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PAU Applied Mathematics for Social Sciences II - Balearic Islands / UIB study materials
PAU Applied Mathematics for Social Sciences II - Balearic Islands / UIB study resources

PAU Applied Mathematics for Social Sciences II - Balearic Islands / UIB practice questions

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Question 1Análisis, límites, derivadas y optimización

Find the derivative f'(x) of the polynomial function f(x) = 3x^3 - 5x^2 + 7x - 12.

  1. 9x^2 - 10x
  2. 9x^2 - 10x + 7 Correct answer.
  3. 3x^2 - 5x + 7
  4. 9x^3 - 10x^2 + 7x

Explanation

Using power rule d/dx(x^n) = n*x^(n-1): f'(x) = 3*(3x^2) - 5*(2x) + 7*(1) - 0 = 9x^2 - 10x + 7.

Question 2Análisis, límites, derivadas y optimización

Evaluate the limit at infinity: lim (x -> infinity) (4x^2 - 3x + 1) / (2x^2 + 5).

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

Explanation

Since the degree of numerator and denominator are equal (both degree 2), the limit as x -> infinity is the ratio of leading coefficients: lim = 4 / 2 = 2.

Question 3Análisis, límites, derivadas y optimización

Find the slope of the tangent line to the curve f(x) = x^2 - 4x + 5 at x = 3.

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

Explanation

Slope m = f'(3). f'(x) = 2x - 4. f'(3) = 2(3) - 4 = 6 - 4 = 2.

Question 4Análisis, límites, derivadas y optimización

Calculate the indefinite integral integral (6x^2 - 4x + 3) dx.

  1. 2x^3 - 2x^2 + 3x + C Correct answer.
  2. 12x - 4 + C
  3. 6x^3 - 4x^2 + 3x + C
  4. 2x^3 - 4x^2 + 3x + C

Explanation

Using power rule for integration integral x^n dx = (x^(n+1))/(n+1): integral 6x^2 dx = 6*(x^3/3) = 2x^3. integral -4x dx = -4*(x^2/2) = -2x^2. integral 3 dx = 3x. Result: 2x^3 - 2x^2 + 3x + C.

Question 5Análisis, límites, derivadas y optimización

Evaluate the definite integral integral_1^3 (2x) dx.

  1. 9
  2. 8 Correct answer.
  3. 4
  4. 6

Explanation

Antiderivative F(x) = x^2. By Fundamental Theorem of Calculus: integral_1^3 (2x) dx = F(3) - F(1) = 3^2 - 1^2 = 9 - 1 = 8.

Question 6Análisis, límites, derivadas y optimización

A firm's total revenue function is R(x) = 50x - 0.5x^2. What is the marginal revenue function MR(x)?

  1. 50 - 0.5x
  2. 50x - x
  3. 50 - x Correct answer.
  4. 25x

Explanation

Marginal revenue is the derivative of total revenue MR(x) = R'(x). MR(x) = d/dx(50x - 0.5x^2) = 50 - 0.5*(2x) = 50 - x.

Question 7Análisis, límites, derivadas y optimización

Find the critical points of the function f(x) = x^3 - 12x.

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

Explanation

Critical points occur where f'(x) = 0. f'(x) = 3x^2 - 12 = 0 => 3x^2 = 12 => x^2 = 4 => x = 2 or x = -2.

Question 8Análisis, límites, derivadas y optimización

Find the value of 'a' so that the piecewise function f(x) = { a*x + 3 if x <= 2; x^2 - a if x > 2 } is continuous at x = 2.

  1. 1/3 Correct answer.
  2. 1
  3. 3
  4. 1/2

Explanation

For continuity at x = 2, left limit must equal right limit: lim (x -> 2-) f(x) = a(2) + 3 = 2a + 3. lim (x -> 2+) f(x) = 2^2 - a = 4 - a. Set equal: 2a + 3 = 4 - a => 3a = 1 => a = 1/3.

Question 9Análisis, límites, derivadas y optimización

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

  1. y = 5x + 3
  2. y = 5x - 7 Correct answer.
  3. y = 8x - 13
  4. y = 5x - 3

Explanation

1) Point of tangency: y = f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3. Point is (2, 3). 2) Slope: f'(x) = 4x - 3 => m = f'(2) = 4(2) - 3 = 5. 3) Equation: y - 3 = 5(x - 2) => y - 3 = 5x - 10 => y = 5x - 7.

Question 10Análisis, límites, derivadas y optimización

A company's revenue is R(x) = 100x - x^2 and cost is C(x) = 20x + 300. Find the production quantity x that maximizes profit P(x) = R(x) - C(x), and the maximum profit.

  1. x = 50 units with max profit €2,200
  2. x = 40 units with max profit €1,600
  3. x = 40 units with max profit €1,300 Correct answer.
  4. x = 30 units with max profit €1,200

Explanation

Profit P(x) = (100x - x^2) - (20x + 300) = 80x - x^2 - 300. P'(x) = 80 - 2x = 0 => 2x = 80 => x = 40. P''(x) = -2 < 0 (maximum). Max profit P(40) = 80(40) - 40^2 - 300 = 3200 - 1600 - 300 = 1300.

Question 11Análisis, límites, derivadas y optimización

A company's total cost function is C(x) = 2x^2 + 50x + 800. Find the production level x that minimizes average cost AC(x) = C(x)/x.

  1. 40
  2. 10
  3. 25
  4. 20 Correct answer.

Explanation

Average cost AC(x) = 2x + 50 + 800/x. AC'(x) = 2 - 800/x^2 = 0 => 2 = 800/x^2 => 2x^2 = 800 => x^2 = 400 => x = 20.

Question 12Análisis, límites, derivadas y optimización

Find the inflection point of the cubic curve f(x) = x^3 - 6x^2 + 9x + 4.

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

Explanation

f'(x) = 3x^2 - 12x + 9. f''(x) = 6x - 12 = 0 => 6x = 12 => x = 2. f(2) = 2^3 - 6(2^2) + 9(2) + 4 = 8 - 24 + 18 + 4 = 6. Thus the inflection point is (2, 6).

Question 13Análisis, límites, derivadas y optimización

On which interval is the function f(x) = -x^2 + 6x - 5 strictly increasing?

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

Explanation

f'(x) = -2x + 6. f(x) is strictly increasing when f'(x) > 0 => -2x + 6 > 0 => 2x < 6 => x < 3. Thus interval is (-infinity, 3).

Question 14Análisis, límites, derivadas y optimización

Calculate the area of the region bounded by the parabola y = 4 - x^2 and the x-axis.

  1. 16/3 (approx 5.33)
  2. 16
  3. 32/3 (approx 10.67) Correct answer.
  4. 8

Explanation

Find roots: 4 - x^2 = 0 => x = -2 and x = 2. Area = integral_{-2}^2 (4 - x^2) dx = [4x - x^3/3]_{-2}^2 = (4(2) - 8/3) - (4(-2) - (-8)/3) = (8 - 8/3) - (-8 + 8/3) = (16/3) - (-16/3) = 32/3.

Question 15Análisis, límites, derivadas y optimización

Calculate the area of the region enclosed between the curves y = x^2 and y = 2x.

  1. 2/3
  2. 2
  3. 4
  4. 4/3 (approx 1.33) Correct answer.

Explanation

Find intersection points: x^2 = 2x => x^2 - 2x = 0 => x(x - 2) = 0 => x = 0 and x = 2. On [0, 2], 2x >= x^2. Area = integral_0^2 (2x - x^2) dx = [x^2 - x^3/3]_0^2 = (2^2 - 2^3/3) - 0 = 4 - 8/3 = 4/3.

Question 16Análisis, límites, derivadas y optimización

Evaluate the limit with indeterminate form 0/0: lim (x -> 3) (x^2 - 9) / (x^2 - 5x + 6).

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

Explanation

Factor numerator and denominator: x^2 - 9 = (x - 3)(x + 3). x^2 - 5x + 6 = (x - 3)(x - 2). lim (x -> 3) (x - 3)(x + 3) / [(x - 3)(x - 2)] = lim (x -> 3) (x + 3) / (x - 2) = (3 + 3) / (3 - 2) = 6 / 1 = 6.

Question 17Análisis, límites, derivadas y optimización

Given f(x) = 5 * e^(2x - 4), evaluate the value of the derivative f'(2).

  1. 5
  2. 10 Correct answer.
  3. 20
  4. 0

Explanation

Using chain rule: f'(x) = 5 * e^(2x - 4) * d/dx(2x - 4) = 5 * 2 * e^(2x - 4) = 10 * e^(2x - 4). Evaluating at x = 2: f'(2) = 10 * e^(2(2) - 4) = 10 * e^0 = 10 * 1 = 10.

Question 18Análisis, límites, derivadas y optimización

Given f(x) = ln(3x^2 + 1), evaluate the value of the derivative f'(1).

  1. 6
  2. 0.75
  3. 1.5 Correct answer.
  4. 3

Explanation

Using chain rule d/dx[ln(u)] = u'/u: f'(x) = d/dx(3x^2 + 1) / (3x^2 + 1) = 6x / (3x^2 + 1). f'(1) = 6(1) / (3(1)^2 + 1) = 6 / 4 = 1.5.

Question 19Análisis, límites, derivadas y optimización

Find the absolute maximum value of f(x) = x^3 - 3x + 2 on the closed interval [-2, 2].

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

Explanation

1) Critical points: f'(x) = 3x^2 - 3 = 0 => x^2 = 1 => x = 1, x = -1. 2) Evaluate f at endpoints and critical points: f(-2) = (-8) - 3(-2) + 2 = -8 + 6 + 2 = 0 f(-1) = (-1) - 3(-1) + 2 = -1 + 3 + 2 = 4 f(1) = (1) - 3(1) + 2 = 1 - 3 + 2 = 0 f(2) = (8) - 3(2) + 2 = 8 - 6 + 2 = 4. Absolute maximum value is 4 (achieved at x = -1 and x = 2).

Question 20Análisis, límites, derivadas y optimización

Evaluate the definite integral integral_0^2 (3x * sqrt(x^2 + 5)) dx using substitution.

  1. 27 - 5*sqrt(5) (approx 15.82) Correct answer.
  2. 27
  3. 19
  4. 38

Explanation

Let u = x^2 + 5, du = 2x dx => x dx = du / 2. Limits: x = 0 => u = 5; x = 2 => u = 4 + 5 = 9. integral_5^9 3 * (du/2) * u^(1/2) = (3/2) * [ (2/3) u^(3/2) ]_5^9 = [ u^(3/2) ]_5^9 = 9^(3/2) - 5^(3/2) = 27 - 5*sqrt(5).

Question 21Análisis, límites, derivadas y optimización

A product's demand function is q(p) = 200 - 4p. Calculate the price elasticity of demand E(p) = - (p/q) * (dq/dp) at price p = 25.

  1. 0.5 (inelastic)
  2. 1.0 (unit elastic) Correct answer.
  3. 2.0 (elastic)
  4. 4.0 (elastic)

Explanation

dq/dp = -4. At p = 25: q(25) = 200 - 4(25) = 100. E(25) = - (25 / 100) * (-4) = (25 / 100) * 4 = 100 / 100 = 1.0 (unit elastic).

Question 22Análisis, límites, derivadas y optimización

Identify the vertical and horizontal asymptotes of the rational function f(x) = (3x - 6) / (x + 4).

  1. Vertical asymptote x = 4, Horizontal asymptote y = 3
  2. Vertical asymptote x = -4, Horizontal asymptote y = -6
  3. Vertical asymptote x = -4, Horizontal asymptote y = 3 Correct answer.
  4. Vertical asymptote x = 2, Horizontal asymptote y = 0

Explanation

1) Vertical asymptote: denominator = 0 => x + 4 = 0 => x = -4. 2) Horizontal asymptote: lim (x -> infinity) (3x - 6)/(x + 4) = 3/1 => y = 3.

Question 23Análisis, límites, derivadas y optimización

Evaluate the definite integral integral_0^1 (4 * e^(2x)) dx.

  1. 4*e^2 - 4
  2. 2*e^2
  3. e^2 - 1
  4. 2*e^2 - 2 (approx 12.78) Correct answer.

Explanation

Antiderivative F(x) = 4 * (1/2) e^(2x) = 2 e^(2x). By Fundamental Theorem of Calculus: integral_0^1 (4 e^(2x)) dx = F(1) - F(0) = 2 e^(2(1)) - 2 e^(2(0)) = 2 e^2 - 2(1) = 2 e^2 - 2.

Question 24Análisis, límites, derivadas y optimización

A rectangular garden alongside a straight river wall requires no fence along the river. Fencing parallel to the river costs €10 per meter, and side fencing perpendicular to the river costs €5 per meter. Total fencing budget is €400. Find the maximum garden area.

  1. 400 m^2 Correct answer.
  2. 800 m^2
  3. 200 m^2
  4. 500 m^2

Explanation

Let x = length parallel to river, y = side length. Cost constraint: 10x + 5(2y) = 400 => 10x + 10y = 400 => x + y = 40 => x = 40 - y. Area A(y) = x*y = (40 - y)y = 40y - y^2. A'(y) = 40 - 2y = 0 => y = 20 m. Then x = 40 - 20 = 20 m. Max Area = 20 * 20 = 400 m^2.

Question 25Análisis, límites, derivadas y optimización

The function f(x) = a*x^3 + b*x^2 + 9x has a local minimum at the point (1, 4). Find the coefficients a and b.

  1. a = 2, b = -7
  2. a = 1, b = -6 Correct answer.
  3. a = -1, b = 4
  4. a = 3, b = -8

Explanation

1) Point condition f(1) = 4: a(1)^3 + b(1)^2 + 9(1) = 4 => a + b + 9 = 4 => a + b = -5. 2) Local minimum condition f'(1) = 0: f'(x) = 3a x^2 + 2b x + 9. f'(1) = 3a + 2b + 9 = 0 => 3a + 2b = -9. 3) Solve system: From a + b = -5 => 2a + 2b = -10. Subtract from 3a + 2b = -9 => a = 1. Then 1 + b = -5 => b = -6.

Question 26Análisis, límites, derivadas y optimización

A firm's marginal cost function is MC(x) = 3x^2 - 12x + 15. Find the total increase in production cost when output is expanded from x = 2 units to x = 5 units.

  1. €50
  2. €14
  3. €36 Correct answer.
  4. €64

Explanation

Total cost increase = integral_2^5 (3x^2 - 12x + 15) dx. Antiderivative F(x) = x^3 - 6x^2 + 15x. F(5) = 5^3 - 6(5^2) + 15(5) = 125 - 150 + 75 = 50. F(2) = 2^3 - 6(2^2) + 15(2) = 8 - 24 + 30 = 14. Cost increase = F(5) - F(2) = 50 - 14 = 36.

Question 27Análisis, límites, derivadas y optimización

Calculate the total area enclosed between the cubic curve f(x) = x^3 - 3x^2 + 2x and the x-axis.

  1. 0.25 (1/4)
  2. 0
  3. 1.0
  4. 0.5 (1/2) Correct answer.

Explanation

Find roots: x(x^2 - 3x + 2) = x(x - 1)(x - 2) = 0 => roots x = 0, x = 1, x = 2. Region 1 [0, 1]: f(x) >= 0. integral_0^1 (x^3 - 3x^2 + 2x) dx = [x^4/4 - x^3 + x^2]_0^1 = 1/4 - 1 + 1 = 1/4. Region 2 [1, 2]: f(x) <= 0. integral_1^2 (x^3 - 3x^2 + 2x) dx = [x^4/4 - x^3 + x^2]_1^2 = (4 - 8 + 4) - (1/4) = -1/4. Area 2 = |-1/4| = 1/4. Total Area = 1/4 + 1/4 = 1/2 = 0.5.

Question 28Análisis, límites, derivadas y optimización

Determine the concavity of the function f(x) = x^2 / (x^2 + 3) on the interval (-1, 1).

  1. Concave up (f''(x) > 0) Correct answer.
  2. Concave down (f''(x) < 0)
  3. Linear (f''(x) = 0)
  4. Alternating concavity at x = 0

Explanation

f'(x) = [2x(x^2+3) - x^2(2x)] / (x^2+3)^2 = 6x / (x^2+3)^2. f''(x) = [6(x^2+3)^2 - 6x * 2(x^2+3)(2x)] / (x^2+3)^4 = [6(x^2+3) - 24x^2] / (x^2+3)^3 = (18 - 18x^2) / (x^2+3)^3. On (-1, 1), x^2 < 1 => 18(1 - x^2) > 0. Since denominator (x^2+3)^3 > 0, f''(x) > 0 for all x in (-1, 1). Thus concave up.

Question 29Análisis, límites, derivadas y optimización

Find the value of the constant k such that lim (x -> 0) [ sqrt(x + 4) - 2 ] / (k * x) = 1/8.

  1. 4
  2. 2 Correct answer.
  3. 1
  4. 8

Explanation

Rationalize numerator: [sqrt(x+4) - 2][sqrt(x+4) + 2] / [k*x*(sqrt(x+4) + 2)] = (x + 4 - 4) / [k*x*(sqrt(x+4) + 2)] = x / [k*x*(sqrt(x+4) + 2)] = 1 / [k*(sqrt(x+4) + 2)]. Limit as x -> 0: 1 / [k*(2 + 2)] = 1 / (4k). Set 1 / (4k) = 1/8 => 4k = 8 => k = 2.

Question 30Análisis, límites, derivadas y optimización

Given a demand function p(q) = 50 - 2q, market equilibrium occurs at quantity q_0 = 10 and price p_0 = 30. Calculate the Consumer Surplus (CS).

  1. 200
  2. 50
  3. 100 Correct answer.
  4. 150

Explanation

Consumer Surplus CS = integral_0^{q_0} [p(q) - p_0] dq = integral_0^{10} [(50 - 2q) - 30] dq = integral_0^{10} (20 - 2q) dq = [20q - q^2]_0^{10} = 20(10) - 10^2 = 200 - 100 = 100.

Question 31Álgebra matricial y sistemas

Given matrices A = [[1, 3], [-2, 4]] and B = [[2, -1], [0, 5]], calculate the matrix C = A + 2B.

  1. [[5, 1], [-2, 14]] Correct answer.
  2. [[3, 2], [-2, 9]]
  3. [[5, 5], [-2, 14]]
  4. [[4, 1], [-4, 14]]

Explanation

To compute C = A + 2B: 2B = 2 * [[2, -1], [0, 5]] = [[4, -2], [0, 10]]. Then C = [[1, 3], [-2, 4]] + [[4, -2], [0, 10]] = [[1+4, 3+(-2)], [-2+0, 4+10]] = [[5, 1], [-2, 14]].

Question 32Álgebra matricial y sistemas

Evaluate the determinant of the 3x3 matrix A = [[1, 2, 0], [3, -1, 4], [0, 2, 1]].

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

Explanation

Expand by Row 1: det(A) = 1 * det([[-1, 4], [2, 1]]) - 2 * det([[3, 4], [0, 1]]) + 0 = 1 * ((-1)(1) - (4)(2)) - 2 * ((3)(1) - (4)(0)) = 1 * (-1 - 8) - 2 * (3 - 0) = -9 - 6 = -15.

Question 33Álgebra matricial y sistemas

Consider the linear system: x + 2y - z = 3, 2x + 4y - 2z = 6, 3x + 6y - 3z = 9. According to the Rouché-Capelli Theorem, how is this system classified?

  1. Compatible Determinate (unique solution)
  2. Incompatible (no solution)
  3. Compatible Indeterminate (infinitely many solutions) Correct answer.
  4. Incomplete system

Explanation

Notice that Row 2 = 2 * Row 1 and Row 3 = 3 * Row 1. Rank(A) = 1 and Rank(A|B) = 1. Since Rank(A) = Rank(A|B) = 1 < number of variables (3), the system is Compatible Indeterminate with 2 free parameters.

Question 34Álgebra matricial y sistemas

For what value of the parameter k does the system x + k*y = 4, 2x + 6y = 8 have infinitely many solutions?

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

Explanation

For infinitely many solutions, the equations must be proportional. Comparing coefficients: 2/1 = 6/k = 8/4 = 2. So 6/k = 2 => k = 3.

Question 35Álgebra matricial y sistemas

Solve the matrix equation A · X + B = C for X, where A = [[1, 2], [0, 1]], B = [[3, 1], [2, 4]], and C = [[7, 5], [4, 6]].

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

Explanation

A · X = C - B. C - B = [[7-3, 5-1], [4-2, 6-4]] = [[4, 4], [2, 2]]. A^(-1) for A = [[1, 2], [0, 1]] is [[1, -2], [0, 1]]. X = A^(-1) · (C - B) = [[1, -2], [0, 1]] · [[4, 4], [2, 2]] = [[1(4)-2(2), 1(4)-2(2)], [0(4)+1(2), 0(4)+1(2)]] = [[0, 0], [2, 2]].

Question 36Álgebra matricial y sistemas

Use Cramer's Rule to find the value of y in the linear system: 3x + 2y = 12, 5x - y = 7.

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

Explanation

Main determinant D = det([[3, 2], [5, -1]]) = (3)(-1) - (2)(5) = -3 - 10 = -13. Determinant for y: Dy = det([[3, 12], [5, 7]]) = (3)(7) - (12)(5) = 21 - 60 = -39. By Cramer's Rule, y = Dy / D = -39 / -13 = 3.

Question 37Álgebra matricial y sistemas

In a Leontief input-output model, consumption matrix C = [[0.2, 0.2], [0.3, 0.1]] and final demand vector D = [[110], [66]]. Find total output vector X = (I - C)^(-1) · D.

  1. [[150], [120]]
  2. [[200], [150]]
  3. [[170], [130]] Correct answer.
  4. [[160], [140]]

Explanation

I - C = [[0.8, -0.2], [-0.3, 0.9]]. det(I - C) = (0.8)(0.9) - (-0.2)(-0.3) = 0.72 - 0.06 = 0.66. (I - C)^(-1) = (1/0.66) * [[0.9, 0.2], [0.3, 0.8]]. X = (1/0.66) * [[0.9(110) + 0.2(66)], [0.3(110) + 0.8(66)]] = (1/0.66) * [[99 + 13.2], [33 + 52.8]] = (1/0.66) * [[112.2], [85.8]] = [[170], [130]].

Question 38Álgebra matricial y sistemas

Let A be a 3x3 invertible matrix with det(A) = -4. Calculate the value of det(A^(-1) · A^T · A^2).

  1. -16
  2. 64
  3. -4
  4. 16 Correct answer.

Explanation

Using properties of determinants: det(A^(-1) · A^T · A^2) = det(A^(-1)) * det(A^T) * (det(A))^2 = (1 / det(A)) * det(A) * (det(A))^2 = (det(A))^2. Since det(A) = -4, (det(A))^2 = (-4)^2 = 16.

Question 39Álgebra matricial y sistemas

Determine the value of the parameter m for which the matrix A = [[1, 1, 2], [2, 3, 5], [1, 2, m]] has rank equal to 2.

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

Explanation

The rank is 2 when the 3x3 determinant is 0. det(A) = 1 * (3m - 10) - 1 * (2m - 5) + 2 * (4 - 3) = 3m - 10 - 2m + 5 + 2 = m - 3. Setting det(A) = 0 yields m = 3.

Question 40Álgebra matricial y sistemas

Solve the system using Gauss elimination: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2. What is the value of x?

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

Explanation

Subtract 2*Eq1 from Eq2: -3y - z = -9 => 3y + z = 9. Subtract Eq1 from Eq3: y - 2z = -4. From y - 2z = -4 => y = 2z - 4. Substitute into 3(2z - 4) + z = 9 => 6z - 12 + z = 9 => 7z = 21 => z = 3. Then y = 2(3) - 4 = 2. Then x + 2 + 3 = 6 => x = 1.

Question 41Álgebra matricial y sistemas

If the matrix A = [[2, x + 1], [5, 4]] is symmetric, what is the value of x?

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

Explanation

A matrix is symmetric if A = A^T, which means off-diagonal elements are equal: a_12 = a_21. Here x + 1 = 5, so x = 4.

Question 42Álgebra matricial y sistemas

Compute the determinant of the 2x2 matrix A = [[4, -3], [2, 5]].

  1. 14
  2. 26 Correct answer.
  3. -26
  4. 23

Explanation

For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here det(A) = (4)(5) - (-3)(2) = 20 - (-6) = 20 + 6 = 26.

Question 43Álgebra matricial y sistemas

Given the matrix A = [[1, 1], [0, 1]], find the matrix A^10.

  1. [[1, 1], [0, 1]]
  2. [[1, 100], [0, 1]]
  3. [[10, 10], [0, 10]]
  4. [[1, 10], [0, 1]] Correct answer.

Explanation

A^2 = [[1, 1], [0, 1]] · [[1, 1], [0, 1]] = [[1, 2], [0, 1]]. By mathematical induction, A^n = [[1, n], [0, 1]]. Thus A^10 = [[1, 10], [0, 1]].

Question 44Álgebra matricial y sistemas

For which value of the parameter 'a' is the system ax + y + z = 1, x + ay + z = a, x + y + az = a^2 inconsistent (incompatible)?

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

Explanation

det(A) = (a - 1)^2 * (a + 2). det(A) = 0 at a = 1 and a = -2. For a = 1, all three equations become x + y + z = 1, which is compatible indeterminate. For a = -2, adding the three equations gives 0 = 3 (contradiction), so the system is incompatible.

Question 45Álgebra matricial y sistemas

Given matrix A = [[2, -1], [1, 3]], find the scalar k such that A^2 - 5A + k*I = 0, where I is the 2x2 identity matrix.

  1. 5
  2. 7 Correct answer.
  3. -5
  4. 1

Explanation

By the Cayley-Hamilton theorem, any square matrix satisfies its characteristic polynomial λ^2 - tr(A)λ + det(A) = 0. tr(A) = 2 + 3 = 5. det(A) = (2)(3) - (-1)(1) = 6 + 1 = 7. Thus A^2 - 5A + 7I = 0, so k = 7.

Question 46Álgebra matricial y sistemas

In an economy with two sectors, (I - C)^(-1) = [[1.25, 0.5], [0.25, 1.0]]. Labor coefficients per unit output are L = [0.3, 0.5]. If final demand for Sector 1 increases by 20 units (and Sector 2 demand is unchanged), how many additional worker-hours are required?

  1. 7.5
  2. 15
  3. 10 Correct answer.
  4. 12

Explanation

Change in total output ΔX = (I - C)^(-1) · ΔD = [[1.25, 0.5], [0.25, 1.0]] · [[20], [0]] = [[25], [5]]. Sector 1 output increases by 25 units, Sector 2 output increases by 5 units. Additional labor = L · ΔX = 0.3(25) + 0.5(5) = 7.5 + 2.5 = 10 worker-hours.

Question 47Álgebra matricial y sistemas

Find the sum of all roots x of the equation det([[x-1, 5, 0, 3], [0, x+2, 4, -1], [0, 0, x-3, 2], [0, 0, 0, 2]]) = 0.

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

Explanation

The matrix is upper triangular, so its determinant is the product of its diagonal entries: det = (x - 1)(x + 2)(x - 3)(2) = 0. Roots: x = 1, x = -2, x = 3. Sum of roots = 1 + (-2) + 3 = 2.

Question 48Álgebra matricial y sistemas

Solve the matrix equation A · X · B = C for X, where A = [[1, 0], [2, 1]], B = [[2, 1], [0, 1]], and C = [[4, 2], [10, 5]].

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

Explanation

X = A^(-1) · C · B^(-1). A^(-1) = [[1, 0], [-2, 1]]. B^(-1) = [[0.5, -0.5], [0, 1]]. A^(-1) · C = [[1, 0], [-2, 1]] · [[4, 2], [10, 5]] = [[4, 2], [2, 1]]. X = [[4, 2], [2, 1]] · [[0.5, -0.5], [0, 1]] = [[4(0.5)+0, 4(-0.5)+2], [2(0.5)+0, 2(-0.5)+1]] = [[2, 0], [1, 0]].

Question 49Álgebra matricial y sistemas

If matrix A has dimensions 2x3 and matrix B has dimensions 3x4, what are the dimensions of the transposed product matrix (A · B)^T?

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

Explanation

The product matrix A · B has dimensions (2x3) · (3x4) = 2x4. Taking the transpose swaps the dimensions, so (A · B)^T has dimensions 4x2.

Question 50Álgebra matricial y sistemas

Calculate the matrix product A · B for A = [[2, 1], [0, 3]] and B = [[1, 4], [2, -1]].

  1. [[2, 4], [0, -3]]
  2. [[4, 9], [6, -3]]
  3. [[3, 5], [2, 2]]
  4. [[4, 7], [6, -3]] Correct answer.

Explanation

Matrix multiplication row by column: Row 1: [2(1)+1(2), 2(4)+1(-1)] = [2+2, 8-1] = [4, 7]. Row 2: [0(1)+3(2), 0(4)+3(-1)] = [0+6, 0-3] = [6, -3]. Thus A · B = [[4, 7], [6, -3]].