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Basque PAU Applied Math Social Sci study materials
Basque PAU Applied Math Social Sci study resources

Basque PAU Applied Math Social Sci practice questions

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Question 1Álgebra matricial y sistemas

Given the matrices A = [[2, -1], [4, 3]] and B = [[1, 5], [-2, 0]], what is the result of the linear combination 2A - 3B?

  1. [[1, -17], [2, 9]]
  2. [[1, 13], [2, 6]]
  3. [[-1, -17], [14, 3]]
  4. [[1, -17], [14, 6]] Correct answer.

Explanation

Compute 2A = [[4, -2], [8, 6]] and 3B = [[3, 15], [-6, 0]]. Subtracting component-wise yields [[4-3, -2-15], [8-(-6), 6-0]] = [[1, -17], [14, 6]].

Question 2Álgebra matricial y sistemas

Matrix A has dimension 3 x 2 and matrix B has dimension 2 x 4. What are the dimensions of the product matrix A · B?

  1. 2 x 2
  2. 4 x 3
  3. 3 x 4 Correct answer.
  4. Matrix multiplication is undefined for these dimensions.

Explanation

Matrix multiplication A · B is valid because the number of columns in A (2) equals the number of rows in B (2). The outer dimensions give the size of the product matrix: 3 rows by 4 columns (3 x 4).

Question 3Álgebra matricial y sistemas

Which of the following matrix algebra properties is FALSE in general for square matrices A and B of order n?

  1. (A + B)^T = A^T + B^T
  2. (A · B)^T = B^T · A^T
  3. A · B = B · A Correct answer.
  4. (A^T)^T = A

Explanation

Matrix multiplication is generally non-commutative (A · B ≠ B · A). The other three identities are standard theorems of matrix algebra.

Question 4Álgebra matricial y sistemas

What is the determinant of the 2 x 2 matrix A = [[4, 7], [2, 5]]?

  1. 34
  2. -6
  3. 27
  4. 6 Correct answer.

Explanation

The determinant of a 2 x 2 matrix [[a, b], [c, d]] is computed as ad - bc. For matrix A, det(A) = (4)(5) - (7)(2) = 20 - 14 = 6.

Question 5Álgebra matricial y sistemas

What is the inverse of the matrix A = [[2, 1], [5, 3]]?

  1. [[3, -1], [-5, 2]] Correct answer.
  2. [[3, 1], [5, 2]]
  3. [[-2, 5], [1, -3]]
  4. [[1/2, 1], [1/5, 1/3]]

Explanation

First calculate det(A) = (2)(3) - (1)(5) = 1. For a 2 x 2 matrix, A^-1 = (1/det(A)) * [[d, -b], [-c, a]]. Substituting yields [[3, -1], [-5, 2]].

Question 6Álgebra matricial y sistemas

What is the determinant of the 3 x 3 matrix A = [[1, 2, 0], [0, 3, 1], [2, 1, 4]]?

  1. 11
  2. 15 Correct answer.
  3. 7
  4. 9

Explanation

Expanding along the first row: det(A) = 1*(3*4 - 1*1) - 2*(0*4 - 1*2) + 0 = 1*(12 - 1) - 2*(-2) = 11 + 4 = 15.

Question 7Álgebra matricial y sistemas

For what value of the parameter k is the matrix A = [[1, 0, k], [2, 1, 3], [0, 2, 1]] singular (non-invertible)?

  1. k = -5/4
  2. k = 5
  3. k = 5/4 Correct answer.
  4. k = 4/5

Explanation

A matrix is singular when det(A) = 0. Expanding along row 1: det(A) = 1*(1*1 - 3*2) - 0 + k*(2*2 - 1*0) = 1*(-5) + 4k = -5 + 4k. Setting -5 + 4k = 0 gives k = 5/4.

Question 8Álgebra matricial y sistemas

Given invertible square matrix A and matrix B, how is the unknown matrix X isolated in the matrix equation A · X = B?

  1. X = B · A^-1
  2. X = B / A
  3. X = A · B^-1
  4. X = A^-1 · B Correct answer.

Explanation

Multiplying both sides on the left by A^-1 yields A^-1 · (A · X) = A^-1 · B => (A^-1 · A) · X = A^-1 · B => X = A^-1 · B. Left-multiplication is mandatory because matrix multiplication is non-commutative.

Question 9Álgebra matricial y sistemas

Solve for the matrix X in the matrix equation X · A + B = C, assuming matrix A is invertible.

  1. X = (C - B) · A^-1 Correct answer.
  2. X = A^-1 · (C - B)
  3. X = (B - C) · A^-1
  4. X = C · A^-1 - B

Explanation

First isolate X · A by subtracting B from both sides: X · A = C - B. Then right-multiply both sides by A^-1 to get X = (C - B) · A^-1.

Question 10Álgebra matricial y sistemas

According to the Rouché-Capelli theorem, what is the classification of a system of 3 linear equations in 3 unknowns if rank(A) = 2 and rank(A|B) = 2?

  1. Consistent determined (unique solution)
  2. Inconsistent (no solution)
  3. Consistent undetermined (infinitely many solutions depending on 1 parameter) Correct answer.
  4. Consistent undetermined (infinitely many solutions depending on 2 parameters)

Explanation

Since rank(A) = rank(A|B) = 2, the system is consistent. Because the rank (2) is strictly less than the number of unknowns (n = 3), it has infinitely many solutions depending on n - rank = 3 - 2 = 1 free parameter.

Question 11Álgebra matricial y sistemas

What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]]?

  1. 1
  2. 3
  3. 2 Correct answer.
  4. 0

Explanation

Notice that Row 2 is exactly twice Row 1 (R2 = 2*R1), making row 2 linearly dependent. Row 1 and Row 3 are linearly independent since they are not proportional. Thus, there are 2 linearly independent rows, so rank(A) = 2.

Question 12Álgebra matricial y sistemas

A square system of linear equations A · X = B has a unique solution if and only if:

  1. det(A) = 0
  2. rank(A) < rank(A|B)
  3. The system is homogeneous with non-zero right hand side
  4. det(A) ≠ 0 Correct answer.

Explanation

A square linear system has a unique solution (consistent determined) if and only if the coefficient matrix A is invertible, which occurs precisely when det(A) ≠ 0.

Question 13Álgebra matricial y sistemas

Consider the linear system: x + y + z = 6 and x + 2y + 3z = 10. Expressing the solution in terms of parameter z = λ, what is x?

  1. x = 2 + λ Correct answer.
  2. x = 4 - λ
  3. x = 6 - 2λ
  4. x = λ - 4

Explanation

Subtracting equation 1 from equation 2 yields (x+2y+3z) - (x+y+z) = 10 - 6 => y + 2z = 4 => y = 4 - 2λ. Substituting y and z into equation 1: x + (4 - 2λ) + λ = 6 => x + 4 - λ = 6 => x = 2 + λ.

Question 14Álgebra matricial y sistemas

When a system of linear equations has rank(A) = 2 and rank(A|B) = 3, what can be concluded about the system?

  1. It has a unique solution.
  2. It is inconsistent and has no solution. Correct answer.
  3. It has infinitely many solutions depending on 1 parameter.
  4. It has a trivial solution x = y = z = 0.

Explanation

By the Rouché-Capelli theorem, if the rank of the coefficient matrix rank(A) is not equal to the rank of the augmented matrix rank(A|B), the system of equations is inconsistent and has no solution.

Question 15Álgebra matricial y sistemas

Which elementary row operation alters the determinant value of a matrix?

  1. Adding a scalar multiple of one row to another row
  2. Swapping two columns
  3. Replacing a row with itself plus 5 times another row
  4. Multiplying a single row by a non-zero scalar k Correct answer.

Explanation

Multiplying a single row of a matrix by a non-zero scalar k multiplies the determinant of the matrix by k. Adding multiples of rows leaves the determinant unchanged, while swapping rows/columns multiplies it by -1.

Question 16Álgebra matricial y sistemas

What is the determinant of the lower triangular matrix L = [[3, 0, 0], [4, 2, 0], [-1, 5, 4]]?

  1. 24 Correct answer.
  2. 12
  3. 0
  4. 48

Explanation

The determinant of any triangular (upper or lower) matrix equals the product of its main diagonal entries: det(L) = 3 * 2 * 4 = 24.

Question 17Álgebra matricial y sistemas

In Leontief's input-output model, if A is the technological matrix, X is the total output vector, and D is the final demand vector, what is the governing matrix equation?

  1. (A - I) · X = D
  2. (I + A) · D = X
  3. A · D = X
  4. (I - A) · X = D Correct answer.

Explanation

Total production X must satisfy internal consumption A · X plus final demand D: X = A · X + D => X - A · X = D => (I - A) · X = D.

Question 18Álgebra matricial y sistemas

Given technology matrix A = [[0.2, 0.3], [0.4, 0.1]] and final demand D = [[50], [100]], what is matrix (I - A)?

  1. [[1.2, 0.3], [0.4, 1.1]]
  2. [[-0.8, 0.3], [0.4, -0.9]]
  3. [[0.8, -0.3], [-0.4, 0.9]] Correct answer.
  4. [[0.2, -0.7], [-0.6, 0.1]]

Explanation

Compute (I - A) = [[1, 0], [0, 1]] - [[0.2, 0.3], [0.4, 0.1]] = [[1-0.2, 0-0.3], [0-0.4, 1-0.1]] = [[0.8, -0.3], [-0.4, 0.9]].

Question 19Álgebra matricial y sistemas

Which statement regarding the n x n identity matrix I_n is FALSE?

  1. A · I_n = A for any n x n matrix A.
  2. det(I_n) = 1.
  3. I_n is equal to its own transpose (I_n^T = I_n).
  4. det(k · I_n) = k for any scalar k. Correct answer.

Explanation

Multiplying an n x n matrix by scalar k scales every row by k, so det(k · I_n) = k^n * det(I_n) = k^n. For n > 1, k^n ≠ k in general.

Question 20Álgebra matricial y sistemas

Given A = [[1, 1], [0, 1]], what is A^n for any positive integer n?

  1. [[1, n], [0, 1]] Correct answer.
  2. [[1, 1], [0, 1]]
  3. [[n, n], [0, n]]
  4. [[1, 2^n], [0, 1]]

Explanation

Computing powers: A^2 = [[1, 2], [0, 1]], A^3 = [[1, 3], [0, 1]]. By mathematical induction, A^n = [[1, n], [0, 1]].

Question 21Álgebra matricial y sistemas

A bakery makes cakes (x) needing 2 eggs each and tarts (y) needing 3 eggs each. If 100 eggs are used in total and the number of cakes is twice the number of tarts, which linear system models this?

  1. 2x + 3y = 100 and y = 2x
  2. 2x + 3y = 100 and x = 2y Correct answer.
  3. 3x + 2y = 100 and x = 2y
  4. x + y = 100 and x = 2y

Explanation

The total egg constraint is 2x + 3y = 100. 'Number of cakes (x) is twice the number of tarts (y)' translates to x = 2y.

Question 22Álgebra matricial y sistemas

An investor puts €10,000 into two accounts: Account A paying 5% interest and Account B paying 8%. If total annual interest earned is €650, how much was invested in Account A?

  1. €4,000
  2. €6,000
  3. €5,000 Correct answer.
  4. €3,000

Explanation

Let x be amount in A and y in B. x + y = 10,000 and 0.05x + 0.08y = 650. From first eq, y = 10,000 - x. Substituting: 0.05x + 0.08(10,000 - x) = 650 => 0.05x + 800 - 0.08x = 650 => -0.03x = -150 => x = 5,000.

Question 23Álgebra matricial y sistemas

By definition, a real square matrix A is skew-symmetric if and only if:

  1. A^T = A
  2. A^-1 = A
  3. det(A) = 1
  4. A^T = -A Correct answer.

Explanation

A matrix is skew-symmetric (or anti-symmetric) if its transpose equals its negative: A^T = -A. (As a consequence, all diagonal entries must be zero).

Question 24Álgebra matricial y sistemas

If A is a 3 x 3 matrix with det(A) = 4, what is det(2A)?

  1. 32 Correct answer.
  2. 8
  3. 24
  4. 16

Explanation

For an n x n matrix, det(k · A) = k^n * det(A). For n = 3 and k = 2, det(2A) = 2^3 * det(A) = 8 * 4 = 32.

Question 25Programación lineal

Which region of the xy-plane satisfies the linear inequality 2x + 3y <= 12?

  1. The half-plane strictly above the line 2x + 3y = 12
  2. The half-plane below the line 2x + 3y = 12 excluding the boundary line
  3. Only points located in the first quadrant x >= 0, y >= 0
  4. The half-plane containing the origin (0,0) and the boundary line 2x + 3y = 12 Correct answer.

Explanation

Testing origin (0,0): 2(0) + 3(0) = 0 <= 12 is true. Because the inequality is non-strict (<=), it includes the boundary line 2x + 3y = 12 and the half-plane containing (0,0).

Question 26Programación lineal

Why are non-negativity constraints (x >= 0, y >= 0) included in almost all social science linear programming models?

  1. To guarantee that the objective function always has a positive maximum
  2. To ensure the system of inequalities has a unique solution
  3. Because physical quantities like products, time, or decision variables cannot be negative Correct answer.
  4. To make all constraint lines perpendicular to the coordinate axes

Explanation

In practical economic, social, or industrial contexts, decision variables x and y represent tangible quantities (e.g. units produced, hours worked, items purchased) which cannot physically be negative.

Question 27Programación lineal

What geometric property is ALWAYS shared by linear programming feasible regions?

  1. They are always bounded closed circles.
  2. They are always non-convex polygons.
  3. They are open regions with curved boundaries.
  4. They are convex sets formed by the intersection of half-planes. Correct answer.

Explanation

Each linear inequality defines a convex half-plane. The intersection of finitely many convex half-planes is always a convex set (polygon or polyhedral region).

Question 28Programación lineal

Find the corner vertices of the feasible region defined by constraints: x + y <= 6, 2x + y <= 8, x >= 0, y >= 0.

  1. (0,0), (4,0), (2,4), (0,6) Correct answer.
  2. (0,0), (6,0), (2,4), (0,8)
  3. (0,0), (8,0), (4,2), (0,6)
  4. (0,0), (4,0), (3,3), (0,6)

Explanation

x-intercepts when y=0: x <= 6 and 2x <= 8 => x <= 4. So x-axis vertex is (4,0). y-intercepts when x=0: y <= 6 and y <= 8 => y <= 6. So y-axis vertex is (0,6). Intersection of x+y=6 and 2x+y=8: subtracting gives x = 2, then y = 4 => (2,4). Vertices are (0,0), (4,0), (2,4), (0,6).

Question 29Programación lineal

According to the Fundamental Theorem of Linear Programming, where does an optimal (maximum or minimum) solution of a linear objective function occur over a bounded non-empty feasible region?

  1. At the exact geometric centroid of the region
  2. At at least one vertex (corner point) of the feasible region Correct answer.
  3. Strictly in the interior of the feasible region
  4. At the origin (0,0) exclusively

Explanation

Because the objective function is linear, its level sets are straight lines. Over a convex polygonal region, maximum and minimum values are always attained at one or more corner vertices.

Question 30Programación lineal

Maximize the objective function Z = 3x + 5y over the feasible region with vertices (0,0), (4,0), (2,4), and (0,5).

  1. Maximum Z = 12 at (4,0)
  2. Maximum Z = 26 at (2,4) Correct answer.
  3. Maximum Z = 25 at (0,5)
  4. Maximum Z = 20 at (0,4)

Explanation

Evaluate Z at each vertex: Z(0,0) = 0; Z(4,0) = 3(4)+0 = 12; Z(2,4) = 3(2)+5(4) = 6+20 = 26; Z(0,5) = 3(0)+5(5) = 25. The maximum value is 26 at vertex (2,4).

Question 31Programación lineal

Minimize the cost function Z = 2x + 4y over the feasible region with vertices (1,5), (3,2), (6,1), and (0,8).

  1. Minimum Z = 22 at (1,5)
  2. Minimum Z = 16 at (6,1)
  3. Minimum Z = 32 at (0,8)
  4. Minimum Z = 14 at (3,2) Correct answer.

Explanation

Evaluate Z at vertices: Z(1,5) = 2(1)+4(5) = 22; Z(3,2) = 2(3)+4(2) = 14; Z(6,1) = 2(6)+4(1) = 16; Z(0,8) = 2(0)+4(8) = 32. The minimum cost is 14 at (3,2).

Question 32Programación lineal

Consider an unbounded feasible region in Quadrant I defined by x + y >= 4. If we wish to MAXIMIZE Z = 2x + 3y over this region, what occurs?

  1. There is no maximum solution (Z can increase indefinitely to infinity). Correct answer.
  2. The maximum occurs at vertex (4,0) with Z = 8.
  3. The maximum occurs at vertex (0,4) with Z = 12.
  4. The maximum is Z = 0 at the origin (0,0).

Explanation

Because x and y can grow infinitely large while satisfying x + y >= 4, Z = 2x + 3y can grow arbitrarily large. Thus, no finite maximum exists.

Question 33Programación lineal

Under what condition does a linear programming problem possess INFINITELY MANY optimal solutions?

  1. When the feasible region has no vertices
  2. When the objective function line is parallel to one of the active boundary constraint segments Correct answer.
  3. When all constraint equations pass through the origin
  4. When the objective function coefficients are both zero

Explanation

If the slope of the objective function matches the slope of a boundary constraint line segment containing two optimal vertices, every point along that entire boundary segment yields the exact same optimal value.

Question 34Programación lineal

A craftsman makes wooden tables (x) and chairs (y). Tables require 4 hours of carpentry and 2 hours of finishing. Chairs require 1 hour of carpentry and 1 hour of finishing. Total available hours are 40 for carpentry and 24 for finishing. What are the resource constraints?

  1. 4x + 2y <= 40 and x + y <= 24
  2. 4x + y <= 24 and 2x + y <= 40
  3. 4x + y <= 40 and 2x + y <= 24 Correct answer.
  4. x + 4y <= 40 and x + 2y <= 24

Explanation

Carpentry constraint: 4x + 1y <= 40. Finishing constraint: 2x + 1y <= 24. Non-negativity constraints x >= 0, y >= 0.

Question 35Programación lineal

A farm diet must contain at least 12 units of Vitamin A and at least 15 units of Vitamin B. Food X provides 2 units of A and 3 of B. Food Y provides 3 units of A and 1 of B. Which system represents the nutritional constraints?

  1. 2x + 3y <= 12 and 3x + y <= 15
  2. 2x + 3y >= 12 and 3x + y >= 15 Correct answer.
  3. 2x + 3y >= 15 and 3x + y >= 12
  4. 3x + 2y >= 12 and x + 3y >= 15

Explanation

Vitamin A requirement (at least 12): 2x + 3y >= 12. Vitamin B requirement (at least 15): 3x + 1y >= 15. 'At least' translates to >= inequalities.

Question 36Programación lineal

Warehouse A has 50 units available. It ships x units to Store 1 and y units to Store 2. What inequality represents Warehouse A's capacity constraint?

  1. x + y <= 50 Correct answer.
  2. x + y >= 50
  3. x - y <= 50
  4. 50x + 50y <= 1

Explanation

The total number of units shipped from Warehouse A is x + y. This total cannot exceed the available inventory of 50 units, yielding x + y <= 50.

Question 37Programación lineal

Determine whether the point P(3, 2) belongs to the feasible region defined by: x + 2y <= 8, 3x - y >= 4, x >= 0, y >= 0.

  1. No, because it violates x + 2y <= 8.
  2. Yes, because it satisfies all four constraints simultaneously. Correct answer.
  3. No, because it violates 3x - y >= 4.
  4. No, because point P lies outside Quadrant I.

Explanation

Test P(3,2): Constraint 1: 3 + 2(2) = 7 <= 8 (True). Constraint 2: 3(3) - 2 = 7 >= 4 (True). Non-negativity: 3 >= 0 and 2 >= 0 (True). All constraints are satisfied.

Question 38Programación lineal

What is the slope of the level curves (iso-profit lines) for the objective function Z = 4x + 5y?

  1. m = 4/5
  2. m = 5/4
  3. m = -4/5 Correct answer.
  4. m = -5/4

Explanation

Setting Z = C (constant): 4x + 5y = C => 5y = -4x + C => y = (-4/5)x + C/5. The slope of the straight lines is m = -4/5.

Question 39Programación lineal

If a continuous linear programming optimal vertex yields fractional output x = 3.8 tables and y = 4.2 chairs, how should a manager determine the optimal whole-number production plan?

  1. Always round up both numbers to x = 4, y = 5.
  2. Always round down both numbers to x = 3, y = 4.
  3. Ignore constraints and pick any nearby integers.
  4. Test all feasible integer coordinate points neighboring the vertex to find the feasible point yielding maximum Z. Correct answer.

Explanation

Naively rounding fractional vertices can violate constraints or yield non-optimal integer solutions. One must evaluate neighboring integer grid points within the feasible region to select the true maximum.

Question 40Programación lineal

In a system with x >= 0, y >= 0, x + y <= 5, and x + y <= 8, which constraint is redundant?

  1. x + y <= 8 Correct answer.
  2. x + y <= 5
  3. x >= 0
  4. None of the constraints are redundant.

Explanation

Any point satisfying x + y <= 5 automatically satisfies x + y <= 8 (since 5 < 8). Therefore, x + y <= 8 does not restrict the feasible region further and is redundant.

Question 41Programación lineal

A clothing company maximizes profit P = 40x + 30y. The feasible region vertices are (0,0), (0,10), (12,8), and (15,0). What is the maximum profit?

  1. €600
  2. €720 Correct answer.
  3. €300
  4. €450

Explanation

Evaluate P at all vertices: P(0,0) = 0; P(0,10) = 300; P(15,0) = 40(15) = 600; P(12,8) = 40(12) + 30(8) = 480 + 240 = 720. Maximum profit is €720 at vertex (12,8).

Question 42Programación lineal

If a bounded feasible region is defined by 4 non-redundant linear inequality constraints (including non-negativity), what shape is the region?

  1. A triangle
  2. A quadrilateral (4-sided polygon) Correct answer.
  3. A pentagon
  4. A line segment

Explanation

A set of 4 non-redundant linear inequality constraints (which includes non-negativity bounds) forms 4 distinct straight boundary lines. A bounded region bounded by 4 non-redundant straight lines forms a 4-sided polygon (quadrilateral).

Question 43Programación lineal

Suppose the optimal vertex for profit function Z = c1*x + c2*y is currently at (10, 5). If the market price c1 increases significantly while c2 remains fixed, how will the optimal objective line behave?

  1. Its slope becomes steeper, potentially shifting the optimal solution to a vertex with a higher x-coordinate. Correct answer.
  2. Its slope becomes flatter, shifting optimal solution to y-axis.
  3. The feasible region expands in size.
  4. The optimal vertex remains guaranteed to stay at (10,5) regardless of how large c1 becomes.

Explanation

The slope of Z = c1*x + c2*y is m = -c1/c2. Increasing c1 increases the magnitude of the negative slope (steeper line), making production of product x relatively more profitable and favoring vertices with higher x values.

Question 44Programación lineal

What happens when a linear programming problem has constraints x + y <= 2, x + y >= 5, x >= 0, y >= 0?

  1. The problem has no feasible region (empty set) and thus no solution exists. Correct answer.
  2. The optimal solution is at (0,0).
  3. There are infinitely many solutions between 2 and 5.
  4. The maximum occurs at x = 2.5, y = 2.5.

Explanation

No point (x,y) can simultaneously satisfy x + y <= 2 and x + y >= 5. The intersection of these half-planes is empty, so the problem is infeasible.

Question 45Programación lineal

A set S in the plane is defined as CONVEX if:

  1. All points in S lie within a circle of radius R.
  2. For any two points A and B in S, the entire straight line segment joining A and B lies completely within S. Correct answer.
  3. The boundary of S consists exclusively of right angles.
  4. The area of S can be computed without using integrals.

Explanation

By standard geometric definition, a set S is convex if for any pair of points A, B ∈ S, the line segment connecting them is entirely contained in S.

Question 46Programación lineal

A firm spends €x thousand on online ads and €y thousand on TV ads. Total budget is at most €10 thousand (x + y <= 10). Online ads must be at least twice TV ads (x >= 2y). Which vertex maximizes total reach R = 50x + 30y?

  1. (0,0)
  2. (10,0) Correct answer.
  3. (6.67, 3.33)
  4. (0,5)

Explanation

Vertices of feasible region bounded by x+y<=10, x>=2y, x>=0, y>=0: (0,0), (10,0), and intersection of x+y=10 & x=2y => (6.67, 3.33). Evaluate reach R: R(0,0)=0; R(6.67,3.33) = 50(6.67)+30(3.33) = 333.3+100 = 433.3; R(10,0) = 50(10)+0 = 500. Maximum reach is 500 at (10,0).

Question 47Programación lineal

In linear programming economics, what does the SHADOW PRICE (dual value) of a resource constraint represent?

  1. The cost of disposing of excess unused inventory
  2. The market discount percentage applied to bulk orders
  3. The tax rate levied on profit above the threshold
  4. The change in optimal objective value per unit increase in the constraint's right-hand side limit Correct answer.

Explanation

The shadow price measures the marginal value of a binding constraint-specifically, how much maximum profit would increase if one additional unit of that resource were made available.

Question 48Programación lineal

Maximize Z = 4x + 2y subject to: x + 2y <= 8, 3x + y <= 9, x >= 0, y >= 0.

  1. Z = 12 at (3,0)
  2. Z = 8 at (0,4)
  3. Z = 14 at (2,3) Correct answer.
  4. Z = 10 at (1,3)

Explanation

The vertices of the feasible region are (0,0), (3,0), (0,4), and the intersection of x + 2y = 8 and 3x + y = 9 at (2,3). Evaluating Z = 4x + 2y at each vertex yields Z(0,0) = 0, Z(0,4) = 8, Z(3,0) = 12, and Z(2,3) = 4(2) + 2(3) = 14. The maximum value is Z = 14 at (2,3).

Question 49Programación lineal

Find the constant profit C if the objective line 5x + 10y = C passes through the optimal vertex P(4, 3).

  1. C = 35
  2. C = 50 Correct answer.
  3. C = 45
  4. C = 70

Explanation

Substitute x = 4 and y = 3 into the objective equation: C = 5(4) + 10(3) = 20 + 30 = 50.

Question 50Análisis, límites, derivadas y optimización

Evaluate the limit lim_{x -> 2} (x^2 - 4) / (x - 2).

  1. 0
  2. 2
  3. 4 Correct answer.
  4. Undefined / Infinite

Explanation

Factoring the numerator: (x^2 - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2). For x ≠ 2, this simplifies to x + 2. Taking the limit as x -> 2 gives 2 + 2 = 4.