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PAU Mathematics II (Aragón) study materials
PAU Mathematics II (Aragón) study resources

PAU Mathematics II (Aragón) practice questions

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Question 1Análisis, límites, derivadas y optimización

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

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

Explanation

This is an indeterminate form 0/0. Factoring numerator gives (x - 2)(x + 2) / (x - 2) = x + 2. Substituting x = 2 yields 2 + 2 = 4.

Question 2Análisis, límites, derivadas y optimización

Evaluate the trigonometric limit lim_{x -> 0} sin(3x) / x.

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

Explanation

Using the standard limit lim_{u->0} sin(u)/u = 1: lim_{x->0} 3 * (sin(3x) / (3x)) = 3 * 1 = 3.

Question 3Análisis, límites, derivadas y optimización

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

  1. infinity
  2. 2/3 Correct answer.
  3. 0
  4. 5/-1

Explanation

Dividing numerator and denominator by highest power x^2 gives (2 + 5/x) / (3 - 1/x^2). As x -> infinity, 5/x -> 0 and 1/x^2 -> 0, leaving 2/3.

Question 4Análisis, límites, derivadas y optimización

Using L'Hôpital's Rule, calculate lim_{x -> 0} (e^(2x) - 1 - 2x) / x^2.

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

Explanation

Direct substitution yields 0/0. Applying L'Hôpital once: lim (2 e^(2x) - 2) / (2x) (still 0/0). Applying L'Hôpital second time: lim (4 e^(2x)) / 2 = (4 * 1) / 2 = 2.

Question 5Análisis, límites, derivadas y optimización

Calculate the limit of the exponential indeterminate form 1^infinity: lim_{x -> infinity} (1 + 3/x)^(2x).

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

Explanation

Using the standard e formula lim_{x->inf} (1 + a/x)^(b x) = e^(a b). Here a = 3 and b = 2, so the limit is e^(3 * 2) = e^6.

Question 6Análisis, límites, derivadas y optimización

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

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

Explanation

Form 0/0. Applying L'Hôpital: lim_{x->0} sin(x) / (2x) = (1/2) * lim_{x->0} (sin(x)/x) = 1/2 * 1 = 1/2.

Question 7Análisis, límites, derivadas y optimización

Evaluate the radical limit at infinity: lim_{x -> infinity} (sqrt(x^2 + 4x) - x).

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

Explanation

Multiply and divide by conjugate: [(x^2 + 4x) - x^2] / [sqrt(x^2 + 4x) + x] = 4x / [x(sqrt(1 + 4/x) + 1)]. Taking x -> infinity yields 4 / (1 + 1) = 4 / 2 = 2.

Question 8Análisis, límites, derivadas y optimización

Evaluate the one-sided limit lim_{x -> 0^+} x * ln(x).

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

Explanation

Form 0 * (-infinity). Rewrite as lim_{x->0^+} ln(x) / (1/x) (form -inf / inf). Apply L'Hôpital: lim (1/x) / (-1/x^2) = lim (-x) = 0.

Question 9Análisis, límites, derivadas y optimización

Classify the discontinuity of f(x) = (x^2 - 9) / (x - 3) at x = 3.

  1. Essential infinite discontinuity (salto infinito)
  2. Continuous at x = 3
  3. Jump discontinuity of finite size (salto finito)
  4. Removable discontinuity (discontinuidad evitable) Correct answer.

Explanation

f(3) is undefined, but lim_{x->3} (x-3)(x+3)/(x-3) = lim_{x->3} (x+3) = 6 exists. Since the limit exists and is finite, x = 3 is a removable discontinuity.

Question 10Análisis, límites, derivadas y optimización

Find parameter 'a' such that f(x) is continuous at x = 1, where f(x) = { 2x + a for x <= 1; x^2 + 3 for x > 1 }.

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

Explanation

Left limit and function value at 1: lim_{x->1^-} f(x) = 2(1) + a = 2 + a. Right limit: lim_{x->1^+} f(x) = 1^2 + 3 = 4. Equating for continuity: 2 + a = 4 => a = 2.

Question 11Análisis, límites, derivadas y optimización

For f(x) = { 3x - 1 for x < 2; x^2 + 2 for x >= 2 }, calculate the jump size at the discontinuity x = 2.

  1. 6
  2. 5
  3. 1 Correct answer.
  4. 0

Explanation

Left limit: lim_{x->2^-} (3x - 1) = 5. Right limit: lim_{x->2^+} (x^2 + 2) = 6. Jump size = |lim_{x->2^+} f(x) - lim_{x->2^-} f(x)| = |6 - 5| = 1.

Question 12Análisis, límites, derivadas y optimización

Using Bolzano's Theorem, show that f(x) = x^3 + 2x - 5 has at least one real root in which interval?

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

Explanation

f(1) = 1^3 + 2(1) - 5 = -2 < 0. f(2) = 2^3 + 2(2) - 5 = 7 > 0. Since f is continuous on [1, 2] and f(1)*f(2) < 0, Bolzano's Theorem guarantees a root in (1, 2).

Question 13Análisis, límites, derivadas y optimización

For what values of k > 0 does Bolzano's Theorem guarantee a solution to cos(x) - k x = 0 in [0, pi/2]?

  1. Only k > pi/2
  2. Only 0 < k < 1
  3. All k > 0 Correct answer.
  4. No positive k

Explanation

Let g(x) = cos(x) - k x. g(0) = cos(0) - 0 = 1 > 0. g(pi/2) = cos(pi/2) - k(pi/2) = -k(pi/2). Since k > 0, g(pi/2) < 0. g is continuous, so g(0)*g(pi/2) < 0 for any k > 0, guaranteeing a root in [0, pi/2].

Question 14Análisis, límites, derivadas y optimización

Find parameters (a, b) so that f(x) = { a x + b for x <= 0; cos(x) for x > 0 } is differentiable at x = 0.

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

Explanation

Continuity at x=0: lim_{x->0^-} (ax + b) = b = cos(0) = 1 => b = 1. Differentiability at x=0: Left derivative is a. Right derivative of cos(x) at 0 is -sin(0) = 0. Equating derivatives gives a = 0. Thus (a, b) = (0, 1).

Question 15Análisis, límites, derivadas y optimización

Compute the derivative f'(x) for f(x) = x^3 * ln(x).

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

Explanation

By the product rule: f'(x) = (x^3)' ln(x) + x^3 (ln(x))' = 3x^2 ln(x) + x^3 (1/x) = 3x^2 ln(x) + x^2 = x^2 (3 ln(x) + 1).

Question 16Análisis, límites, derivadas y optimización

Using the chain rule, find the derivative of f(x) = e^(x^2 - 3x).

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

Explanation

Chain rule for e^u(x): f'(x) = u'(x) * e^u(x). Here u(x) = x^2 - 3x, so u'(x) = 2x - 3, giving f'(x) = (2x - 3) * e^(x^2 - 3x).

Question 17Análisis, límites, derivadas y optimización

Find the equation of the tangent line to the curve y = x^2 - 4x + 5 at the point x = 3.

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

Explanation

At x = 3, y(3) = 9 - 12 + 5 = 2. Derivative y' = 2x - 4, so slope m = y'(3) = 6 - 4 = 2. Tangent line: y - 2 = 2(x - 3) => y = 2x - 4.

Question 18Análisis, límites, derivadas y optimización

Using implicit differentiation, find the slope of the tangent line to x^2 + y^2 = 25 at the point (3, 4).

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

Explanation

Differentiating both sides with respect to x: 2x + 2y y' = 0 => y' = -x / y. At (3, 4), y' = -3 / 4.

Question 19Análisis, límites, derivadas y optimización

If f(x) = x^3 + x, evaluate the derivative of the inverse function (f^-1)'(2).

  1. 1/13
  2. 1/3
  3. 4
  4. 1/4 Correct answer.

Explanation

Notice f(1) = 1^3 + 1 = 2, so f^-1(2) = 1. f'(x) = 3x^2 + 1, so f'(1) = 3(1)^2 + 1 = 4. By the inverse derivative theorem, (f^-1)'(2) = 1 / f'(1) = 1 / 4.

Question 20Análisis, límites, derivadas y optimización

Find the equation of the normal line (recta normal) to y = ln(x) at x = 1.

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

Explanation

Point: (1, ln(1)) = (1, 0). Derivative y' = 1/x, so tangent slope m_T = y'(1) = 1. Normal slope m_N = -1 / m_T = -1. Normal line: y - 0 = -1(x - 1) => y = -x + 1 => x + y = 1.

Question 21Análisis, límites, derivadas y optimización

Find the point c in (1, 3) guaranteed by Rolle's Theorem for f(x) = x^2 - 4x + 3 on [1, 3].

  1. c = 0
  2. c = 1.5
  3. c = 2.5
  4. c = 2 Correct answer.

Explanation

f(1) = 0 and f(3) = 0. f is continuous on [1, 3] and differentiable on (1, 3). Rolle's theorem guarantees c in (1, 3) where f'(c) = 0. f'(x) = 2x - 4 = 0 => c = 2.

Question 22Análisis, límites, derivadas y optimización

Find the point c in (0, 4) satisfying Lagrange's Mean Value Theorem for f(x) = x^2 on [0, 4].

  1. c = 1
  2. c = 2 Correct answer.
  3. c = 3
  4. c = sqrt(8)

Explanation

Average rate of change = [f(4) - f(0)] / (4 - 0) = (16 - 0) / 4 = 4. Set f'(c) = 2c = 4 => c = 2.

Question 23Análisis, límites, derivadas y optimización

Verify Rolle's Theorem for f(x) = x^3 - 3x on [-sqrt(3), sqrt(3)]. What are the points c in (-sqrt(3), sqrt(3)) where f'(c) = 0?

  1. c = -sqrt(3) and c = sqrt(3)
  2. c = -1/2 and c = 1/2
  3. c = -1 and c = 1 Correct answer.
  4. c = 0 only

Explanation

f(-sqrt(3)) = 0 and f(sqrt(3)) = 0. f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 0 => x = -1 or x = 1. Both points lie inside (-sqrt(3), sqrt(3)).

Question 24Análisis, límites, derivadas y optimización

Apply Lagrange's Mean Value Theorem to f(x) = 1/x on [1, 3]. Find c in (1, 3).

  1. c = 1.5
  2. c = 9/4
  3. c = 2
  4. c = sqrt(3) Correct answer.

Explanation

f(1) = 1, f(3) = 1/3. Average slope = (1/3 - 1) / (3 - 1) = (-2/3) / 2 = -1/3. Derivative f'(c) = -1 / c^2. Setting -1/c^2 = -1/3 => c^2 = 3 => c = sqrt(3) ≈ 1.732 in (1, 3).

Question 25Análisis, límites, derivadas y optimización

Find the vertical asymptote of the rational function f(x) = (2x + 1) / (x - 4).

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

Explanation

Vertical asymptotes occur where the denominator is zero while numerator is non-zero. x - 4 = 0 => x = 4. Since 2(4) + 1 = 9 != 0, x = 4 is a vertical asymptote.

Question 26Análisis, límites, derivadas y optimización

Determine the horizontal asymptote of f(x) = (3x^2 + 2) / (x^2 - 1) as x -> +/- infinity.

  1. x = 1
  2. y = 3 Correct answer.
  3. No horizontal asymptote
  4. y = 0

Explanation

lim_{x->inf} (3x^2 + 2) / (x^2 - 1) = 3. Therefore, y = 3 is the horizontal asymptote.

Question 27Análisis, límites, derivadas y optimización

Find the equation of the oblique asymptote (asíntota oblicua) for f(x) = (x^2 + 3x + 1) / (x - 1).

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

Explanation

Polynomial long division of (x^2 + 3x + 1) by (x - 1): (x^2 + 3x + 1) = (x - 1)(x + 4) + 5. Thus f(x) = x + 4 + 5/(x - 1). As x -> inf, 5/(x-1) -> 0, so the oblique asymptote is y = x + 4.

Question 28Análisis, límites, derivadas y optimización

Find the local extrema of f(x) = x^3 - 3x^2 + 4.

  1. Local max at (0, 4), local min at (2, 0) Correct answer.
  2. Local min at (0, 4), local max at (2, 0)
  3. Local max at (1, 2), local min at (-1, 0)
  4. Inflection point at (0, 4)

Explanation

f'(x) = 3x^2 - 6x = 3x(x - 2) = 0 => critical points at x = 0 and x = 2. Second derivative f''(x) = 6x - 6. f''(0) = -6 < 0 => local max at x=0, y=f(0)=4. f''(2) = 6 > 0 => local min at x=2, y=f(2)=0.

Question 29Análisis, límites, derivadas y optimización

Find the inflection point (punto de inflexión) of f(x) = x^3 - 6x^2 + 9x + 1.

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

Explanation

f'(x) = 3x^2 - 12x + 9. f''(x) = 6x - 12 = 0 => x = 2. Since f'' changes sign across x = 2, it is an inflection point. f(2) = 8 - 24 + 18 + 1 = 3. Point: (2, 3).

Question 30Análisis, límites, derivadas y optimización

On which interval is the function f(x) = x * e^x concave upwards (cóncava hacia arriba)?

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

Explanation

f'(x) = (x + 1) e^x. f''(x) = (x + 2) e^x. Concave up where f''(x) > 0. Since e^x > 0 for all x, x + 2 > 0 => x > -2, so (-2, infinity).

Question 31Análisis, límites, derivadas y optimización

A rectangular plot of land is to be enclosed on 3 sides using 200 meters of fencing (the 4th side is a straight river bank). What is the maximum area that can be enclosed?

  1. 5000 m^2 Correct answer.
  2. 2500 m^2
  3. 10000 m^2
  4. 4000 m^2

Explanation

Let x be the width (perpendicular to river) and y be the length (parallel to river). Constraint: 2x + y = 200 => y = 200 - 2x. Area A(x) = x(200 - 2x) = 200x - 2x^2. A'(x) = 200 - 4x = 0 => x = 50 m. Then y = 100 m. Max Area = 50 * 100 = 5000 m^2.

Question 32Análisis, límites, derivadas y optimización

Find two positive numbers x and y such that x + y = 12 and the product P = x * y^2 is maximized.

  1. x = 8, y = 4
  2. x = 3, y = 9
  3. x = 4, y = 8 Correct answer.
  4. x = 6, y = 6

Explanation

x = 12 - y. Product P(y) = (12 - y) y^2 = 12y^2 - y^3. P'(y) = 24y - 3y^2 = 3y(8 - y) = 0. For y > 0, y = 8. Then x = 12 - 8 = 4. Max product P = 4 * 64 = 256.

Question 33Análisis, límites, derivadas y optimización

Find the minimum distance from the point (0, 3) to the parabola y = x^2.

  1. sqrt(11)/2 Correct answer.
  2. sqrt(5)/2
  3. 3
  4. sqrt(7)/2

Explanation

Distance squared D^2 = x^2 + (y - 3)^2. Since x^2 = y on the parabola: f(y) = y + (y - 3)^2 = y^2 - 5y + 9. f'(y) = 2y - 5 = 0 => y = 5/2. Then D^2 = (5/2)^2 - 5(5/2) + 9 = 25/4 - 50/4 + 36/4 = 11/4. Min distance D = sqrt(11)/2.

Question 34Análisis, límites, derivadas y optimización

An open box is constructed from a 12 cm by 12 cm square sheet of cardboard by cutting equal square corners of side x and folding up the sides. What value of x maximizes the volume?

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

Explanation

Dimensions of box: x by (12 - 2x) by (12 - 2x). Volume V(x) = x(12 - 2x)^2 = 4x^3 - 48x^2 + 144x. V'(x) = 12x^2 - 96x + 144 = 12(x^2 - 8x + 12) = 12(x - 2)(x - 6) = 0. Since 0 < x < 6, x = 2 cm gives max volume V(2) = 2(8)^2 = 128 cm^3.

Question 35Análisis, límites, derivadas y optimización

Evaluate the indefinite integral integral (3x^2 - 4x + 5) dx.

  1. x^3 - 2x^2 + 5x + C Correct answer.
  2. 6x - 4 + C
  3. 3x^3 - 4x^2 + 5x + C
  4. x^3 - 4x^2 + 5x + C

Explanation

Applying power rule for integrals term-by-term: integral 3x^2 dx = x^3, integral -4x dx = -2x^2, integral 5 dx = 5x. Result: x^3 - 2x^2 + 5x + C.

Question 36Análisis, límites, derivadas y optimización

Evaluate integral 2x * e^(x^2) dx using u-substitution.

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

Explanation

Let u = x^2, then du = 2x dx. The integral transforms to integral e^u du = e^u + C = e^(x^2) + C.

Question 37Análisis, límites, derivadas y optimización

Evaluate integral x * e^x dx using integration by parts.

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

Explanation

Choose u = x => du = dx, and dv = e^x dx => v = e^x. By parts formula integral u dv = u v - integral v du: x e^x - integral e^x dx = x e^x - e^x + C = (x - 1) e^x + C.

Question 38Análisis, límites, derivadas y optimización

Evaluate integral 1 / (x^2 - 5x + 6) dx using partial fractions.

  1. ln |x^2 - 5x + 6| + C
  2. ln |(x - 3) / (x - 2)| + C Correct answer.
  3. 1 / (2x - 5) + C
  4. ln |(x - 2) / (x - 3)| + C

Explanation

Factor denominator: (x - 2)(x - 3). Decompose: 1 / [(x - 2)(x - 3)] = A / (x - 2) + B / (x - 3). 1 = A(x - 3) + B(x - 2). For x = 3 => B = 1; for x = 2 => A = -1. Integral = integral [-1/(x-2) + 1/(x-3)] dx = -ln|x-2| + ln|x-3| + C = ln |(x - 3) / (x - 2)| + C.

Question 39Análisis, límites, derivadas y optimización

Using Barrow's Rule (Fundamental Theorem of Calculus), calculate the definite integral integral_0^2 (3x^2 + 1) dx.

  1. 14
  2. 12
  3. 10 Correct answer.
  4. 8

Explanation

Antiderivative F(x) = x^3 + x. By Barrow's Rule: F(2) - F(0) = (2^3 + 2) - (0^3 + 0) = (8 + 2) - 0 = 10.

Question 40Análisis, límites, derivadas y optimización

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

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

Explanation

Intersection points: x^2 = x => x(x - 1) = 0 => x = 0 and x = 1. On [0, 1], x >= x^2. Area = integral_0^1 (x - x^2) dx = [x^2/2 - x^3/3]_0^1 = (1/2 - 1/3) - 0 = 1/6.

Question 41Álgebra matricial y sistemas

What is the determinant of the 2x2 matrix A = [[3, -1], [5, 2]]?

  1. 1
  2. -11
  3. 11 Correct answer.
  4. 6

Explanation

For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here det(A) = (3)(2) - (-1)(5) = 6 - (-5) = 6 + 5 = 11.

Question 42Álgebra matricial y sistemas

Given matrices A = [[1, 4], [-2, 3]] and B = [[3, -1], [5, 0]], what is the matrix sum A + B?

  1. [[4, 3], [3, 3]] Correct answer.
  2. [[3, -4], [-10, 0]]
  3. [[4, 5], [3, 3]]
  4. [[2, 5], [-7, 3]]

Explanation

Matrix addition is performed entrywise: A + B = [[1+3, 4+(-1)], [-2+5, 3+0]] = [[4, 3], [3, 3]].

Question 43Álgebra matricial y sistemas

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

  1. [[5, -2], [-5, 5]]
  2. [[4, 2], [5, 5]]
  3. [[4, -2], [11, 13]]
  4. [[4, 2], [-5, 5]] Correct answer.

Explanation

Compute 3A = [[6, 0], [3, 9]] and 2B = [[2, -2], [8, 4]]. Then 3A - 2B = [[6-2, 0-(-2)], [3-8, 9-4]] = [[4, 2], [-5, 5]].

Question 44Álgebra matricial y sistemas

For matrices A = [[1, 2], [3, 0]] and B = [[2, -1], [1, 4]], what is the matrix product A * B?

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

Explanation

Row-column multiplication: top-left = 1(2)+2(1)=4; top-right = 1(-1)+2(4)=7; bottom-left = 3(2)+0(1)=6; bottom-right = 3(-1)+0(4)=-3. Thus A*B = [[4, 7], [6, -3]].

Question 45Álgebra matricial y sistemas

If A = [[1, 2], [0, 5]], what is the transpose matrix A^T?

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

Explanation

The transpose of a matrix swaps its rows and columns. The first row (1, 2) becomes the first column, and the second row (0, 5) becomes the second column, yielding [[1, 0], [2, 5]].

Question 46Álgebra matricial y sistemas

Solve the matrix equation A X = B for column vector X, where A = [[1, 2], [3, 5]] and B = [[1], [2]].

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

Explanation

det(A) = 1(5) - 2(3) = -1. A^-1 = (1/-1) [[5, -2], [-3, 1]] = [[-5, 2], [3, -1]]. Then X = A^-1 B = [[-5(1) + 2(2)], [3(1) - 1(2)]] = [[-1], [1]].

Question 47Álgebra matricial y sistemas

If the matrix A = [[2, x - 1], [3, 5]] is symmetric, what is the value of x?

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

Explanation

A matrix is symmetric if A^T = A, meaning off-diagonal elements are equal: a_12 = a_21. Here x - 1 = 3 => x = 4.

Question 48Álgebra matricial y sistemas

Given A = [[1, 0], [2, 1]], find the matrix power A^5.

  1. [[1, 0], [2, 1]]
  2. [[5, 0], [10, 5]]
  3. [[1, 0], [32, 1]]
  4. [[1, 0], [10, 1]] Correct answer.

Explanation

A^2 = [[1, 0], [4, 1]], A^3 = [[1, 0], [6, 1]], and by induction A^n = [[1, 0], [2n, 1]]. For n = 5, A^5 = [[1, 0], [10, 1]].

Question 49Álgebra matricial y sistemas

Using Sarrus' rule, calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [2, -1, 1], [0, 3, 1]].

  1. 11
  2. 14
  3. 8 Correct answer.
  4. -8

Explanation

det(A) = 1(-1)(1) + 0(1)(0) + 2(2)(3) - [0(-1)(2) + 3(1)(1) + 1(2)(0)] = (-1 + 0 + 12) - (0 + 3 + 0) = 11 - 3 = 8.

Question 50Álgebra matricial y sistemas

If A is a 3x3 matrix with determinant det(A) = 5, what is the determinant of the matrix 2A?

  1. 30
  2. 40 Correct answer.
  3. 20
  4. 10

Explanation

For an n x n matrix, det(k A) = k^n det(A). Here n = 3 and k = 2, so det(2A) = 2^3 * det(A) = 8 * 5 = 40.