{2n} for n ≥ 2.

Consider the following statements:

P: If z_{0} ∈ 𝔻, then f is one-one in some neighbourhood of z_{0}.

Q: If E = {z ∈ ℂ : |z| ≤ 1/2}, then f(E) is a closed subset of ℂ.

Which of the following statements is/are correct?

(A) P is TRUE

(B) Q is TRUE

(C) Q is FALSE

(D) P is FALSE

43. Let Ω be an open connected subset of ℂ containing U = {z ∈ ℂ : |z| ≤ 1/2}.

Let 𝔍 = {f : Ω → ℂ : f is analytic and

Consider the following statements:

P : There exists f ∈ 𝔍 such that |fʹ(0)| ≥ 2.

Q: |f^{(3)}(0)| ≤ 48 for all f ∈ 𝔍, where f^{(3)} denotes the third derivative of f.

Then

(A) P is TRUE

(B) Q is FALSE

(C) P is FALSE

(D) Q is TRUE

44. Let (ℝ, τ) be a topological space, where the topology τ is defined as

τ = {U ⊆ ℝ : U = ∅ or 1 ∈ U}.

Which of the following statements is/are correct?

(A) (ℝ, τ) is first countable

(B) (ℝ, τ) is Hausdorff

(C) (ℝ, τ) is separable

(D) The closure of (1, 5) is [1, 5]

45. Let ℛ = {p(x) ∈ ℚ[x] : p(0) ∈ ℤ}, where ℚ denotes the set of rational numbers and ℤ denotes the set of integers. For a ∈ ℛ, let ‹a› denote the ideal generated by a in ℛ.

Which of the following statements is/are correct?

(A) If p(x) is an irreducible element in ℛ, then ‹p(x)› is a prime ideal in ℛ

(B) ℛ is a unique factorization domain

(C) ‹x› is a prime ideal in ℛ

(D) ℛ is NOT a principal ideal domain

46. Consider the rings

where ‹2, x^{3}› denotes the ideal generated by {2, x^{3}} in ℤ[x] and ‹x^{2}› denotes the ideal generated by x^{2} in ℤ_{2}[x].

Which of the following statements is/are correct?

(A) Every prime ideal of 𝒮_{1} is a maximal ideal

(B) 𝒮_{2} has exactly one maximal ideal

(C) Every element of 𝒮_{1} is either nilpotent or a unit

(D) There exists an element in 𝒮_{2} which is NEITHER nilpotent NOR a unit

47. Consider the sequence of Lebesgue measurable functions f_{n} : ℝ → ℝ given by

For a measurable subset E of ℝ, denote m(E) to be the Lebesgue measure of E.

Which of the following statements is/are correct?

48. Define the characteristic function χ_{E} of a subset E in ℝ by

For 1 ≤ p < 2, let

L^{p}[0, 1] = {f: [0, 1] → ℝ : f is Lebesgue measurable and

Let f : [0, 1] → ℝ be defined by

Consider the following two statements:

P : f ∈ L^{p} [0, 1] for every p ∈ (1, 2).

Q: f ∈ L^{1}[0, 1].

Then

(A) P is TRUE

(B) Q is TRUE

(C) Q is FALSE

(D) P is FALSE

49. Let x(t), y(t), t ∈ ℝ, be two functions satisfying the following system of differential equations:

xʹ(t) = y(t).

yʹ(t) = x(t).

and x(0) = α, y(0) = β, where α, β are real numbers.

Which of the following statements is/are correct?

(A) If α = 1, β = −1, then |x(t)| + |y(t)| → 0 as t → ∞

(B) If α = 1, β = 1, then |x(t)| + |y(t)| → 0 as t → ∞

(C) If α = 1.01, β = −1, then |x(t)| + |y(t)| → 0 as t → ∞

(D) If α = 1, β = 1.01, then |x(t)| + |y(t)| → 0 as t → ∞

50. For h > 0, and α, β, γ ∈ ℝ, let

be a three-point formula to approximate f ʹ (a) for any differentiable function f : ℝ → ℝ and a ∈ ℝ.

If D_{h}f(a) = f ʹ(a) for every polynomial f of degree less than or equal to 2 and for all a ∈ ℝ, then

(A) α + 2γ = −2

(B) α + 2β – 2γ = 0

(C) α + 2γ = 2

(D) α + 2β – 2γ = 1

51. Let f be a twice continuously differentiable function on [a, b] such that f ʹ(x) < 0 and f ʹʹ(x) < 0 for all x ∈ (a, b). Let f(ζ) is given by

for an initial guess x_{0}.

If x_{k} ∈ (ζ, b) for some k ≥ 0, then which of the following statements is/are correct

(A) x_{k+1} > ζ

(B) x_{k+1} < ζ

(C) x_{k+1} < x_{k}

(D) For every

52. Let f : ℝ^{2} → ℝ be defined by

Then

(A) the directional derivative of f at (0,0) in the direction of (1/√2, 1/√2) is 1/√2

(B) the directional derivative of f at (0,0) in the direction of (0,1 ) is 1

(C) the directional derivative of f at (0,0) in the direction of (1,0) is 0

(D) f is NOT differentiable at (0,0)

53. Let C[0, 1] = {f : [0, 1] → ℝ : f is continuous} and

d_{∞}(f, g) = sup{|f(x) – g(x)|: x ∈ [0, 1]} for f, g ∈ C[0, 1].

For each n ∈ ℕ, define f_{n} : [0, 1] → ℝ by f_{n}(x) = x^{n} for all x ∈ [0, 1].

Let P = {f_{n} : n ∈ ℕ}.

Which of the following statements is/are correct?

(A) P is totally bounded in (C[0, 1], d_{∞})

(B) P is bounded in (C[0, 1], d_{∞})

(C) P is closed in (C[0, 1], d_{∞})

(D) P is open in (C[0, 1], d_{∞})

54. Let G be an abelian group and Φ : G → (ℤ, +) be a surjective group homomorphism. Let 1 = Φ(a) for some a ∈

Consider the following statements.

P: For every g ∈ G, there exists an n ∈ ℤ such that ga^{n} ∈ ker(Φ).

Q: Let e be the identity of G and <a> be the subgroup generated by a. Then G = ker(Φ) < a > and ker(Φ) ∩ < a > = {e}.

Which of the following statements is/are correct?

(A) P is TRUE

(B) P is FALSE

(C) Q is TRUE

(D) Q is FALSE

55. Let C be the curve of intersection of the cylinder x^{2} + y^{2} = 4 and the plane

z − 2 =0. Suppose C is oriented in the counterclockwise direction around the

𝑧-axis, when viewed from above. If

|∫_{C}(sin x + e^{x})dx + 4x dy + e^{z} cos^{2} z dz| = απ,

Then the value of α equals ________.

56. Let ℓ^{2} = {(x_{1}, x_{2}, x_{3}…) : x_{n} ∈ ℝ for all n ∈ ℕ and

For a sequence (x_{1}, x_{2}, x_{3}, …} ∈ ℓ^{2}, define

Consider the subspace

Let M^{⊥} denote the orthogonal complement of M in the Hilbert space (ℓ^{2}, ||∙||_{2}).

Consider (1, 1/2, 1/3, 1/4, …) ∈ ℓ^{2}.

If the orthogonal projection of (1, 1/2, 1/3, 1/4, …) onto M^{⊥} is given by

for some α ∈ ℝ, then α equals ________.

57. Consider the transportation problem between five sources and four destinations as given in the cost table below. The supply and demand at each of the source and destination are also provided:

Let C_{N} and C_{L} be the total cost of the initial basic feasible solution obtained from the North-West corner method and the Least-Cost method, respectively. Then C_{𝑁 }− C_{L} equals _________.

58. Let σ ∈ S_{8}, where S_{8} is the permutation group on 8 elements. Suppose σ is the product of σ_{1} and σ_{2}, where σ_{1} is a 4-cycle and σ_{2} is a 3-cycle in S_{8}. If σ_{1} and σ_{2} are disjoint cycles, then the number of elements in S_{8} which are conjugate to σ is _________.

59. Let A be a 3 × 3 real matrix with det(A + iI) = 0, where i = √−1 and I is the 3 × 3 identity matrix. If det(A) = 3, then the trace of A^{2} is _________.

60. Let A = [a_{aj}] be a 3 × 3 real matrix such that

If m is the degree of the minimal polynomial of A, then a_{11} + a_{21} + a_{31} + m equals _______.

61. Let Ω be the disk x^{2} + y^{2} < 4 in ℝ^{2} with boundary ∂Ω. If u(x, y) is the solution of the Dirichlet problem

then the value of u(0,1) is __________.

62. For every k ∈ ℕ ∪ {0}, let y_{k}(x) be a polynomial of degree k with y_{k}(1) = 5. Further, let y_{k}(x) satisfy the Legendre equation

(1 – x^{2})yʹʹ − 2xyʹ + k(k + 1)y = 0.

If

for some positive integer n, then the value of n is ___________.

63. Consider the ordinary differential equation (ODE)

4(ln x) yʹʹ + 3yʹ + y = 0, x > 1.

If r_{1} and r_{2} are the roots of the indicial equation of the above ODE at the regular singular point x=1, then |r_{1} − r_{2}| is equal to _________ (rounded off to 2 decimal places).

64. Let u(x, t) be the solution of the non-homogeneous wave equation

Then the value of u(π/2, 3π/2) is _______ (rounded off to 2 decimal places).

65. Consider the Linear Programming Problem P:

Maximize 3x_{1} + 2x_{2} + 5x_{3}

subject to

x_{1} + 2x_{2} + x_{3} ≤ 44,

x_{1} + 2x_{3} ≤ 48,

x_{1} + 4x_{2} ≤ 52,

x_{1} ≥ 0, x_{2} ≥ 0, x_{3} ≥ 0.

The optimal value of the problem P is equal to __________.

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