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PAU Matemáticas II (Extremadura) study materials
PAU Matemáticas II (Extremadura) study resources

PAU Matemáticas II (Extremadura) practice questions

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Question 1Álgebra matricial y sistemas

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

  1. [[4, 8], [-14, 3]] Correct answer.
  2. [[4, -4], [10, 3]]
  3. [[2, 3], [-5, 2]]
  4. [[4, 8], [10, 9]]

Explanation

Compute scalar multiples: 2A = [[4, 2], [-2, 6]] and 3B = [[0, -6], [12, 3]]. Subtracting component-wise yields 2A - 3B = [[4-0, 2-(-6)], [-2-12, 6-3]] = [[4, 8], [-14, 3]].

Question 2Álgebra matricial y sistemas

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

  1. 15
  2. 24 Correct answer.
  3. 0
  4. 12

Explanation

For any triangular matrix (upper or lower), the determinant is the product of its main diagonal entries: det(A) = 1 * 4 * 6 = 24.

Question 3Álgebra matricial y sistemas

Find the inverse of the 2x2 matrix A = [[3, 1], [5, 2]].

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

Explanation

The determinant of A is det(A) = 3*2 - 1*5 = 6 - 5 = 1. The inverse formula for a 2x2 matrix [[a, b], [c, d]] is (1/det(A)) * [[d, -b], [-c, a]]. Thus A^-1 = [[2, -1], [-5, 3]].

Question 4Álgebra matricial y sistemas

Solve the linear system 2x + 3y = 8 and x - y = -1 using Cramer's rule. What is the value of x?

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

Explanation

The main determinant is D = |[2, 3], [1, -1]| = -2 - 3 = -5. Replacing the first column with constants yields Dx = |[8, 3], [-1, -1]| = -8 - (-3) = -5. By Cramer's rule, x = Dx / D = -5 / -5 = 1.

Question 5Álgebra matricial y sistemas

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

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

Explanation

Row 2 is exactly twice Row 1 (R2 = 2*R1), making R2 linearly dependent. Row 1 [1, 2, 3] and Row 3 [0, 1, 1] are linearly independent because no scalar multiple of [0, 1, 1] gives [1, 2, 3]. Therefore, the maximum number of linearly independent rows is 2, so rank(M) = 2.

Question 6Álgebra matricial y sistemas

If matrix A has dimensions 2x3 and matrix B has dimensions 3x4, what are the dimensions of the product matrix C = A * B?

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

Explanation

Matrix multiplication of an (m x n) matrix and an (n x p) matrix produces an (m x p) matrix. Here m = 2, n = 3, and p = 4, so C has dimensions 2x4.

Question 7Álgebra matricial y sistemas

For what values of the real parameter k is the matrix A = [[k, 1, 1], [1, k, 1], [1, 1, k]] singular (det(A) = 0)?

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

Explanation

Computing the determinant det(A) = k(k^2 - 1) - 1(k - 1) + 1(1 - k) = k^3 - 3k + 2. Factoring the polynomial yields (k - 1)^2 * (k + 2) = 0. The roots are k = 1 and k = -2.

Question 8Álgebra matricial y sistemas

Consider the system with parameter a: x + y + z = 1, x + ay + z = 1, x + y + az = a. For a = 1, how many solutions does the system possess?

  1. Unique solution
  2. No solution
  3. Infinitely many solutions with 2 degrees of freedom Correct answer.
  4. Infinitely many solutions with 1 degree of freedom

Explanation

Substituting a = 1 reduces all three equations to x + y + z = 1. This represents a single plane in R^3. The rank of the coefficient matrix and augmented matrix are both 1. With n = 3 variables, the number of free parameters is 3 - 1 = 2 degrees of freedom.

Question 9Álgebra matricial y sistemas

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

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

Explanation

Subtract B from both sides: A * X = C - B. Left-multiply both sides by A^-1: A^-1 * (A * X) = A^-1 * (C - B) => X = A^-1 * (C - B). Left-multiplication is mandatory because matrix multiplication is non-commutative.

Question 10Álgebra matricial y sistemas

Using Cramer's rule on the system x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2, find the value of z.

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

Explanation

Determinant D = |[1, 1, 1], [2, -1, 1], [1, 2, -1]| = 1(1 - 2) - 1(-2 - 1) + 1(4 + 1) = -1 + 3 + 5 = 7. Dz = |[1, 1, 6], [2, -1, 3], [1, 2, 2]| = 1(-2 - 6) - 1(4 - 3) + 6(4 + 1) = -8 - 1 + 30 = 21. Thus z = Dz / D = 21 / 7 = 3.

Question 11Álgebra matricial y sistemas

Apply the Rouché-Capelli theorem to classify the system: x + y - z = 2, 2x + 2y - 2z = 5, and x - y + z = 1.

  1. Inconsistent system (no solution) Correct answer.
  2. Consistent determined system (unique solution)
  3. Consistent indeterminate system with 1 parameter
  4. Consistent indeterminate system with 2 parameters

Explanation

Notice equations 1 and 2: x + y - z = 2 implies 2x + 2y - 2z = 4, but equation 2 states 2x + 2y - 2z = 5. Computing ranks: rank(A) = 2 while rank(A|B) = 3. Since rank(A) != rank(A|B), the system is inconsistent (no solution).

Question 12Álgebra matricial y sistemas

Given the matrix A = [[1, 2], [0, 1]], compute A^n for any positive integer n.

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

Explanation

Compute A^2 = [[1, 2], [0, 1]] * [[1, 2], [0, 1]] = [[1, 4], [0, 1]]. By mathematical induction, A^n = [[1, 2n], [0, 1]]. The upper right entry grows linearly as 2n.

Question 13Álgebra matricial y sistemas

If A is a 3x3 matrix with determinant det(A) = 4, calculate the determinant det(2 * A^-1).

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

Explanation

For an n x n matrix, det(k M) = k^n det(M). Here n = 3, so det(2 * A^-1) = 2^3 * det(A^-1) = 8 * (1 / det(A)) = 8 * (1 / 4) = 2.

Question 14Álgebra matricial y sistemas

Find the inverse of matrix B = [[1, 0, 2], [2, -1, 3], [4, 1, 8]].

  1. [[-11, 2, 2], [-4, 0, 1], [6, -1, -1]] Correct answer.
  2. [[11, -2, -2], [4, 0, -1], [-6, 1, 1]]
  3. [[-11, -4, 6], [2, 0, -1], [2, 1, -1]]
  4. [[1, 0, 1/2], [1/2, -1, 1/3], [1/4, 1, 1/8]]

Explanation

First det(B) = 1(-8 - 3) - 0 + 2(2 - (-4)) = -11 + 12 = 1. Cofactor matrix C has entries C11 = -11, C12 = -4, C13 = 6; C21 = 2, C22 = 0, C23 = -1; C31 = 2, C32 = 1, C33 = -1. Transposing C gives B^-1 = [[-11, 2, 2], [-4, 0, 1], [6, -1, -1]].

Question 15Álgebra matricial y sistemas

Find the general solution of the homogeneous linear system x + 2y - z = 0 and 2x - y + 3z = 0.

  1. (t, 2t, 3t)
  2. (2t, -5t, -8t) Correct answer.
  3. (-2t, 5t, 8t)
  4. (t, -t, 0)

Explanation

Add 2 * eq1 + eq2: 2(x + 2y - z) + (2x - y + 3z) = 4x + z = 0 => z = -4x. Substitute z = -4x into eq1: x + 2y - (-4x) = 5x + 2y = 0 => y = -5x/2. Setting parameter x = 2t yields (x, y, z) = (2t, -5t, -8t) for any real t.

Question 16Álgebra matricial y sistemas

If A and B are 3x3 matrices with det(A) = 3 and det(B) = -2, calculate det(A^T * B^2).

  1. -12
  2. 12 Correct answer.
  3. -6
  4. 36

Explanation

Using determinant properties: det(A^T) = det(A) = 3. det(B^2) = (det(B))^2 = (-2)^2 = 4. Since det(M * N) = det(M) * det(N), det(A^T * B^2) = 3 * 4 = 12.

Question 17Álgebra matricial y sistemas

Determine the value of m for which the system mx + y + z = 1, x + my + z = m, x + y + mz = m^2 is INCONSISTENT (has no solution).

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

Explanation

det(A) = (m - 1)^2 * (m + 2) = 0 for m = 1 and m = -2. For m = 1: all equations are x + y + z = 1 (consistent, infinitely many solutions). For m = -2: rank(A) = 2. Adding the three equations gives 0x + 0y + 0z = 1 + (-2) + 4 = 3, which is impossible (0 = 3). Thus rank(A|B) = 3 != rank(A) = 2, so the system is inconsistent for m = -2.

Question 18Álgebra matricial y sistemas

Solve the matrix equation X * A + 2B = X for X, where A = [[2, 0], [1, 3]] and B = [[1, -1], [0, 2]].

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

Explanation

Rearrange X * A - X = -2B => X * (A - I) = -2B. Here A - I = [[1, 0], [1, 2]] and -2B = [[-2, 2], [0, -4]]. The inverse (A - I)^-1 = [[1, 0], [-1/2, 1/2]]. Right-multiplying yields X = -2B * (A - I)^-1 = [[-2, 2], [0, -4]] * [[1, 0], [-1/2, 1/2]] = [[-3, 1], [2, -2]].

Question 19Álgebra matricial y sistemas

For the system with parameter lambda: lambda*x + y = 1 and x + lambda*y = lambda, find the value of y when lambda != 1 and lambda != -1 under Cramer's rule.

  1. y = lambda / (lambda + 1)
  2. y = 1 Correct answer.
  3. y = 1 / (lambda - 1)
  4. y = 0

Explanation

Main determinant D = lambda^2 - 1. Dy = |[lambda, 1], [1, lambda]| = lambda^2 - 1. By Cramer's rule, y = Dy / D = (lambda^2 - 1) / (lambda^2 - 1) = 1 for any lambda != +/- 1.

Question 20Álgebra matricial y sistemas

Compute the determinant of the 4x4 Vandermonde matrix V = [[1, 1, 1, 1], [1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]].

  1. 6
  2. 24
  3. 12 Correct answer.
  4. 0

Explanation

The determinant of a Vandermonde matrix with nodes x1=1, x2=2, x3=3, x4=4 is det(V) = product_{1 <= i < j <= 4} (xj - xi) = (2-1)(3-1)(4-1)(3-2)(4-2)(4-3) = 1 * 2 * 3 * 1 * 2 * 1 = 12.

Question 21Análisis, límites, derivadas y optimización

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

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

Explanation

Factoring the numerator yields (x^2 - 4) = (x - 2)(x + 2). Canceling the factor (x - 2) for x != 2 gives lim_{x -> 2} (x + 2) = 2 + 2 = 4.

Question 22Análisis, límites, derivadas y optimización

Find the derivative of the function f(x) = x^3 * e^(2x).

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

Explanation

Apply product rule f'(x) = u'v + uv' with u = x^3 and v = e^(2x): u' = 3x^2, v' = 2e^(2x). So f'(x) = 3x^2 * e^(2x) + x^3 * 2e^(2x) = x^2 * e^(2x) * (3 + 2x).

Question 23Análisis, límites, derivadas y optimización

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

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

Explanation

Applying L'Hôpital's rule on the 0/0 form yields lim_{x -> 0} [3 cos(3x) / 1] = 3 cos(0) = 3. Alternatively, use standard limit lim_{u->0} sin(u)/u = 1 with u = 3x.

Question 24Análisis, límites, derivadas y optimización

Find the x-coordinates of the critical points (local extrema) for f(x) = x^3 - 3x.

  1. x = 0
  2. x = +/- 1 Correct answer.
  3. x = +/- 3
  4. x = +/- sqrt(3)

Explanation

Compute first derivative f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives 3(x^2 - 1) = 0 => x^2 = 1 => x = 1 and x = -1.

Question 25Análisis, límites, derivadas y optimización

Bolzano's theorem guarantees at least one root for a continuous function f(x) on a closed interval [a, b] if which condition holds?

  1. f'(a) * f'(b) < 0
  2. f(a) * f(b) < 0 Correct answer.
  3. f(a) = f(b)
  4. f''(x) > 0 on [a, b]

Explanation

Bolzano's theorem requires f to be continuous on [a, b] and take values of opposite signs at endpoints, i.e., f(a) * f(b) < 0.

Question 26Análisis, límites, derivadas y optimización

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

  1. y = 2x - 4 Correct answer.
  2. y = 2x + 2
  3. y = -2x + 8
  4. y = 4x - 10

Explanation

Evaluate y(3) = 9 - 12 + 5 = 2. Derivative y' = 2x - 4 => slope m = y'(3) = 6 - 4 = 2. Tangent equation: y - 2 = 2(x - 3) => y = 2x - 4.

Question 27Análisis, límites, derivadas y optimización

Find the value of k such that f(x) = (x^2 - k)/(x - 3) for x != 3, and f(3) = 6, is continuous at x = 3.

  1. k = 3
  2. k = 6
  3. k = 9 Correct answer.
  4. k = 0

Explanation

For the limit at x = 3 to exist and equal 6, the numerator x^2 - k must evaluate to 0 at x = 3, giving 3^2 - k = 0 => k = 9. Then lim_{x->3} (x^2 - 9)/(x - 3) = lim_{x->3} (x + 3) = 6 = f(3).

Question 28Análisis, límites, derivadas y optimización

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

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

Explanation

Form 0/0. Apply L'Hôpital once: lim_{x->0} (e^x - 1)/(2x). Form 0/0 again. Apply L'Hôpital a second time: lim_{x->0} (e^x / 2) = 1/2.

Question 29Análisis, límites, derivadas y optimización

A rectangle has perimeter 40 meters. Find its dimensions (length x and width y) that maximize its area, and state the maximum area.

  1. 50 m^2
  2. 80 m^2
  3. 100 m^2 Correct answer.
  4. 400 m^2

Explanation

Perimeter 2x + 2y = 40 => y = 20 - x. Area A(x) = x(20 - x) = 20x - x^2. Derivative A'(x) = 20 - 2x = 0 => x = 10, y = 10. Maximum area = 10 * 10 = 100 m^2.

Question 30Análisis, límites, derivadas y optimización

Find all inflection points of the polynomial function f(x) = x^4 - 4x^3.

  1. x = 0 and x = 3
  2. x = 0 and x = 2 Correct answer.
  3. x = 1 and x = 2
  4. No inflection points exist

Explanation

Compute f'(x) = 4x^3 - 12x^2 and f''(x) = 12x^2 - 24x = 12x(x - 2). Setting f''(x) = 0 yields x = 0 and x = 2. Since f''(x) changes sign around both x = 0 and x = 2, both are inflection points.

Question 31Análisis, límites, derivadas y optimización

Find the derivative of f(x) = ln((x + 1) / (x - 1)) for x > 1.

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

Explanation

Use logarithm properties f(x) = ln(x + 1) - ln(x - 1). Differentiating gives f'(x) = 1/(x + 1) - 1/(x - 1) = [(x - 1) - (x + 1)] / (x^2 - 1) = -2 / (x^2 - 1).

Question 32Análisis, límites, derivadas y optimización

Verify the Mean Value Theorem for f(x) = x^2 on [1, 3]. Find the value of c in (1, 3) satisfying f'(c) = (f(3) - f(1)) / (3 - 1).

  1. c = 1.5
  2. c = 2 Correct answer.
  3. c = 2.5
  4. c = sqrt(3)

Explanation

Average rate of change = (f(3) - f(1))/(3 - 1) = (9 - 1)/2 = 4. Derivative f'(x) = 2x. Setting 2c = 4 gives c = 2, which lies strictly in (1, 3).

Question 33Análisis, límites, derivadas y optimización

Evaluate the limit lim_{x -> 0^+} (1 + 2x)^(1/x).

  1. 1
  2. e
  3. e^2 Correct answer.
  4. Infinity

Explanation

Indeterminate form 1^infinity. Let L = lim_{x->0} (1 + 2x)^(1/x). Taking logarithm: ln(L) = lim_{x->0} ln(1 + 2x)/x = lim_{x->0} [2/(1 + 2x) / 1] = 2. Exponentiating gives L = e^2.

Question 34Análisis, límites, derivadas y optimización

Find the minimum distance from the origin (0, 0) to the line 3x + 4y - 10 = 0 using calculus optimization.

  1. 1
  2. 2 Correct answer.
  3. 2.5
  4. 5

Explanation

The distance formula from point (x0, y0) to line Ax + By + C = 0 gives d = |3(0) + 4(0) - 10| / sqrt(3^2 + 4^2) = |-10| / 5 = 2. Optimization of d^2 = x^2 + y^2 subject to y = (10 - 3x)/4 yields the exact same minimum distance 2.

Question 35Análisis, límites, derivadas y optimización

Determine all asymptotes of the rational function f(x) = (2x^2 + 3) / (x - 1).

  1. Vertical asymptote x = 1, Oblique asymptote y = 2x
  2. Vertical asymptote x = 1, Oblique asymptote y = 2x + 2 Correct answer.
  3. Vertical asymptote x = -1, Oblique asymptote y = 2x - 2
  4. Horizontal asymptote y = 2

Explanation

Vertical asymptote occurs at denominator zero: x = 1. Perform polynomial division: (2x^2 + 3) / (x - 1) = 2x + 2 + 5/(x - 1). As x -> +/- infinity, 5/(x - 1) -> 0, so the oblique asymptote is y = 2x + 2.

Question 36Análisis, límites, derivadas y optimización

Using implicit differentiation, find dy/dx for the circle x^2 + y^2 - 4x + 6y = 0 at the point (4, 0).

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

Explanation

Differentiate implicitly with respect to x: 2x + 2y y' - 4 + 6y' = 0 => y'(2y + 6) = 4 - 2x => y' = (4 - 2x) / (2y + 6). At (4, 0): y' = (4 - 8) / (0 + 6) = -4/6 = -2/3.

Question 37Análisis, límites, derivadas y optimización

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

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

Explanation

Form 0/0. Apply L'Hôpital 1: lim_{x->0} (1 - cos(x))/(3x^2). Apply L'Hôpital 2: lim_{x->0} sin(x)/(6x). Apply L'Hôpital 3: lim_{x->0} cos(x)/6 = 1/6.

Question 38Análisis, límites, derivadas y optimización

A closed cylindrical container must hold volume V = 16*pi cm^3. Find the base radius r (in cm) that minimizes total surface area S = 2*pi*r^2 + 2*pi*r*h.

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

Explanation

Volume V = pi*r^2*h = 16*pi => h = 16/r^2. Surface area S(r) = 2*pi*r^2 + 2*pi*r*(16/r^2) = 2*pi*r^2 + 32*pi/r. Derivative S'(r) = 4*pi*r - 32*pi/r^2 = 0 => 4*pi*r^3 = 32*pi => r^3 = 8 => r = 2 cm.

Question 39Análisis, límites, derivadas y optimización

Which theorems are combined to prove that x^5 + 2x - 4 = 0 has EXACTLY ONE real root?

  1. Bolzano's theorem for existence and strict monotonicity (f'(x) > 0) for uniqueness Correct answer.
  2. Rolle's theorem only
  3. Mean Value Theorem only
  4. L'Hôpital's rule and Weierstrass theorem

Explanation

f(0) = -4 < 0 and f(2) = 32 + 4 - 4 = 32 > 0. By Bolzano's theorem on continuous f(x), at least one root exists in (0, 2). Derivative f'(x) = 5x^4 + 2 > 0 for all x, so f is strictly increasing, guaranteeing uniqueness (no second root by Rolle's theorem).

Question 40Análisis, límites, derivadas y optimización

Find the absolute minimum value of f(x) = x * ln(x) on the interval [0.1, 2].

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

Explanation

Derivative f'(x) = ln(x) + 1 = 0 => ln(x) = -1 => x = e^-1 = 1/e. Second derivative f''(x) = 1/x > 0, confirming a local/global minimum. Value f(1/e) = (1/e) * ln(1/e) = (1/e) * (-1) = -1/e.

Question 41Cálculo integral

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

Integrate term-by-term using power rule: integral 3x^2 dx = x^3, integral -4x dx = -2x^2, integral 5 dx = 5x. Total integral = x^3 - 2x^2 + 5x + C.

Question 42Cálculo integral

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

  1. 10
  2. 12 Correct answer.
  3. 14
  4. 9

Explanation

Antiderivative F(x) = x^2 + x. Evaluate from 0 to 3: F(3) - F(0) = (3^2 + 3) - (0^2 + 0) = (9 + 3) - 0 = 12.

Question 43Cálculo integral

Compute integral 2x * e^(x^2) dx.

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

Explanation

Substitute u = x^2 => du = 2x dx. The integral becomes integral e^u du = e^u + C = e^(x^2) + C.

Question 44Cálculo integral

Calculate the area under the curve y = x^2 bounded by the x-axis, x = 0, and x = 2.

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

Explanation

Area = integral_0^2 x^2 dx = [x^3 / 3]_0^2 = 2^3 / 3 - 0 = 8/3.

Question 45Cálculo integral

Evaluate integral (1 / (2x + 3)) dx.

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

Explanation

Substitute u = 2x + 3 => du = 2 dx => dx = du / 2. Integral = (1/2) integral (1/u) du = (1/2) ln|u| + C = (1/2) ln|2x + 3| + C.

Question 46Cálculo integral

Identify the standard formula for integration by parts for integral u dv.

  1. u*v + integral v du
  2. u*v - integral v du Correct answer.
  3. u'*v - u*v'
  4. (u/v) - integral u du

Explanation

Integration by parts is derived from the product rule of differentiation: integral u dv = u*v - integral v du.

Question 47Cálculo integral

Compute integral x * cos(x) dx.

  1. x * sin(x) - cos(x) + C
  2. x * sin(x) + cos(x) + C Correct answer.
  3. -x * sin(x) + cos(x) + C
  4. x * cos(x) - sin(x) + C

Explanation

Apply integration by parts with u = x, dv = cos(x) dx => du = dx, v = sin(x). Integral = x * sin(x) - integral sin(x) dx = x * sin(x) - (-cos(x)) + C = x * sin(x) + cos(x) + C.

Question 48Cálculo integral

Evaluate integral (1 / (x^2 - 4)) dx.

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

Explanation

Decompose into partial fractions: 1/((x-2)(x+2)) = A/(x-2) + B/(x+2). Solving gives A = 1/4, B = -1/4. Integrating yields (1/4) ln|x-2| - (1/4) ln|x+2| + C = (1/4) ln|(x-2)/(x+2)| + C.

Question 49Cálculo integral

Calculate the total enclosed area between the parabola y = x^2 and the line y = x + 2.

  1. 9/2 Correct answer.
  2. 7/6
  3. 16/3
  4. 3

Explanation

Intersection points: x^2 = x + 2 => x^2 - x - 2 = 0 => (x - 2)(x + 1) = 0 => x = -1 and x = 2. Area = integral_{-1}^2 (x + 2 - x^2) dx = [x^2/2 + 2x - x^3/3]_{-1}^2 = (2 + 4 - 8/3) - (1/2 - 2 + 1/3) = 10/3 - (-7/6) = 20/6 + 7/6 = 27/6 = 9/2 = 4.5.

Question 50Cálculo integral

According to the Fundamental Theorem of Calculus, if F(x) = integral_1^x sqrt(t^3 + 1) dt, compute F'(2).

  1. 3 Correct answer.
  2. 9
  3. sqrt(8)
  4. 0

Explanation

By the Fundamental Theorem of Calculus, F'(x) = d/dx [integral_1^x sqrt(t^3 + 1) dt] = sqrt(x^3 + 1). Evaluating at x = 2 gives F'(2) = sqrt(2^3 + 1) = sqrt(8 + 1) = sqrt(9) = 3.