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Andalusia PAU Applied Mathematics for Social Sciences II study materials
Andalusia PAU Applied Mathematics for Social Sciences II study resources

Andalusia PAU Applied Mathematics for Social Sciences II practice questions

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Question 1Álgebra matricial y sistemas

Given the 2x2 matrix A = [[2, 1], [4, 3]], what is the determinant of matrix A?

  1. 2 Correct answer.
  2. 10
  3. -2
  4. 6

Explanation

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. For matrix A = [[2, 1], [4, 3]], det(A) = (2 * 3) - (1 * 4) = 6 - 4 = 2.

Question 2Álgebra matricial y sistemas

If matrix A has dimensions 3x2 and matrix B has dimensions 2x4, what are the dimensions of the product matrix A * B?

  1. 2x2
  2. 3x4 Correct answer.
  3. 4x3
  4. 3x2

Explanation

Matrix multiplication A * B is defined when the number of columns in A (2) equals the number of rows in B (2). The resulting matrix product has the number of rows of A (3) and the number of columns of B (4), giving dimensions 3x4.

Question 3Álgebra matricial y sistemas

What is the transpose matrix A^T of matrix A = [[1, 5, -2], [0, 4, 3]]?

  1. [[1, 0], [5, 4], [-2, 3]] Correct answer.
  2. [[0, 1], [4, 5], [3, -2]]
  3. [[1, 5, -2], [0, 4, 3]]
  4. [[-2, 3], [5, 4], [1, 0]]

Explanation

The transpose of a matrix is formed by swapping its rows and columns. The 2x3 matrix A becomes a 3x2 matrix A^T where row 1 [1, 5, -2] becomes column 1 and row 2 [0, 4, 3] becomes column 2, yielding [[1, 0], [5, 4], [-2, 3]].

Question 4Álgebra matricial y sistemas

A square matrix A is invertible (regular) if and only if which condition is met?

  1. det(A) = 0
  2. det(A) != 0 Correct answer.
  3. A is a symmetric matrix
  4. All elements of A are positive

Explanation

A square matrix A is invertible (non-singular) if and only if its determinant is non-zero (det(A) != 0). If det(A) = 0, the matrix is singular and does not possess a inverse matrix.

Question 5Álgebra matricial y sistemas

Given matrices A = [[3, 1], [0, 2]] and B = [[1, 4], [2, 1]], compute the matrix 2A - B.

  1. [[5, -2], [-2, 3]] Correct answer.
  2. [[5, -2], [2, 3]]
  3. [[7, 6], [2, 5]]
  4. [[4, -3], [-2, 1]]

Explanation

First, scalar multiply 2A = [[6, 2], [0, 4]]. Next, subtract B: 2A - B = [[6 - 1, 2 - 4], [0 - 2, 4 - 1]] = [[5, -2], [-2, 3]].

Question 6Álgebra matricial y sistemas

Which matrix represents the 3x3 identity matrix I_3?

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

Explanation

The identity matrix of order 3 is a square 3x3 matrix with ones along its main diagonal (top-left to bottom-right) and zeros in all other entries.

Question 7Álgebra matricial y sistemas

According to the Rouché-Frobenius theorem, a system of linear equations AX = B is consistent (has at least one solution) if and only if:

  1. rank(A) = rank(A|B) Correct answer.
  2. rank(A) < rank(A|B)
  3. rank(A) = number of variables + 1
  4. det(A) = 0

Explanation

The Rouché-Frobenius theorem states that a system of linear equations is consistent (compatible) if and only if the rank of the coefficient matrix A equals the rank of the augmented matrix (A|B).

Question 8Álgebra matricial y sistemas

In a linear system with n variables, if rank(A) = rank(A|B) = n, how is the system classified under Rouché-Frobenius?

  1. Inconsistent (no solution)
  2. Consistent determined (unique solution) Correct answer.
  3. Consistent undetermined (infinitely many solutions)
  4. Homogeneous trivial system

Explanation

When rank(A) = rank(A|B) = n (where n is the number of unknown variables), the system is consistent determined (compatible determinado) and has exactly one unique solution.

Question 9Álgebra matricial y sistemas

Calculate the inverse matrix A^-1 for A = [[1, 2], [3, 4]].

  1. [[-2, 1], [1.5, -0.5]] Correct answer.
  2. [[-2, -1], [-1.5, -0.5]]
  3. [[4, -2], [-3, 1]]
  4. [[2, -1], [-1.5, 0.5]]

Explanation

det(A) = (1*4) - (2*3) = 4 - 6 = -2. Using A^-1 = (1/det(A)) * [[d, -b], [-c, a]], A^-1 = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].

Question 10Álgebra matricial y sistemas

Solve the matrix equation A * X = B for X, where A = [[2, 0], [0, 3]] and B = [[4, 6], [8, 12]].

  1. [[2, 3], [8/3, 4]] Correct answer.
  2. [[8, 18], [24, 36]]
  3. [[2, 3], [4, 4]]
  4. [[1/2, 2], [3, 4]]

Explanation

Multiply both sides by A^-1 on the left: X = A^-1 * B. A^-1 = [[1/2, 0], [0, 1/3]]. Thus X = [[1/2, 0], [0, 1/3]] * [[4, 6], [8, 12]] = [[(1/2)*4, (1/2)*6], [(1/3)*8, (1/3)*12]] = [[2, 3], [8/3, 4]].

Question 11Álgebra matricial y sistemas

Calculate the determinant of the 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [1, 0, 6]].

  1. 22 Correct answer.
  2. 18
  3. 26
  4. 12

Explanation

Expanding across row 1: det(A) = 1*(4*6 - 5*0) - 2*(0*6 - 5*1) + 3*(0*0 - 4*1) = 1*(24) - 2*(-5) + 3*(-4) = 24 + 10 - 12 = 22.

Question 12Álgebra matricial y sistemas

For what value of parameter k is the matrix A = [[k, 2], [3, 6]] singular (non-invertible)?

  1. 1 Correct answer.
  2. 0
  3. 6
  4. -1

Explanation

A matrix is singular when its determinant is zero. det(A) = (k * 6) - (2 * 3) = 6k - 6 = 0. Solving 6k = 6 gives k = 1.

Question 13Álgebra matricial y sistemas

Solve the system of linear equations using Gauss elimination: x + y = 5 and 2x - y = 1.

  1. (2, 3) Correct answer.
  2. (3, 2)
  3. (1, 4)
  4. (4, 1)

Explanation

Add the two equations: (x + y) + (2x - y) = 5 + 1 => 3x = 6 => x = 2. Substitute x = 2 into x + y = 5 => 2 + y = 5 => y = 3. Solution is (2, 3).

Question 14Álgebra matricial y sistemas

Compute A^2 for the matrix A = [[1, 2], [0, 1]].

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

Explanation

A^2 = A * A = [[1, 2], [0, 1]] * [[1, 2], [0, 1]] = [[1*1 + 2*0, 1*2 + 2*1], [0*1 + 1*0, 0*2 + 1*1]] = [[1, 4], [0, 1]].

Question 15Álgebra matricial y sistemas

Determine the rank of matrix A = [[1, 2, 3], [2, 4, 6], [0, 1, 2]].

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

Explanation

Row 2 [2, 4, 6] is twice Row 1 [1, 2, 3], so row reduction produces a zero row. Rows 1 [1, 2, 3] and 3 [0, 1, 2] are linearly independent. Thus, rank(A) = 2.

Question 16Álgebra matricial y sistemas

Calculate the trace of matrix A = [[5, -1, 2], [3, 4, 0], [1, 2, -2]].

  1. 7 Correct answer.
  2. 9
  3. 5
  4. 11

Explanation

The trace of a square matrix is the sum of the elements on its main diagonal. tr(A) = 5 + 4 + (-2) = 7.

Question 17Álgebra matricial y sistemas

If A is a 3x3 matrix with det(A) = 4, what is the value of det(2A)?

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

Explanation

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

Question 18Álgebra matricial y sistemas

Solve for matrix X in A + X = B^T where A = [[1, 3], [2, 4]] and B = [[5, 2], [1, 6]].

  1. [[4, -2], [0, 2]] Correct answer.
  2. [[4, -1], [-1, 2]]
  3. [[6, 5], [3, 10]]
  4. [[4, 2], [0, 2]]

Explanation

First, transpose B: B^T = [[5, 1], [2, 6]]. Then X = B^T - A = [[5 - 1, 1 - 3], [2 - 2, 6 - 4]] = [[4, -2], [0, 2]].

Question 19Álgebra matricial y sistemas

Classify the system of equations: x + 2y - z = 3, 2x + 4y - 2z = 6, 3x + 6y - 3z = 9.

  1. Consistent undetermined with 2 degrees of freedom Correct answer.
  2. Consistent determined with a unique solution
  3. Inconsistent with no solutions
  4. Consistent undetermined with 1 degree of freedom

Explanation

Equations 2 and 3 are scalar multiples of equation 1 (multiplied by 2 and 3). Thus, rank(A) = 1 and rank(A|B) = 1. Since n = 3 variables, n - rank(A) = 3 - 1 = 2 degrees of freedom (infinitely many^2 solutions).

Question 20Álgebra matricial y sistemas

If A and B are 3x3 square matrices with det(A) = 3 and det(B) = -2, what is det(A * B^-1)?

  1. -1.5 Correct answer.
  2. -6
  3. 1.5
  4. -0.67

Explanation

Using determinant properties: det(A * B^-1) = det(A) * det(B^-1) = det(A) / det(B) = 3 / (-2) = -1.5.

Question 21Álgebra matricial y sistemas

Discuss the system depending on parameter m: mx + y = 1 and x + my = 1.

  1. Unique solution for m != +-1, infinitely many for m = 1, no solution for m = -1 Correct answer.
  2. Unique solution for all values of m
  3. No solution for m = 1 and m = -1
  4. Infinitely many solutions for m = -1

Explanation

det(A) = m^2 - 1 = (m-1)(m+1). If m != +-1, det(A) != 0 => unique solution. If m = 1: x + y = 1, x + y = 1 => rank(A)=1, rank(A|B)=1 => infinitely many solutions. If m = -1: -x + y = 1, x - y = 1 => adding gives 0 = 2 => rank(A)=1, rank(A|B)=2 => no solution.

Question 22Álgebra matricial y sistemas

Compute the inverse matrix A^-1 for the 3x3 matrix A = [[1, 0, 1], [0, 2, 0], [1, 0, 3]].

  1. [[1.5, 0, -0.5], [0, 0.5, 0], [-0.5, 0, 0.5]] Correct answer.
  2. [[1.5, 0, 0.5], [0, 0.5, 0], [0.5, 0, 1.5]]
  3. [[3, 0, -1], [0, 1, 0], [-1, 0, 1]]
  4. [[0.5, 0, -0.5], [0, 1, 0], [-0.5, 0, 1.5]]

Explanation

det(A) = 2*(3 - 1) = 4. The adjugate matrix adj(A) = [[6, 0, -2], [0, 2, 0], [-2, 0, 2]]. Dividing adj(A) by det(A) = 4 gives A^-1 = [[1.5, 0, -0.5], [0, 0.5, 0], [-0.5, 0, 0.5]].

Question 23Álgebra matricial y sistemas

Find the matrix X that satisfies A * X * B = C, where A, B, and C are invertible square matrices of order n.

  1. X = A^-1 * C * B^-1 Correct answer.
  2. X = C * A^-1 * B^-1
  3. X = B^-1 * C * A^-1
  4. X = A^-1 * B^-1 * C

Explanation

Multiply by A^-1 on the left: A^-1 * (A * X * B) = A^-1 * C => I * X * B = A^-1 * C. Then multiply by B^-1 on the right: (X * B) * B^-1 = (A^-1 * C) * B^-1 => X = A^-1 * C * B^-1.

Question 24Álgebra matricial y sistemas

Find all values of parameter 'a' for which the rank of matrix A = [[1, a, 1], [1, 1, a], [a, 1, 1]] is less than 3.

  1. a = 1 or a = -2 Correct answer.
  2. a = 0 or a = 1
  3. a = -1 or a = 2
  4. a = 3 or a = -3

Explanation

rank(A) < 3 when det(A) = 0. det(A) = 1(1 - a) - a(1 - a^2) + 1(1 - a) = a^3 - 3a + 2 = (a - 1)^2 * (a + 2). Setting det(A) = 0 yields a = 1 or a = -2.

Question 25Álgebra matricial y sistemas

A bakery makes cakes X, Y, Z. Type X needs 1kg flour, 2 eggs, 100g sugar. Type Y needs 2kg flour, 1 egg, 200g sugar. Type Z needs 1.5kg flour, 3 eggs, 150g sugar. Which matrix M computes total flour, eggs, and sugar as M * [x, y, z]^T?

  1. [[1, 2, 1.5], [2, 1, 3], [100, 200, 150]] Correct answer.
  2. [[1, 2, 100], [2, 1, 200], [1.5, 3, 150]]
  3. [[1.5, 2, 1], [3, 1, 2], [150, 200, 100]]
  4. [[1, 1, 1], [2, 2, 2], [100, 200, 150]]

Explanation

Row 1 must represent flour per cake type: [1, 2, 1.5]. Row 2 represents eggs per cake type: [2, 1, 3]. Row 3 represents sugar per cake type: [100, 200, 150]. Multiplying M by column vector [x, y, z]^T produces totals.

Question 26Programación lineal

In a linear programming problem, what is the definition of the feasible region?

  1. The set of all points (x, y) that satisfy all constraint inequalities simultaneously Correct answer.
  2. The set of points where the objective function is equal to zero
  3. The region outside all constraint boundaries
  4. The single corner point where profit is maximized

Explanation

The feasible region (región factible) is the set of all points (x, y) in the plane that satisfy all given system constraint inequalities simultaneously.

Question 27Programación lineal

Where does the optimal value of a linear objective function occur on a bounded convex feasible region?

  1. At at least one vertex (corner point) of the feasible region Correct answer.
  2. Always at the origin (0, 0)
  3. At the geometric midpoint of the region
  4. Outside the feasible region

Explanation

According to the Fundamental Theorem of Linear Programming, if an optimal solution exists on a bounded convex polygonal region, it occurs at at least one vertex (corner point) of the feasible region.

Question 28Programación lineal

What do the non-negativity constraints x >= 0 and y >= 0 represent in social science optimization problems?

  1. Physical and economic reality where quantities of goods or resources cannot be negative Correct answer.
  2. A requirement that profit must always be positive
  3. A condition that forces the feasible region to be unbounded
  4. A mathematical constraint to make lines parallel

Explanation

In real-world applications (production, investment, labor), negative quantities of items or hours are impossible. The non-negativity constraints x >= 0 and y >= 0 restrict the feasible region to the first quadrant.

Question 29Programación lineal

Which inequality models: 'Total hours spent on product A (x hours) and product B (y hours) cannot exceed 40 hours'?

  1. x + y <= 40 Correct answer.
  2. x + y >= 40
  3. x - y <= 40
  4. x * y <= 40

Explanation

'Cannot exceed' means 'less than or equal to'. The sum of hours spent on A and B is x + y, so the correct constraint is x + y <= 40.

Question 30Programación lineal

Given objective function Z = 3x + 5y, evaluate Z at vertex P(2, 4).

  1. 26 Correct answer.
  2. 22
  3. 14
  4. 30

Explanation

Substitute x = 2 and y = 4 into the objective function: Z = 3(2) + 5(4) = 6 + 20 = 26.

Question 31Programación lineal

For a bounded feasible region with vertices (0,0), (5,0), (3,4), and (0,6), find the maximum value of Z = 4x + 2y.

  1. 20 Correct answer.
  2. 26
  3. 12
  4. 24

Explanation

Evaluate Z at all vertices: Z(0,0) = 0; Z(5,0) = 4(5)+2(0) = 20; Z(3,4) = 4(3)+2(4) = 12+8 = 20; Z(0,6) = 4(0)+2(6) = 12. Maximum value is 20 (achieved at both (5,0) and (3,4)).

Question 32Programación lineal

Find the vertex coordinates formed by the intersection of boundary lines x + y = 10 and 2x - y = 5.

  1. (5, 5) Correct answer.
  2. (4, 6)
  3. (6, 4)
  4. (3, 7)

Explanation

Add equations: (x + y) + (2x - y) = 10 + 5 => 3x = 15 => x = 5. Substitute x = 5 into x + y = 10 => 5 + y = 10 => y = 5. Intersection vertex is (5, 5).

Question 33Programación lineal

Minimize Z = 2x + 3y subject to x + y >= 6, x >= 1, y >= 2.

  1. 14 at (4, 2) Correct answer.
  2. 17 at (1, 5)
  3. 18 at (6, 2)
  4. 12 at (3, 2)

Explanation

Vertices of the feasible region: (1, 5) from x=1 and x+y=6 -> Z = 2(1)+3(5) = 17; (4, 2) from y=2 and x+y=6 -> Z = 2(4)+3(2) = 14. For larger x and y, Z increases. Minimum value is 14 at (4, 2).

Question 34Programación lineal

Chair A needs 2h carpentry and 1h finishing. Chair B needs 1h carpentry and 2h finishing. Total hours available: 100 carpentry, 80 finishing. Profit: €20/Chair A, €30/Chair B. What is the objective function P?

  1. P = 20x + 30y Correct answer.
  2. P = 100x + 80y
  3. P = 2x + y
  4. P = 30x + 20y

Explanation

Let x = units of Chair A and y = units of Chair B. The objective is to maximize total profit P = 20x + 30y.

Question 35Programación lineal

For the chair production problem (Chair A: 2h carp, 1h fin; Chair B: 1h carp, 2h fin; limits: 100 carp, 80 fin), what is the correct constraint system?

  1. 2x + y <= 100, x + 2y <= 80, x >= 0, y >= 0 Correct answer.
  2. x + 2y <= 100, 2x + y <= 80, x >= 0, y >= 0
  3. 20x + 30y <= 100, x + y <= 80, x >= 0, y >= 0
  4. 2x + y >= 100, x + 2y >= 80, x >= 0, y >= 0

Explanation

Carpentry hours used: 2x + y <= 100 available. Finishing hours used: x + 2y <= 80 available. Non-negativity: x >= 0, y >= 0.

Question 36Programación lineal

Find the vertex where 2x + y = 100 and x + 2y = 80 intersect.

  1. (40, 20) Correct answer.
  2. (30, 25)
  3. (20, 40)
  4. (50, 0)

Explanation

Multiply second equation by 2: 2x + 4y = 160. Subtract first: 3y = 60 => y = 20. Substitute y = 20: 2x + 20 = 100 => 2x = 80 => x = 40. Intersection vertex is (40, 20).

Question 37Programación lineal

For feasible vertices (0,0), (50,0), (40,20), (0,40), find the maximum profit P = 20x + 30y.

  1. EUR 1,400 at (40, 20) Correct answer.
  2. EUR 1,200 at (0, 40)
  3. EUR 1,000 at (50, 0)
  4. EUR 1,500 at (30, 30)

Explanation

P(0,0) = 0; P(50,0) = 20(50) = 1,000; P(40,20) = 20(40) + 30(20) = 800 + 600 = 1,400; P(0,40) = 30(40) = 1,200. Maximum profit is €1,400 at (40, 20).

Question 38Programación lineal

What occurs when the objective function level curves are parallel to a boundary segment of a bounded feasible region?

  1. There are infinitely many optimal solutions along that entire boundary segment Correct answer.
  2. No optimal solution exists
  3. The unique optimal solution is at the origin
  4. The objective function becomes undefined

Explanation

When an objective function line has the same slope as a boundary segment of the feasible region, all points along that line segment connecting two optimal vertices yield the exact same maximum or minimum value.

Question 39Programación lineal

Minimize Z = 5x + 4y subject to x + y >= 10, x >= 0, y >= 0.

  1. 40 at (0, 10) Correct answer.
  2. 50 at (10, 0)
  3. 0 at (0, 0)
  4. 45 at (5, 5)

Explanation

Boundary vertices on x + y = 10 (first quadrant) are (0, 10) and (10, 0). Z(0, 10) = 5(0) + 4(10) = 40. Z(10, 0) = 5(10) + 4(0) = 50. Minimum value is 40 at (0, 10).

Question 40Programación lineal

Which test point lies strictly inside the feasible region defined by x + 2y <= 12, 3x + y <= 15, x >= 0, y >= 0?

  1. (2, 3) Correct answer.
  2. (4, 5)
  3. (5, 4)
  4. (6, 2)

Explanation

Check (2, 3): 2 + 2(3) = 8 <= 12 (True); 3(2) + 3 = 9 <= 15 (True); 2 >= 0, 3 >= 0 (True). Point (2, 3) satisfies all constraints.

Question 41Programación lineal

An investor allocates up to EUR 20,000 between Stock A (x) and Stock B (y). At least EUR 5,000 in A, at most EUR 10,000 in B. Which system represents this?

  1. x + y <= 20000, x >= 5000, 0 <= y <= 10000 Correct answer.
  2. x + y >= 20000, x <= 5000, y >= 10000
  3. x + y <= 20000, x <= 5000, y >= 10000
  4. x + y = 20000, x >= 5000, y <= 10000

Explanation

Total capital: x + y <= 20,000. Minimum in Stock A: x >= 5,000. Maximum in Stock B: y <= 10,000 (with y >= 0). Thus, x + y <= 20000, x >= 5000, 0 <= y <= 10000.

Question 42Programación lineal

In an unbounded region (first quadrant) defined by x + y >= 4 and 2x + y >= 6, find the minimum value of Z = 3x + 2y.

  1. 10 at (2, 2) Correct answer.
  2. 12 at (0, 6)
  3. 12 at (4, 0)
  4. 8 at (0, 4)

Explanation

Boundary intersection of x + y = 4 and 2x + y = 6 is (2, 2). Vertices of the unbounded feasible region: (0, 6), (2, 2), (4, 0). Evaluate Z: Z(0,6) = 12; Z(2,2) = 3(2)+2(2) = 10; Z(4,0) = 12. Minimum value is 10 at (2, 2).

Question 43Programación lineal

For the same unbounded region (x + y >= 4, 2x + y >= 6, x >= 0, y >= 0), what is the MAXIMUM value of Z = 3x + 2y?

  1. Maximum value does not exist (unbounded) Correct answer.
  2. 12
  3. 24
  4. 10

Explanation

Because the feasible region extends infinitely into the first quadrant, x and y can increase without bound, causing Z = 3x + 2y to approach positive infinity. No finite maximum value exists.

Question 44Programación lineal

Farm has 50 ha. Crop 1 yields EUR 400/ha (needs 2 labor days/ha). Crop 2 yields EUR 500/ha (needs 4 labor days/ha). Total labor: 140 days. Maximize total profit P.

  1. 30 ha Crop 1 and 20 ha Crop 2 (EUR 22,000) Correct answer.
  2. 50 ha Crop 1 and 0 ha Crop 2 (EUR 20,000)
  3. 0 ha Crop 1 and 35 ha Crop 2 (EUR 17,500)
  4. 25 ha Crop 1 and 25 ha Crop 2 (EUR 22,500)

Explanation

Constraints: x + y <= 50, 2x + 4y <= 140 (or x + 2y <= 70), x>=0, y>=0. Max P = 400x + 500y. Vertices: (0,0) P=0; (50,0) P=20,000; (30,20) P=400(30)+500(20)=22,000; (0,35) P=17,500. Maximum profit is €22,000 at (30, 20).

Question 45Programación lineal

What is concluded if the constraint system of a linear programming problem produces an empty feasible region?

  1. The problem has no feasible solution Correct answer.
  2. The optimal solution is at the origin (0, 0)
  3. The optimal solution is infinite
  4. The problem has a unique optimal solution

Explanation

If no points satisfy all constraint inequalities simultaneously, the feasible set is empty (empty region), meaning no feasible solution exists.

Question 46Análisis, límites, derivadas y optimización

Find the derivative of polynomial function f(x) = 4x^3 - 5x^2 + 7x - 9.

  1. 12x^2 - 10x + 7 Correct answer.
  2. 12x^2 - 10x
  3. 4x^2 - 5x + 7
  4. 12x^3 - 10x^2 + 7x

Explanation

Apply power rule d/dx(x^n) = n x^(n-1): f'(x) = 4(3x^2) - 5(2x) + 7(1) - 0 = 12x^2 - 10x + 7.

Question 47Análisis, límites, derivadas y optimización

Find the derivative of exponential function f(x) = e^(3x).

  1. 3 e^(3x) Correct answer.
  2. e^(3x)
  3. 3x e^(3x-1)
  4. 1/3 e^(3x)

Explanation

Using chain rule d/dx(e^(u(x))) = u'(x) e^(u(x)): f'(x) = d/dx(3x) * e^(3x) = 3 e^(3x).

Question 48Análisis, límites, derivadas y optimización

Find the derivative of logarithmic function f(x) = ln(2x + 5).

  1. 2 / (2x + 5) Correct answer.
  2. 1 / (2x + 5)
  3. 2 ln(2x + 5)
  4. 1 / (2x)

Explanation

Using chain rule d/dx(ln(u(x))) = u'(x) / u(x): u(x) = 2x + 5, u'(x) = 2. Thus f'(x) = 2 / (2x + 5).

Question 49Análisis, límites, derivadas y optimización

Find the slope of the tangent line to f(x) = x^2 - 4x + 3 at point x = 3.

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

Explanation

The slope of the tangent line equals the first derivative evaluated at x = 3. f'(x) = 2x - 4. f'(3) = 2(3) - 4 = 6 - 4 = 2.

Question 50Análisis, límites, derivadas y optimización

A differentiable function f(x) is strictly increasing on an open interval (a, b) if for all x in (a, b):

  1. f'(x) > 0 Correct answer.
  2. f'(x) < 0
  3. f''(x) = 0
  4. f(x) > 0

Explanation

A positive first derivative f'(x) > 0 on an interval implies that the slope of tangent lines is positive, meaning f(x) is strictly increasing.