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Valencian Community PAU Applied Mathematics for Social Sciences II study materials
Valencian Community PAU Applied Mathematics for Social Sciences II study resources

Valencian Community PAU Applied Mathematics for Social Sciences II practice questions

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Question 1Análisis, límites, derivadas y optimización

Evaluate the limit lim_{x -> infinity} (4x^2 - 3x + 1) / (2x^2 + 5).

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

Explanation

Dividing numerator and denominator by highest power x^2: lim_{x -> inf} (4 - 3/x + 1/x^2) / (2 + 5/x^2) = (4 - 0 + 0) / (2 + 0) = 4 / 2 = 2.

Question 2Análisis, límites, derivadas y optimización

Determine the value of k such that the piecewise function f(x) = x^2 + k for x < 2, and f(x) = 3x - 1 for x >= 2, is continuous at x = 2.

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

Explanation

For continuity at x = 2, left limit equals right limit and value: lim_{x -> 2^-} (x^2 + k) = 2^2 + k = 4 + k. lim_{x -> 2^+} (3x - 1) = 3(2) - 1 = 5. Setting 4 + k = 5 yields k = 1.

Question 3Análisis, límites, derivadas y optimización

Find the horizontal asymptote of the rational function f(x) = (6x - 5) / (3x + 2) as x -> infinity.

  1. y = 0
  2. y = -2.5
  3. y = 2 Correct answer.
  4. x = -2/3

Explanation

As x -> infinity, lim_{x -> inf} (6x - 5) / (3x + 2) = 6/3 = 2. The horizontal asymptote is the horizontal line y = 2.

Question 4Análisis, límites, derivadas y optimización

Identify the vertical asymptote(s) of the function f(x) = (x + 3) / (x^2 - 4).

  1. x = -3 only
  2. y = 0
  3. x = 4 only
  4. x = 2 and x = -2 Correct answer.

Explanation

Vertical asymptotes occur where denominator equals zero and numerator is non-zero. x^2 - 4 = 0 => (x - 2)(x + 2) = 0 => x = 2 and x = -2. At both points, numerator (2+3=5, -2+3=1) is non-zero.

Question 5Análisis, límites, derivadas y optimización

Calculate the derivative f'(x) for f(x) = 5x^3 - 4x^2 + 7x - 9.

  1. 15x^2 - 8x + 7 Correct answer.
  2. 15x^3 - 8x^2 + 7x
  3. 5x^2 - 4x + 7
  4. 15x^2 - 8x

Explanation

Applying the power rule d/dx(x^n) = n x^(n-1): f'(x) = 5(3x^2) - 4(2x) + 7(1) - 0 = 15x^2 - 8x + 7.

Question 6Análisis, límites, derivadas y optimización

Find the derivative f'(x) of the function f(x) = (2x + 1) e^x using the product rule.

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

Explanation

Product rule: (u v)' = u' v + u v'. Here u = 2x + 1 => u' = 2, and v = e^x => v' = e^x. f'(x) = (2) e^x + (2x + 1) e^x = (2 + 2x + 1) e^x = (2x + 3) e^x.

Question 7Análisis, límites, derivadas y optimización

Apply the quotient rule to find f'(x) for f(x) = 3x / (x + 2).

  1. 3 / (x + 2)^2
  2. 6x / (x + 2)^2
  3. 6 / (x + 2)^2 Correct answer.
  4. 3

Explanation

Quotient rule: (u/v)' = (u'v - uv') / v^2. u = 3x, u' = 3; v = x+2, v' = 1. f'(x) = [3(x + 2) - 3x(1)] / (x + 2)^2 = (3x + 6 - 3x) / (x + 2)^2 = 6 / (x + 2)^2.

Question 8Análisis, límites, derivadas y optimización

Find the derivative f'(x) of f(x) = ln(x^2 + 4).

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

Explanation

Chain rule for natural log: d/dx[ln(g(x))] = g'(x) / g(x). With g(x) = x^2 + 4 => g'(x) = 2x. Thus f'(x) = 2x / (x^2 + 4).

Question 9Análisis, límites, derivadas y optimización

Find the slope of the line tangent to the curve f(x) = x^3 - 3x^2 + 5 at the point x = 2.

  1. 12
  2. 6
  3. 1
  4. 0 Correct answer.

Explanation

The slope of the tangent line equals f'(2). Derivative f'(x) = 3x^2 - 6x. Evaluating at x = 2: f'(2) = 3(2^2) - 6(2) = 3(4) - 12 = 12 - 12 = 0. The tangent line is horizontal.

Question 10Análisis, límites, derivadas y optimización

Find the full equation of the tangent line to f(x) = x^2 - 4x + 1 at x = 3.

  1. y = 2x - 2
  2. y = 2x - 8 Correct answer.
  3. y = -2x + 4
  4. y = 3x - 11

Explanation

Point y-value: f(3) = 3^2 - 4(3) + 1 = 9 - 12 + 1 = -2 => point (3, -2). Derivative f'(x) = 2x - 4 => slope m = f'(3) = 2(3) - 4 = 2. Tangent line equation: y - (-2) = 2(x - 3) => y + 2 = 2x - 6 => y = 2x - 8.

Question 11Análisis, límites, derivadas y optimización

Critical points of a smooth, differentiable function f(x) occur at values of x where:

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

Explanation

Critical points of a differentiable function occur where the first derivative f'(x) = 0 (stationary points) or where f'(x) is undefined.

Question 12Análisis, límites, derivadas y optimización

On which interval is the function f(x) = x^2 - 6x + 8 strictly increasing?

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

Explanation

f(x) is strictly increasing where f'(x) > 0. Derivative f'(x) = 2x - 6. Setting 2x - 6 > 0 => 2x > 6 => x > 3, corresponding to the interval (3, infinity).

Question 13Análisis, límites, derivadas y optimización

Find the x-location of the local minimum for f(x) = x^3 - 3x.

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

Explanation

f'(x) = 3x^2 - 3 = 0 => 3(x^2 - 1) = 0 => x = 1 or x = -1. Second derivative f''(x) = 6x. At x = 1: f''(1) = 6 > 0 (concave up => local minimum). At x = -1: f''(-1) = -6 < 0 (concave down => local maximum). Thus local minimum is at x = 1.

Question 14Análisis, límites, derivadas y optimización

Find the point of inflection for f(x) = x^3 - 6x^2 + 9x + 2.

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

Explanation

Inflection points occur where f''(x) = 0 and concavity changes sign. f'(x) = 3x^2 - 12x + 9; f''(x) = 6x - 12. Setting 6x - 12 = 0 => x = 2. Since f''(x) changes from negative (x < 2) to positive (x > 2), x = 2 is an inflection point.

Question 15Análisis, límites, derivadas y optimización

On which interval is the graph of f(x) = x^3 - 6x^2 concave upward (convexa)?

  1. (-infinity, 2)
  2. (0, 4)
  3. (-infinity, 0)
  4. (2, infinity) Correct answer.

Explanation

Concave upward occurs where f''(x) > 0. f'(x) = 3x^2 - 12x; f''(x) = 6x - 12. 6x - 12 > 0 => 6x > 12 => x > 2, which corresponds to (2, infinity).

Question 16Análisis, límites, derivadas y optimización

A retail firm determines revenue R(x) = 120x - 0.5x^2 euros for selling x units. How many units x must be sold to maximize total revenue?

  1. 120 units Correct answer.
  2. 240 units
  3. 60 units
  4. 100 units

Explanation

To maximize revenue, set marginal revenue R'(x) = 0: R'(x) = 120 - 1.0 x = 0 => x = 120 units. Second derivative R''(x) = -1 < 0 confirms a maximum.

Question 17Análisis, límites, derivadas y optimización

A business has revenue R(x) = 200x and total cost C(x) = x^2 + 40x + 1000. Find the production level x that maximizes total profit P(x) = R(x) - C(x).

  1. 40 units
  2. 100 units
  3. 80 units Correct answer.
  4. 120 units

Explanation

Profit function P(x) = 200x - (x^2 + 40x + 1000) = -x^2 + 160x - 1000. Derivative P'(x) = -2x + 160 = 0 => 2x = 160 => x = 80 units.

Question 18Análisis, límites, derivadas y optimización

Using the profit function P(x) = -x^2 + 160x - 1000 from question 042, calculate the maximum profit value obtained at x = 80 units.

  1. EUR 6,400
  2. EUR 5,400 Correct answer.
  3. EUR 12,800
  4. EUR 11,800

Explanation

P(80) = -(80)^2 + 160(80) - 1000 = -6400 + 12800 - 1000 = 6400 - 1000 = EUR 5,400.

Question 19Análisis, límites, derivadas y optimización

A production plant has cost C(x) = x^2 + 16x + 64. At what positive output level x is average cost C_avg(x) = C(x)/x minimized?

  1. 8 units Correct answer.
  2. 16 units
  3. 4 units
  4. 64 units

Explanation

Average cost C_avg(x) = x + 16 + 64/x. Derivative d/dx[C_avg] = 1 - 64/x^2 = 0 => 64/x^2 = 1 => x^2 = 64 => x = 8 units. (Or set marginal cost C'(x) = C_avg(x): 2x + 16 = x + 16 + 64/x => x = 64/x => x = 8).

Question 20Análisis, límites, derivadas y optimización

Find the absolute maximum value of f(x) = -x^2 + 4x + 1 on the closed interval [0, 5].

  1. 1
  2. -4
  3. 4
  4. 5 Correct answer.

Explanation

Find critical point: f'(x) = -2x + 4 = 0 => x = 2 in [0,5]. Evaluate at critical point and endpoints: f(0) = 1; f(2) = -4 + 8 + 1 = 5; f(5) = -25 + 20 + 1 = -4. The absolute maximum value is 5 (at x = 2).

Question 21Análisis, límites, derivadas y optimización

Annual profit P(t) of a startup in thousands of euros after t years is modeled by P(t) = 10 + 6t - t^2. At what year t does annual profit reach its maximum before declining?

  1. t = 6 years
  2. t = 3 years Correct answer.
  3. t = 5 years
  4. t = 10 years

Explanation

P'(t) = 6 - 2t = 0 => 2t = 6 => t = 3 years. Since P''(t) = -2 < 0, profit is maximized at t = 3 years.

Question 22Análisis, límites, derivadas y optimización

Which statement correctly describes the mathematical requirement for a function f(x) to be differentiable at x = a?

  1. f(x) must be continuous at x = a, and the left-hand derivative equals the right-hand derivative at x = a Correct answer.
  2. f(x) must have a local extremum at x = a
  3. f(x) must be a polynomial of degree <= 2
  4. f(a) must equal zero

Explanation

Differentiability at x = a requires continuity at x = a and the existence of a unique limit of the difference quotient as x -> a (meaning left-hand and right-hand derivatives exist and are equal).

Question 23Análisis, límites, derivadas y optimización

Find the derivative f'(x) for the exponential composite function f(x) = e^(3x^2 - 5).

  1. e^(6x)
  2. 6x e^(3x^2 - 5) Correct answer.
  3. (3x^2 - 5) e^(3x^2 - 5)
  4. 6x e^(6x)

Explanation

Chain rule for exponential functions: d/dx[e^(g(x))] = g'(x) e^(g(x)). Here g(x) = 3x^2 - 5 => g'(x) = 6x. Thus f'(x) = 6x e^(3x^2 - 5).

Question 24Análisis, límites, derivadas y optimización

Evaluate the indeterminate limit lim_{x -> 3} (x^2 - 9) / (x - 3).

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

Explanation

Direct substitution yields 0/0. Factor numerator: (x^2 - 9) = (x - 3)(x + 3). Cancel common factor (x - 3): lim_{x -> 3} (x + 3) = 3 + 3 = 6.

Question 25Análisis, límites, derivadas y optimización

In business economics, what does the derivative of total cost C'(x) represent?

  1. Marginal cost (approximate cost of producing one additional unit) Correct answer.
  2. Average cost per unit produced
  3. Total revenue generated
  4. Fixed overhead costs

Explanation

The derivative of total cost C'(x) represents marginal cost, which measures the instantaneous rate of change of total cost with respect to output x, approximating the cost to produce the next unit.

Question 26Análisis, límites, derivadas y optimización

The number of active app subscribers N(t) in thousands after t months is given by N(t) = 100t / (t + 4). What is the growth rate N'(t) at t = 6 months?

  1. 10 thousand subscribers/month
  2. 6 thousand subscribers/month
  3. 4 thousand subscribers/month Correct answer.
  4. 2.5 thousand subscribers/month

Explanation

Quotient rule derivative: N'(t) = [100(t + 4) - 100t(1)] / (t + 4)^2 = 400 / (t + 4)^2. At t = 6: N'(6) = 400 / (6 + 4)^2 = 400 / 100 = 4 thousand subscribers/month.

Question 27Análisis, límites, derivadas y optimización

Evaluate lim_{x -> 0} (e^(2x) - 1) / x.

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

Explanation

This has indeterminate form 0/0. Applying L'Hopital's Rule: lim_{x -> 0} d/dx[e^(2x) - 1] / d/dx[x] = lim_{x -> 0} (2 e^(2x)) / 1 = 2 e^0 / 1 = 2.

Question 28Análisis, límites, derivadas y optimización

For what values of a and b is the piecewise function f(x) = a x^2 for x <= 1, and f(x) = 2x + b for x > 1, differentiable at x = 1?

  1. a = 2, b = 0
  2. a = 1, b = -1 Correct answer.
  3. a = 1, b = 1
  4. a = 2, b = -1

Explanation

Left derivative at 1: d/dx(a x^2)|_{x=1} = 2a(1) = 2a. Right derivative at 1: d/dx(2x + b)|_{x=1} = 2. Equal derivatives => 2a = 2 => a = 1. Continuity requires left value = right value: a(1)^2 = 2(1) + b => 1 = 2 + b => b = -1.

Question 29Análisis, límites, derivadas y optimización

Calculate the definite integral integral_0^3 (3x^2 - 2x + 1) dx.

  1. 18
  2. 27
  3. 24
  4. 21 Correct answer.

Explanation

Antiderivative F(x) = x^3 - x^2 + x. Evaluate from 0 to 3: F(3) = 3^3 - 3^2 + 3 = 27 - 9 + 3 = 21. F(0) = 0. Definite integral = 21 - 0 = 21.

Question 30Análisis, límites, derivadas y optimización

If marginal revenue is R'(x) = 50 - 2x euros/unit, what is the net total revenue change when sales increase from x = 10 to x = 20 units?

  1. EUR 200 Correct answer.
  2. EUR 300
  3. EUR 400
  4. EUR 150

Explanation

Total revenue change Delta R = integral_{10}^{20} (50 - 2x) dx = [50x - x^2]_{10}^{20} = (50(20) - 20^2) - (50(10) - 10^2) = (1000 - 400) - (500 - 100) = 600 - 400 = EUR 200.

Question 31Álgebra matricial y sistemas

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

  1. 2 x 4 Correct answer.
  2. 3 x 3
  3. 4 x 2
  4. The matrix product A · B is undefined

Explanation

Matrix multiplication A · B is defined when the number of columns in A (3) equals the number of rows in B (3). The resulting product matrix C has dimensions equal to the number of rows of A and the number of columns of B, which is 2 x 4.

Question 32Álgebra matricial y sistemas

Given the matrix A = [[3, -2], [4, 5]], calculate the determinant det(A).

  1. 7
  2. 15
  3. 23 Correct answer.
  4. -7

Explanation

For a 2 x 2 matrix [[a, b], [c, d]], the determinant is det(A) = ad - bc. Here, det(A) = (3)(5) - (-2)(4) = 15 - (-8) = 15 + 8 = 23.

Question 33Álgebra matricial y sistemas

Compute the inverse of matrix A = [[2, 1], [4, 3]].

  1. [[3, -1], [-4, 2]]
  2. [[1.5, -0.5], [-2, 1]] Correct answer.
  3. [[-1.5, 0.5], [2, -1]]
  4. [[0.5, 1], [0.25, 0.33]]

Explanation

The determinant det(A) = (2)(3) - (1)(4) = 6 - 4 = 2. The formula for a 2 x 2 inverse is A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] = (1/2) * [[3, -1], [-4, 2]] = [[1.5, -0.5], [-2, 1]].

Question 34Álgebra matricial y sistemas

Solve the matrix equation A · X = B for X = [[x], [y]], where A = [[1, 2], [3, 4]] and B = [[5], [11]].

  1. x = 1, y = 2 Correct answer.
  2. x = 2, y = 1
  3. x = 3, y = 1
  4. x = 5, y = 0

Explanation

det(A) = (1)(4) - (2)(3) = -2. A^(-1) = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]]. Multiplying X = A^(-1) · B gives [[-2(5) + 1(11)], [1.5(5) - 0.5(11)]] = [[-10 + 11], [7.5 - 5.5]] = [[1], [2]]. Thus x = 1, y = 2.

Question 35Álgebra matricial y sistemas

Which of the following matrix identity properties is correct for conformable matrices A and B?

  1. (A · B)^T = A^T · B^T
  2. A · B = B · A for all square matrices
  3. det(A + B) = det(A) + det(B)
  4. (A · B)^T = B^T · A^T Correct answer.

Explanation

The transpose of a matrix product reverses the order of multiplication: (A · B)^T = B^T · A^T. Matrix multiplication is generally non-commutative (A · B != B · A), and determinants are non-linear under addition (det(A + B) != det(A) + det(B)).

Question 36Álgebra matricial y sistemas

Calculate the determinant of the 3 x 3 matrix A = [[1, 0, 2], [2, -1, 3], [4, 1, 8]].

  1. 1 Correct answer.
  2. 0
  3. -5
  4. 12

Explanation

Expanding along the first row: det(A) = 1*((-1)(8) - (3)(1)) - 0 + 2*((2)(1) - (-1)(4)) = 1*(-8 - 3) + 2*(2 + 4) = 1*(-11) + 2*(6) = -11 + 12 = 1.

Question 37Álgebra matricial y sistemas

For what values of parameter k is the matrix A = [[k, 4], [2, k]] singular (non-invertible)?

  1. k = 2 only
  2. k = +/- 2*sqrt(2) Correct answer.
  3. k = +/- 8
  4. k = 0

Explanation

A matrix is singular when its determinant is zero: det(A) = k^2 - (4)(2) = k^2 - 8 = 0. Solving k^2 = 8 yields k = +/- sqrt(8) = +/- 2*sqrt(2).

Question 38Álgebra matricial y sistemas

If A is a 3 x 3 square matrix with det(A) = 5, what is the determinant of 2A?

  1. 10
  2. 15
  3. 40 Correct answer.
  4. 125

Explanation

For an n x n matrix and scalar c, det(c A) = c^n * det(A). For n = 3 and c = 2: det(2A) = 2^3 * det(A) = 8 * 5 = 40.

Question 39Álgebra matricial y sistemas

A system of 3 linear equations in 3 unknowns has coefficient matrix A with rank(A) = 2 and augmented matrix (A|B) with rank(A|B) = 3. According to the Rouché-Capelli Theorem, how many solutions does the system have?

  1. Zero solutions (Inconsistent system / Sistema incompatible) Correct answer.
  2. Exactly one unique solution (Consistent determined / Sistema compatible determinado)
  3. Infinitely many solutions depending on 1 parameter (Consistent indeterminate)
  4. Infinitely many solutions depending on 2 parameters

Explanation

According to the Rouché-Capelli Theorem, a system is consistent if and only if rank(A) = rank(A|B). When rank(A) = 2 != rank(A|B) = 3, the system has no solution and is classified as an inconsistent system (sistema incompatible).

Question 40Álgebra matricial y sistemas

Consider the system of linear equations: x + y + z = 1, x + 2y + 3z = 2, 2x + 3y + k z = 3. For what value of parameter k does the system have infinitely many solutions?

  1. k = 0
  2. k = 2
  3. k = 4 Correct answer.
  4. k = 6

Explanation

The determinant of A is det(A) = 1(2k - 9) - 1(k - 6) + 1(3 - 4) = k - 4. Setting det(A) = 0 yields k = 4. At k = 4, rank(A) = 2. Notice that Equation 3 = Equation 1 + Equation 2 (2x + 3y + 4z = 3), so rank(A|B) is also 2. Since rank(A) = rank(A|B) = 2 < 3 variables, k = 4 gives infinitely many solutions.

Question 41Álgebra matricial y sistemas

Solve for y using Cramer's rule in the system: 2x + y = 7, 3x - 2y = 0.

  1. y = 2
  2. y = 3 Correct answer.
  3. y = 1
  4. y = 4

Explanation

Main determinant Delta = det([[2, 1], [3, -2]]) = (2)(-2) - (1)(3) = -4 - 3 = -7. Determinant for y is Delta_y = det([[2, 7], [3, 0]]) = (2)(0) - (7)(3) = -21. By Cramer's rule, y = Delta_y / Delta = -21 / -7 = 3.

Question 42Álgebra matricial y sistemas

A company produces x units of product A and y units of product B using resource constraints: 2x + 4y = 100 (raw material constraint) and 3x + y = 90 (labor hour constraint). How many units of A and B should be produced to use all available resources?

  1. x = 26, y = 12 Correct answer.
  2. x = 20, y = 15
  3. x = 24, y = 18
  4. x = 30, y = 0

Explanation

From the second equation, y = 90 - 3x. Substitute into the first equation: 2x + 4(90 - 3x) = 100 => 2x + 360 - 12x = 100 => -10x = -260 => x = 26. Then y = 90 - 3(26) = 90 - 78 = 12.

Question 43Álgebra matricial y sistemas

For square invertible matrices A and B of the same size, what is the inverse of their product, (A · B)^(-1)?

  1. A^(-1) · B^(-1)
  2. B^(-1) · A^(-1) Correct answer.
  3. 1 / (A · B)
  4. A^T · B^T

Explanation

Matrix inversion reverses product order: (A · B)^(-1) = B^(-1) · A^(-1). Verification: (A · B) · (B^(-1) · A^(-1)) = A · (B · B^(-1)) · A^(-1) = A · I · A^(-1) = A · A^(-1) = I.

Question 44Álgebra matricial y sistemas

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

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

Explanation

Row 2 = 2 * Row 1, so Row 2 is linearly dependent on Row 1. Row 1 (1, 2, 3) and Row 3 (0, 1, 1) are linearly independent since neither is a multiple of the other. Thus, there are 2 linearly independent rows, so rank(A) = 2.

Question 45Álgebra matricial y sistemas

A system of 3 linear equations in 2 variables has rank(A) = 2 and rank(A|B) = 2. How many solutions does the system have?

  1. Exactly one unique solution (Sistema compatible determinado) Correct answer.
  2. Zero solutions (Sistema incompatible)
  3. Infinitely many solutions (Sistema compatible indeterminado)
  4. Two distinct solutions

Explanation

By the Rouché-Capelli Theorem, since rank(A) = rank(A|B) = 2 and the number of unknown variables is n = 2, rank equals the number of variables, guaranteeing exactly one unique solution.

Question 46Álgebra matricial y sistemas

For matrix A = [[4, x - 1], [5, 2]] to be symmetric (A^T = A), what must be the value of x?

  1. x = 6 Correct answer.
  2. x = 5
  3. x = 4
  4. x = 1

Explanation

A square matrix is symmetric if entry a_12 equals entry a_21. Here, a_12 = x - 1 and a_21 = 5. Setting x - 1 = 5 yields x = 6.

Question 47Álgebra matricial y sistemas

If A = [[1, 1], [0, 1]], compute the matrix power A^3.

  1. [[1, 1], [0, 1]]
  2. [[1, 8], [0, 1]]
  3. [[1, 3], [0, 1]] Correct answer.
  4. [[3, 3], [0, 3]]

Explanation

A^2 = [[1, 1], [0, 1]] · [[1, 1], [0, 1]] = [[1, 2], [0, 1]]. Then A^3 = A^2 · A = [[1, 2], [0, 1]] · [[1, 1], [0, 1]] = [[1, 3], [0, 1]]. In general, A^n = [[1, n], [0, 1]].

Question 48Programación lineal

In a 2D linear programming problem, what geometric shape is formed by the intersection of a finite set of linear inequality constraints?

  1. A hyperbola
  2. A convex polygonal region (polyhedron) Correct answer.
  3. A circle
  4. A non-convex parabolic arc

Explanation

Each linear inequality defines a closed half-plane, which is a convex set. The intersection of a finite collection of half-planes forms a convex polygonal region (the feasible region).

Question 49Programación lineal

According to the fundamental theorem of linear programming, where must the maximum or minimum value of a linear objective function Z = a x + b y over a bounded feasible region occur?

  1. At one or more vertices (corner points) of the feasible region Correct answer.
  2. At the origin (0,0) exclusively
  3. At the arithmetic mean of all interior points
  4. At the point of highest curvature of the boundary

Explanation

The fundamental theorem of linear programming states that if a linear objective function achieves an optimal value over a bounded, non-empty convex feasible region, that value must occur at least at one of the corner points (vertices) of the region.

Question 50Programación lineal

Find the vertex (corner point) formed by the intersection of the boundary lines x + y = 10 and 2x + y = 14.

  1. (2, 8)
  2. (3, 7)
  3. (5, 5)
  4. (4, 6) Correct answer.

Explanation

Subtract the first equation (x + y = 10) from the second (2x + y = 14): (2x - x) + (y - y) = 14 - 10 => x = 4. Substituting x = 4 into x + y = 10 yields y = 6. The intersection vertex is (4, 6).