# GATE Exam 2023 Mathematics (MA) Question Paper With Answer Key

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 : ℝ2 → ℝ be defined by and g(x, y) = 4x4 – 5x2y + y2

for all (x, y) ∈ ℝ2.

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 ℝ3 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) ∈ ℝ3 ∶ x2 + y2 + z2 = 1}

W ={(x, y, z) ∈ ℝ3 ∶ 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 + e2πix + 2 e3πix, x ∈ ℝ, i = √− Then is equal to

(A) 0

(B) 1

(C) 3

(D) 4

14. Let T : ℝ3 → ℝ3 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 Ek be a measurable subset of [0, 1] with Lebesgue measure 1/k2.

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 ℝ2 with the usual Euclidean metric. Let Consider the following statements:

P: X is a connected subset of ℝ2.

Q: Y is connected subset of ℝ2.

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: M8 + M12 is diagonalizable.

Q: M7 + M9 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 f0(x) = 0 for all x ∈ [0, 1] and

X = {f ∈ (C[0, 1], d): d(f0, f) ≥ 1/2}.

Let f1, f2 ∈ C[0, 1] be defined by f­1(x) = x and f2(x) = 1 – x for all x ∈ [0, 1].

Consider the following statements:

P: f1 is in the interior of X.

Q: f2 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 ρ1 and ρ2 on ℝ, defined by Consider the following statements:

P: The function f : (X ∪ Y, ρ1) → (ℝ, ρ1) is uniformly continuous.

Q: The function f : (X ∪ Y, ρ2) → (ℝ, ρ1) 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 : ℝ4 → ℝ4 be a linear transformation and the null space of T be the subspace of ℝ4 given by

{(x1, x2, x3, x4) ∈ ℝ4 : 4x1 + 3x2 + 2x3 + x4 = 0}.

If Rank(T – 3I) = 3, where I is the identity map of ℝ4, then the minimal polynomial of T is

(A) x(x – 3)

(B) x(x – 3)3

(C) x3(x – 3)

(D) x2(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  SX = {f ∈ X : ||f|| = 1}.

Then the set {f ∈ X : F(f) = ||F||} ⋂ SX 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 ) = x2 – y2

25. Consider the following Linear Programming Problem P:

Minimize 3x1 + 4x2

Subject to             x1 − x2 ≤ 1,

x1 + x2 ≥ 3,

x1 ≥ 0, x2 ≥ 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 c2 is _________.

27. Let for all z in some neighbourhood of 0 in ℂ.

Then the value of a6 + a5  is equal to _______.

28. Let p(x) = x3 – 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 x2 + 𝑦2 = 1. If the surface area of S is απ, then the value of α is equal to ___________.

32. Let L2[−1, 1] = {f : [−1, 1] → ℝ : f is Lebesgue measurable and the norm Let F : (L2[−1, 1], ||∙||2) → ℝ be defined by for all f ∈ L2[−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 ℝ4 with the inner product < x, y > = for x = (x1, x2, x3, x4) and y = (y­1, y2, y3, y4).

Let M = {(x1, x2, x3, x4) ∈ ℝ4 : x1 = x3} and M denote the orthogonal complement of M. The dimension of M is equal to ________.

35. Let and If 6M1 = M2 – 6M + αI for some α ∈ ℝ, then the value of α is equal to _________.

Q.36 – Q.65 Carry TWO marks Each

36. Let GL2(ℂ) denote the group of 2 × 2 invertible complex matrices with usual matrix multiplication. For S, T ∈ GL2(ℂ), < S, T > denotes the subgroup generated by S and T. Let and G1, G2, G3 be three subgroups of GL2(ℂ) given by Let  Z(Gi) denote the center of Gi and i = 1, 2, 3.

Which of the following statements is correct?

(A) G1 is isomorphic to G3

(B) Z(G1) is isomorphic to Z(G2)

(C) (D) Z(G2) is isomorphic to Z(G3)

37. Let ℓ2 = {(x1, x2, x3, …) : xn ∈ ℝ for all n ∈ ℕ and For a sequence (x1, x2, x3, …) ∈ ℓ2, define ||(x1, x2, x3, …||2 Let S : (ℓ2, ||∙||2) → T : (ℓ2, ||∙||2) and T : (ℓ2, ||∙||2) → (ℓ2, ||∙||2) be defined by

S(x1, x2, x3, …) = (y1, y2, y3, …), where T(x1, x2, x3, …) = (y1, y2, y3, …), 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

c00 = {(x1, x2, x3…) : xi ∈ ℝ, i ∈ ℕ, xi ≠ 0 only for finitely many indices i}.

For (x1, x2, x3, …) ∈ c00, let||(x1, x2, x3, …)|| = sup{|xi| : i ∈ ℕ}.

Define F, G : (c00, ||∙||) → (c00, ||∙||) 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 a0 = a1 = 1 and an = 1/22n for n ≥ 2.

Consider the following statements:

P: If z0 ∈ 𝔻, then f is one-one in some neighbourhood of z0.

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, x3› denotes the ideal generated by {2, x3} in ℤ[x] and ‹x2› denotes the ideal generated by x2 in ℤ[x].

Which of the following statements is/are correct?

(A) Every prime ideal of 𝒮1 is a maximal ideal

(B) 𝒮2 has exactly one maximal ideal

(C) Every element of 𝒮1 is either nilpotent or a unit

(D) There exists an element in 𝒮2 which is NEITHER nilpotent NOR a unit

47. Consider the sequence of Lebesgue measurable functions fn : 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 χE of a subset E in ℝ by For 1 ≤ p < 2, let

Lp[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 ∈ Lp [0, 1] for every p ∈ (1, 2).

Q: f ∈ L1[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 Dhf(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 x0.

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

(A) xk+1 > ζ

(B) xk+1 < ζ

(C) xk+1 < xk

(D) For every 52. Let f : ℝ2 → 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 fn : [0, 1] → ℝ by fn(x) = xn for all x ∈ [0, 1].

Let P = {fn : 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 gan ∈ 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 x2 + y2 = 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 + ex)dx + 4x dy + ez cos2 z dz| = απ,

Then the value of α equals ________.

56. Let ℓ2 = {(x1, x2, x3…) : xn ∈ ℝ for all n ∈ ℕ and For a sequence (x1, x2, x3, …} ∈ ℓ2, define Consider the subspace Let M denote the orthogonal complement of M in the Hilbert space (ℓ2, ||∙||2).

Consider (1, 1/2, 1/3, 1/4, …) ∈ ℓ2.

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 CN and CL be the total cost of the initial basic feasible solution obtained from the North-West corner method and the Least-Cost method, respectively. Then C𝑁 − CL equals _________.

58. Let σ ∈ S8, where S8 is the permutation group on 8 elements. Suppose σ is the product of σ1 and σ2, where σ1 is a 4-cycle and σ2 is a 3-cycle in S8. If σ1 and σ2 are disjoint cycles, then the number of elements in S8 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 A2 is _________.

60. Let A = [aaj] be a 3 × 3 real matrix such that If m is the degree of the minimal polynomial of A, then a11 + a21 + a31 + m equals _______.

61. Let Ω be the disk x2 + y2 < 4 in ℝ2 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 yk(x) be a polynomial of degree k with yk(1) = 5. Further, let yk(x) satisfy the Legendre equation

(1 – x2)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 r1 and r2 are the roots of the indicial equation of the above ODE at the regular singular point x=1, then |r1 − r2| 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 3x1 + 2x2 + 5x3

subject to

x1 + 2x2 + x3 ≤ 44,

x1 + 2x3 ≤ 48,

1 + 4x2 ≤ 52,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

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