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Catalonia PAU Applied Mathematics for Social Sciences Exam study materials
Catalonia PAU Applied Mathematics for Social Sciences Exam study resources

Catalonia PAU Applied Mathematics for Social Sciences Exam practice questions

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Question 1Análisis, límites, derivadas y optimización

Find the domain of the real-valued function f(x) = (2x + 1) / (x^2 - 4).

  1. All real numbers except x = -2 and x = 2 Correct answer.
  2. All real numbers except x = 4
  3. x > 2
  4. All real numbers

Explanation

A rational function is defined for all real numbers except where its denominator equals zero. Setting x^2 - 4 = 0 gives x^2 = 4, so x = 2 or x = -2. The domain is R \ {-2, 2}.

Question 2Análisis, límites, derivadas y optimización

Determine the domain of the logarithmic function f(x) = ln(3x - 9).

  1. (3, +infinity) Correct answer.
  2. [3, +infinity)
  3. (0, +infinity)
  4. All real numbers except x = 3

Explanation

The argument of a natural logarithm must be strictly positive. Setting 3x - 9 > 0 yields 3x > 9, so x > 3. In interval notation, the domain is (3, +infinity).

Question 3Análisis, límites, derivadas y optimización

Evaluate the limit at infinity: lim_{x -> infinity} (3x^2 + 5x - 2) / (2x^2 - 7).

  1. 3/2 Correct answer.
  2. 0
  3. +infinity
  4. 2/7

Explanation

For a rational function as x approaches infinity where the numerator and denominator have equal degrees (degree 2), the limit equals the ratio of leading coefficients: 3/2.

Question 4Análisis, límites, derivadas y optimización

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

  1. 6 Correct answer.
  2. 0
  3. Undefined (0/0)
  4. 3

Explanation

Direct substitution yields 0/0 indeterminacy. Factor the numerator as (x - 3)(x + 3). Cancel (x - 3) for x != 3 to get lim_{x -> 3} (x + 3) = 3 + 3 = 6.

Question 5Análisis, límites, derivadas y optimización

Find the value of k that makes f(x) continuous at x = 2: f(x) = 3x + k for x <= 2 and f(x) = x^2 + 2k for x > 2.

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

Explanation

For continuity at x = 2, left limit equals right limit: lim_{x->2^-} (3x + k) = 3(2) + k = 6 + k; lim_{x->2^+} (x^2 + 2k) = 2^2 + 2k = 4 + 2k. Setting 6 + k = 4 + 2k yields k = 2.

Question 6Análisis, límites, derivadas y optimización

Identify the type of discontinuity at x = 1 for piecewise function f(x) = 2x + 1 for x < 1 and f(x) = x^2 + 4 for x >= 1.

  1. Finite jump discontinuity (jump of 2) Correct answer.
  2. Removable discontinuity
  3. Essential infinite discontinuity
  4. Continuous everywhere

Explanation

Left limit is lim_{x->1^-} (2x + 1) = 3. Right limit is lim_{x->1^+} (x^2 + 4) = 5. Both one-sided limits exist and are finite, but they are unequal (3 != 5). This constitutes a finite jump discontinuity of size 5 - 3 = 2.

Question 7Análisis, límites, derivadas y optimización

Find the horizontal and vertical asymptotes of f(x) = (4x - 2) / (x + 3).

  1. Horizontal: y = 4; Vertical: x = -3 Correct answer.
  2. Horizontal: y = -2/3; Vertical: x = 3
  3. Horizontal: y = 0; Vertical: x = -3
  4. Horizontal: y = 4; Vertical: x = 3

Explanation

Vertical asymptote occurs where denominator equals zero: x + 3 = 0 => x = -3. Horizontal asymptote is limit as x -> infinity: ratio of leading coefficients is 4/1 = 4, so y = 4.

Question 8Análisis, límites, derivadas y optimización

For the polynomial function f(x) = 2x^3 - 5x^2 + 4x - 7, calculate derivative value f'(2).

  1. 8 Correct answer.
  2. 12
  3. 4
  4. 16

Explanation

Find derivative: f'(x) = 6x^2 - 10x + 4. Substitute x = 2: f'(2) = 6(2^2) - 10(2) + 4 = 6(4) - 20 + 4 = 24 - 20 + 4 = 8.

Question 9Análisis, límites, derivadas y optimización

Differentiate f(x) = x^2 * e^x with respect to x.

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

Explanation

Use product rule (u*v)' = u'v + uv': let u = x^2 (u' = 2x) and v = e^x (v' = e^x). Then f'(x) = 2x * e^x + x^2 * e^x = (x^2 + 2x) * e^x.

Question 10Análisis, límites, derivadas y optimización

Calculate the derivative value f'(2) for function f(x) = (2x + 3) / (x - 1).

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

Explanation

Use quotient rule f'(x) = [2(x-1) - (2x+3)(1)] / (x-1)^2 = [2x - 2 - 2x - 3] / (x-1)^2 = -5 / (x-1)^2. Substituting x = 2 gives f'(2) = -5 / (2-1)^2 = -5.

Question 11Análisis, límites, derivadas y optimización

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

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

Explanation

Point on curve: f(2) = 2^2 - 3(2) + 5 = 4 - 6 + 5 = 3, so point is (2,3). Derivative: f'(x) = 2x - 3 => slope m = f'(2) = 2(2) - 3 = 1. Tangent line: y - 3 = 1(x - 2) => y = x + 1.

Question 12Análisis, límites, derivadas y optimización

Determine the intervals where function f(x) = x^3 - 3x^2 - 9x + 2 is strictly increasing.

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

Explanation

Derivative: f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1). Setting f'(x) > 0 gives critical roots x = -1 and x = 3. Sign analysis shows f'(x) > 0 for x < -1 and x > 3, so f is increasing on (-infinity, -1) U (3, +infinity).

Question 13Análisis, límites, derivadas y optimización

Find coordinates of relative maximum and relative minimum for f(x) = x^3 - 6x^2 + 9x + 5.

  1. Relative max at (1, 9); Relative min at (3, 5) Correct answer.
  2. Relative max at (3, 5); Relative min at (1, 9)
  3. Relative max at (0, 5); Relative min at (2, 7)
  4. Relative max at (2, 7); Relative min at (4, 9)

Explanation

Derivative f'(x) = 3x^2 - 12x + 9 = 3(x - 1)(x - 3). Critical points at x = 1 and x = 3. Second derivative f''(x) = 6x - 12: f''(1) = -6 < 0 (relative max at x=1, y=1-6+9+5=9); f''(3) = +6 > 0 (relative min at x=3, y=27-54+27+5=5).

Question 14Análisis, límites, derivadas y optimización

Find the inflection point coordinates of function f(x) = x^3 - 3x^2 + 2.

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

Explanation

First derivative f'(x) = 3x^2 - 6x. Second derivative f''(x) = 6x - 6. Setting f''(x) = 0 yields x = 1. Value f(1) = 1^3 - 3(1)^2 + 2 = 1 - 3 + 2 = 0. Since f'' changes sign at x = 1, (1, 0) is the inflection point.

Question 15Análisis, límites, derivadas y optimización

A firm's profit function is P(x) = -x^2 + 120x - 2000 in euros, where x is quantity produced. Find quantity x that maximizes profit.

  1. 60 units Correct answer.
  2. 120 units
  3. 40 units
  4. 50 units

Explanation

To maximize profit, set P'(x) = -2x + 120 = 0, which gives 2x = 120, so x = 60 units. (P''(x) = -2 < 0 confirms a maximum). Maximum profit is P(60) = -3600 + 7200 - 2000 = 1600 euros.

Question 16Análisis, límites, derivadas y optimización

Demand function for a product is p(x) = 100 - 2x. Total revenue is R(x) = x * p(x) = 100x - 2x^2. Find selling price p that maximizes total revenue.

  1. 50 euros Correct answer.
  2. 25 euros
  3. 100 euros
  4. 75 euros

Explanation

Maximize R(x): R'(x) = 100 - 4x = 0 => x = 25 units. Substitute optimal quantity x = 25 into demand function: p(25) = 100 - 2(25) = 50 euros.

Question 17Análisis, límites, derivadas y optimización

Total cost function is C(x) = 2x^2 + 3200. Average cost function is AC(x) = C(x)/x = 2x + 3200/x. Find quantity x > 0 that minimizes average cost.

  1. 40 units Correct answer.
  2. 80 units
  3. 1600 units
  4. 20 units

Explanation

Differentiate AC(x): AC'(x) = 2 - 3200/x^2 = 0 => 2x^2 = 3200 => x^2 = 1600 => x = 40 units. (Minimum average cost is AC(40) = 2(40) + 3200/40 = 80 + 80 = 160 euros/unit).

Question 18Análisis, límites, derivadas y optimización

Calculate the total change in revenue between x = 0 and x = 5 given marginal revenue function MR(x) = 3x^2 + 4.

  1. 145 euros Correct answer.
  2. 125 euros
  3. 79 euros
  4. 150 euros

Explanation

Total change in revenue is definite integral from 0 to 5 of MR(x) dx: integral_0^5 (3x^2 + 4) dx = [x^3 + 4x]_0^5 = (5^3 + 4(5)) - (0) = 125 + 20 = 145 euros.

Question 19Análisis, límites, derivadas y optimización

Find the area bounded by marginal profit curve P'(x) = 12 - 3x^2 and x-axis between x = 0 and positive root x = 2.

  1. 16 Correct answer.
  2. 24
  3. 8
  4. 12

Explanation

Integrate from 0 to 2: integral_0^2 (12 - 3x^2) dx = [12x - x^3]_0^2 = (12(2) - 2^3) - 0 = 24 - 8 = 16.

Question 20Análisis, límites, derivadas y optimización

Given total cost function C(x) = 500 + 10x + 0.05x^2, calculate marginal cost at production level x = 100 units.

  1. 20 euros/unit Correct answer.
  2. 15 euros/unit
  3. 10 euros/unit
  4. 25 euros/unit

Explanation

Marginal cost is the derivative C'(x) = 10 + 0.10x. Substituting x = 100 gives C'(100) = 10 + 0.10(100) = 10 + 10 = 20 euros/unit.

Question 21Análisis, límites, derivadas y optimización

Function f(x) = ax^2 + bx + 1 for x <= 1 and f(x) = 4x - 2 for x > 1. Find parameters a and b such that f(x) is differentiable at x = 1.

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

Explanation

Continuity at x=1: a(1)^2 + b(1) + 1 = 4(1) - 2 => a + b + 1 = 2 => a + b = 1. Differentiability at x=1: Left derivative 2a(1) + b = 2a + b; Right derivative 4 => 2a + b = 4. Subtracting equations: (2a+b) - (a+b) = 4 - 1 => a = 3. Then b = 1 - 3 = -2.

Question 22Análisis, límites, derivadas y optimización

Price elasticity of demand is Ed = -(p/x) * (dx/dp). For demand function x = 200 - 4p, calculate price elasticity at p = 25.

  1. 1.0 (Unitary elasticity) Correct answer.
  2. 0.5 (Inelastic)
  3. 2.0 (Elastic)
  4. 4.0

Explanation

At p = 25: quantity x = 200 - 4(25) = 100 units. Derivative dx/dp = -4. Substitute into elasticity formula: Ed = -(25 / 100) * (-4) = (0.25) * 4 = 1.0. At p = 25, demand has unitary elasticity.

Question 23Análisis, límites, derivadas y optimización

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

  1. 5 at x = 2 Correct answer.
  2. 1 at x = 0
  3. -4 at x = 5
  4. 9 at x = 4

Explanation

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

Question 24Análisis, límites, derivadas y optimización

On which interval is the curve f(x) = x^3 - 6x^2 concave upwards?

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

Explanation

Differentiate twice: f'(x) = 3x^2 - 12x; f''(x) = 6x - 12. A curve is concave upwards where f''(x) > 0 => 6x - 12 > 0 => 6x > 12 => x > 2. Thus concave upwards on (2, +infinity).

Question 25Análisis, límites, derivadas y optimización

A firm sells x items at price p(x) = 120 - x. Production cost is C(x) = 20x + 500. A unit tax of t = 10 euros per item sold is introduced. Find output x maximizing post-tax profit.

  1. 45 units Correct answer.
  2. 50 units
  3. 40 units
  4. 60 units

Explanation

Post-tax profit P(x) = R(x) - C(x) - tx = x(120 - x) - (20x + 500) - 10x = -x^2 + 90x - 500. Differentiate and set to zero: P'(x) = -2x + 90 = 0 => 2x = 90 => x = 45 units.

Question 26Álgebra matricial y sistemas

Given matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]], calculate the result of 2A - 3B.

  1. [[1, -17], [12, 8]] Correct answer.
  2. [[1, -17], [0, 8]]
  3. [[7, 13], [0, 8]]
  4. [[1, 13], [12, 8]]

Explanation

To compute 2A - 3B, first multiply matrix A by 2 to get [[4, -2], [6, 8]] and matrix B by 3 to get [[3, 15], [-6, 0]]. Subtracting 3B from 2A yields [[4-3, -2-15], [6-(-6), 8-0]] = [[1, -17], [12, 8]].

Question 27Álgebra matricial y sistemas

If 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. 3 x 3
  2. 2 x 4 Correct answer.
  3. 4 x 2
  4. The product is undefined

Explanation

Matrix multiplication is defined when the number of columns of the first matrix matches the number of rows of the second matrix (both are 3). The resulting product matrix takes the row dimension of the first matrix (2) and the column dimension of the second matrix (4), giving dimensions 2 x 4.

Question 28Álgebra matricial y sistemas

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

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

Explanation

Multiplying row by column: top-left entry is 1(2) + 2(1) = 4; top-right entry is 1(0) + 2(-1) = -2; bottom-left entry is 3(2) + 4(1) = 10; bottom-right entry is 3(0) + 4(-1) = -4. The product matrix is [[4, -2], [10, -4]].

Question 29Álgebra matricial y sistemas

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

  1. 14
  2. 26 Correct answer.
  3. -26
  4. 20

Explanation

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

Question 30Álgebra matricial y sistemas

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

  1. 9 Correct answer.
  2. -11
  3. 5
  4. 0

Explanation

Expanding along the first row: det(A) = 1 * ((-1)(1) - (4)(0)) - 2 * ((3)(1) - (4)(2)) + 0 = 1(-1) - 2(3 - 8) = -1 - 2(-5) = -1 + 10 = 9.

Question 31Álgebra matricial y sistemas

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

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

Explanation

The determinant of A is (4)(2) - (2)(3) = 8 - 6 = 2. The adjugate matrix is [[2, -2], [-3, 4]]. Multiplying by 1/det(A) = 1/2 gives A^-1 = [[1, -1], [-1.5, 2]].

Question 32Álgebra matricial y sistemas

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

  1. k = 4 Correct answer.
  2. k = -4
  3. k = 9
  4. k = 0

Explanation

A matrix is singular if its determinant equals zero. Setting det(A) = 3k - (6)(2) = 3k - 12 = 0 gives 3k = 12, so k = 4.

Question 33Álgebra matricial y sistemas

Find the value of k for which the matrix A = [[1, 0, k], [2, k, 1], [0, 1, 1]] does NOT have an inverse.

  1. k = 1/3 Correct answer.
  2. k = 3
  3. k = -1
  4. k = 0

Explanation

A matrix fails to have an inverse when its determinant is zero. Expanding along the first row: det(A) = 1(k(1) - 1(1)) - 0 + k(2(1) - k(0)) = k - 1 + 2k = 3k - 1. Setting 3k - 1 = 0 yields k = 1/3.

Question 34Álgebra matricial y sistemas

Solve the matrix equation A * X = B for vector X, given A = [[3, 1], [2, 1]] and B = [[5], [4]].

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

Explanation

The determinant of A is (3)(1) - (1)(2) = 1. The inverse A^-1 is [[1, -1], [-2, 3]]. Multiplying X = A^-1 * B gives [[1, -1], [-2, 3]] * [[5], [4]] = [[1(5) - 1(4)], [-2(5) + 3(4)]] = [[1], [2]]. Thus x = 1 and y = 2.

Question 35Álgebra matricial y sistemas

Given the matrix equation X * A + B = C, where A is an invertible matrix, solve for X.

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

Explanation

First subtract B from both sides to obtain X * A = C - B. Since matrix multiplication is non-commutative and X is multiplied on the right by A, post-multiply both sides by A^-1 to get X = (C - B) * A^-1.

Question 36Álgebra matricial y sistemas

Which of the following algebraic properties holds true for any conformable matrices A and B?

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

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 determinant is not additive.

Question 37Álgebra matricial y sistemas

Classify the system of linear equations: 2x - y = 5 and 4x - 2y = 10.

  1. Dependent consistent (infinitely many solutions) Correct answer.
  2. Independent consistent (unique solution)
  3. Inconsistent (no solution)
  4. Trivial solution only

Explanation

Multiplying the first equation by 2 yields 4x - 2y = 10, which is identical to the second equation. The coefficient rank equals the augmented matrix rank (rank = 1 < 2 variables), so the system is dependent consistent with infinitely many solutions.

Question 38Álgebra matricial y sistemas

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

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

Explanation

The determinant of the coefficient matrix is det(A) = 3 - k. Setting det(A) = 0 gives k = 3. When k = 3, adding Row 1 (x+y+z=1) and Row 3 (x+2y+2z=2) gives 2x+3y+3z=3, which equals Row 2. Thus rank(A) = rank(A|B) = 2 < 3, so the system has infinitely many solutions.

Question 39Álgebra matricial y sistemas

Solve the system 3x + 2y = 12 and 5x - y = 7 using Cramer's Rule. What is the value of x?

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

Explanation

The coefficient determinant is det(A) = (3)(-1) - (2)(5) = -13. Replacing column 1 with constants gives det(A_x) = (12)(-1) - (2)(7) = -12 - 14 = -26. By Cramer's Rule, x = det(A_x) / det(A) = -26 / -13 = 2.

Question 40Álgebra matricial y sistemas

Solve the 3x3 system of linear equations: x + y + z = 6, 2x - y + z = 3, and x + y - z = 0. Find the value of z.

  1. z = 3 Correct answer.
  2. z = 2
  3. z = 1
  4. z = 6

Explanation

Subtracting Equation 3 (x+y-z=0) from Equation 1 (x+y+z=6) gives 2z = 6, so z = 3. Substituting z = 3 gives x + y = 3 and 2x - y = 0, which yields x = 1 and y = 2.

Question 41Álgebra matricial y sistemas

A bakery produces two types of cakes: Cake A requires 200g of flour and 100g of sugar; Cake B requires 150g of flour and 150g of sugar. If 3.2 kg of flour and 2.2 kg of sugar are used in total, how many units of Cake A were produced?

  1. 10 Correct answer.
  2. 8
  3. 12
  4. 6

Explanation

Convert kg to grams: flour equation is 200A + 150B = 3200; sugar equation is 100A + 150B = 2200. Subtracting the second equation from the first gives 100A = 1000, so A = 10 cakes.

Question 42Álgebra matricial y sistemas

A store sells products A, B, and C with unit prices P = [15, 25, 40] in euros. On Day 1, sales quantities are [10, 5, 2]. Calculate total revenue on Day 1.

  1. 355 euros Correct answer.
  2. 275 euros
  3. 400 euros
  4. 310 euros

Explanation

Total revenue is the dot product of price and quantity vectors: R = (10)(15) + (5)(25) + (2)(40) = 150 + 125 + 80 = 355 euros.

Question 43Álgebra matricial y sistemas

A square matrix A is called symmetric if it satisfies which condition?

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

Explanation

By definition, a matrix is symmetric if it equals its transpose, i.e., A^T = A. (If A^T = -A, it is skew-symmetric; if A * A^T = I, it is orthogonal).

Question 44Álgebra matricial y sistemas

Given matrix A = [[1, 1], [0, 1]], calculate the matrix power A^3.

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

Explanation

Computing powers: A^2 = [[1, 1], [0, 1]] * [[1, 1], [0, 1]] = [[1, 2], [0, 1]]. Multiplying by A again yields A^3 = [[1, 2], [0, 1]] * [[1, 1], [0, 1]] = [[1, 3], [0, 1]]. In general, A^n = [[1, n], [0, 1]].

Question 45Álgebra matricial y sistemas

In a 2-sector Leontief input-output model, the technology matrix is A = [[0.2, 0.3], [0.4, 0.1]] and final demand vector is D = [[120], [180]]. Calculate gross output vector X = (I - A)^-1 * D.

  1. Sector 1 = 270, Sector 2 = 320 Correct answer.
  2. Sector 1 = 200, Sector 2 = 250
  3. Sector 1 = 300, Sector 2 = 350
  4. Sector 1 = 150, Sector 2 = 200

Explanation

Compute I - A = [[0.8, -0.3], [-0.4, 0.9]]. Its determinant is (0.8)(0.9) - (-0.3)(-0.4) = 0.72 - 0.12 = 0.60. Inverse (I - A)^-1 = (1/0.60) * [[0.9, 0.3], [0.4, 0.8]] = [[1.5, 0.5], [2/3, 4/3]]. Multiplying by D = [[120], [180]]: Sector 1 output = 1.5(120) + 0.5(180) = 180 + 90 = 270; Sector 2 output = (2/3)(120) + (4/3)(180) = 80 + 240 = 320.

Question 46Programación lineal

A manufacturing problem has constraints x >= 0, y >= 0, 2x + y <= 10, and x + 2y <= 12. Which of the following test points lies INSIDE the feasible region?

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

Explanation

Test (2, 4): 2(2) + 4 = 8 <= 10 (satisfied) and 2 + 2(4) = 10 <= 12 (satisfied). Since both constraints and non-negativity hold, (2, 4) is in the feasible region.

Question 47Programación lineal

Find the coordinates of the vertex formed by the intersection of boundary lines x + y = 10 and 2x + y = 16.

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

Explanation

Subtracting the first equation from the second equation: (2x + y) - (x + y) = 16 - 10 => x = 6. Substituting x = 6 into x + y = 10 yields y = 4. The intersection vertex is (6, 4).

Question 48Programación lineal

Determine all vertices (corner points) of the feasible region bounded by constraints x >= 0, y >= 0, x + y <= 6, and 2x + y <= 8.

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

Explanation

The non-negativity axis intercepts are (0,0), (4,0) from 2x+y=8 on x-axis, and (0,6) from x+y=6 on y-axis. Intersecting x+y=6 and 2x+y=8 gives x=2, y=4. The vertices are (0,0), (4,0), (2,4), and (0,6).

Question 49Programación lineal

Maximize the objective function Z = 3x + 4y over the feasible region with vertices (0,0), (4,0), (2,4), and (0,6).

  1. 24 at vertex (0,6) Correct answer.
  2. 22 at vertex (2,4)
  3. 12 at vertex (4,0)
  4. 28 at vertex (4,4)

Explanation

Evaluate Z at all vertices: Z(0,0) = 0; Z(4,0) = 3(4)+0 = 12; Z(2,4) = 3(2)+4(4) = 6+16 = 22; Z(0,6) = 3(0)+4(6) = 24. The maximum value is 24, occurring at vertex (0,6).

Question 50Programación lineal

Minimize cost function C = 5x + 2y subject to an unbounded feasible region with vertices (1,8), (3,3), and (7,1).

  1. Minimum cost 21 at vertices (1,8) and (3,3) Correct answer.
  2. Minimum cost 37 at vertex (7,1)
  3. Minimum cost 13 at vertex (1,4)
  4. Minimum cost 0 at origin

Explanation

Evaluate C at vertices: C(1,8) = 5(1)+2(8) = 21; C(3,3) = 5(3)+2(3) = 21; C(7,1) = 5(7)+2(1) = 37. The minimum value is 21, achieved at both (1,8) and (3,3) (and along the entire edge connecting them).