I85PREP
⌘ K
PAU Andalucía Matemáticas II - fundamentos de práctica study materials
PAU Andalucía Matemáticas II - fundamentos de práctica study resources

PAU Andalucía Matemáticas II - fundamentos de práctica practice questions

One question at a time. Build your PAU Andalucía Matemáticas II - fundamentos de práctica understanding.

First 50 of 190 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Álgebra matricial y sistemas lineales

For the matrix A = [[2, 1], [4, 3]], what is the determinant det(A)?

  1. 2 Correct answer.
  2. 10
  3. -2
  4. 6

Explanation

For a 2x2 matrix [[a, b], [c, d]], the determinant is computed as ad - bc. Here det(A) = (2)(3) - (1)(4) = 6 - 4 = 2.

Question 2Álgebra matricial y sistemas lineales

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

  1. [[3, 2], [4, 9]] Correct answer.
  2. [[2, 0], [3, 20]]
  3. [[3, 2], [4, 20]]
  4. [[1, 2], [4, 9]]

Explanation

Matrix addition is performed entrywise: A + B = [[1+2, 2+0], [3+1, 4+5]] = [[3, 2], [4, 9]].

Question 3Álgebra matricial y sistemas lineales

What is the transpose of the matrix M = [[1, 5, 9], [2, 6, 0]]?

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

Explanation

The transpose M^T is formed by swapping rows and columns. The 2x3 matrix becomes a 3x2 matrix where column 1 is [1, 5, 9]^T and column 2 is [2, 6, 0]^T.

Question 4Álgebra matricial y sistemas lineales

If det(A) = 5 for a 3x3 matrix A, what is det(2A)?

  1. 40 Correct answer.
  2. 10
  3. 30
  4. 25

Explanation

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

Question 5Álgebra matricial y sistemas lineales

For what value of k is the 2x2 matrix [[k, 3], [2, 6]] singular (non-invertible)?

  1. 1 Correct answer.
  2. 0
  3. 4
  4. 9

Explanation

A matrix is singular if its determinant equals 0. Setting det = (k)(6) - (3)(2) = 6k - 6 = 0 yields 6k = 6, so k = 1.

Question 6Álgebra matricial y sistemas lineales

What is the inverse of the 2x2 matrix A = [[1, 2], [0, 1]]?

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

Explanation

For A = [[a, b], [c, d]] with det(A) = ad - bc = 1 - 0 = 1, A^-1 = (1/det)[[d, -b], [-c, a]] = [[1, -2], [0, 1]].

Question 7Álgebra matricial y sistemas lineales

Given matrices A = [[1, 2], [3, 4]] and B = [[2, 1], [0, 2]], compute the matrix product AB.

  1. [[2, 5], [6, 11]] Correct answer.
  2. [[2, 2], [0, 8]]
  3. [[5, 2], [11, 6]]
  4. [[4, 5], [6, 10]]

Explanation

Matrix multiplication AB = [[1*2 + 2*0, 1*1 + 2*2], [3*2 + 4*0, 3*1 + 4*2]] = [[2, 1+4], [6, 3+8]] = [[2, 5], [6, 11]].

Question 8Álgebra matricial y sistemas lineales

Evaluate the 3x3 determinant det(A) where A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]].

  1. 24 Correct answer.
  2. 15
  3. 0
  4. 14

Explanation

Matrix A is upper triangular. The determinant of an upper triangular matrix is the product of its main diagonal entries: det(A) = 1 * 4 * 6 = 24.

Question 9Álgebra matricial y sistemas lineales

Compute the determinant of the 3x3 matrix A = [[2, 1, 0], [1, 3, 1], [0, 1, 4]] using Sarrus rule or cofactor expansion.

  1. 18 Correct answer.
  2. 24
  3. 12
  4. 20

Explanation

Expanding along row 1: det(A) = 2*(3*4 - 1*1) - 1*(1*4 - 1*0) + 0*(1*1 - 3*0) = 2*(12 - 1) - 1*(4) + 0 = 22 - 4 = 18.

Question 10Álgebra matricial y sistemas lineales

Solve the matrix equation AX = B for X, where A = [[1, 3], [2, 5]] and B = [[4], [7]].

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

Explanation

X = A^-1 B. det(A) = 1*5 - 3*2 = -1. A^-1 = (1/-1)[[5, -3], [-2, 1]] = [[-5, 3], [2, -1]]. Thus X = [[-5, 3], [2, -1]] [[4], [7]] = [[-20 + 21], [8 - 7]] = [[1], [1]].

Question 11Álgebra matricial y sistemas lineales

What is the rank of the 3x3 matrix A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]]?

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

Explanation

Row 2 is twice row 1 (2, 4, 6 = 2*(1, 2, 3)), so det(A) = 0 and rank < 3. A 2x2 minor [[2, 3], [1, 1]] has det = 2 - 3 = -1 != 0. Thus rank(A) = 2.

Question 12Álgebra matricial y sistemas lineales

If matrix A is 3x3 and det(A) = 4, what is the determinant of the inverse matrix A^-1?

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

Explanation

Since A * A^-1 = I, det(A * A^-1) = det(A) * det(A^-1) = 1. Therefore det(A^-1) = 1 / det(A) = 1/4.

Question 13Álgebra matricial y sistemas lineales

If A is a 2x3 matrix and B is a 3x4 matrix, what is the size of the product matrix AB?

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

Explanation

The inner dimensions match (3 = 3), so matrix multiplication is defined. The resulting matrix AB has dimensions equal to the outer dimensions: 2x4.

Question 14Álgebra matricial y sistemas lineales

Which of the following matrix properties is NOT generally true for square matrices A and B of the same size?

  1. AB = BA Correct answer.
  2. det(AB) = det(A)det(B)
  3. (AB)^T = B^T A^T
  4. A(B + C) = AB + AC

Explanation

Matrix multiplication is generally non-commutative (AB != BA in general). The other three properties (determinant of product, transpose of product, and distributivity) are fundamental matrix algebra identities.

Question 15Álgebra matricial y sistemas lineales

If matrix A satisfies A^2 - 3A + 2I = 0, what is the inverse matrix A^-1 expressed in terms of A and I?

  1. (3/2)I - (1/2)A Correct answer.
  2. 3A - 2I
  3. (1/2)A - (3/2)I
  4. A - 3I

Explanation

Rearrange A^2 - 3A = -2I -> A(3I - A) = 2I -> A [ (3/2)I - (1/2)A ] = I. Thus A^-1 = (3/2)I - (1/2)A.

Question 16Álgebra matricial y sistemas lineales

Find the value of parameter k such that the matrix equation (A - k I) X = 0 has non-zero solutions, where A = [[3, 1], [2, 2]].

  1. k = 1 or k = 4 Correct answer.
  2. k = 2 or k = 3
  3. k = 0 or k = 5
  4. k = -1 or k = 4

Explanation

Non-zero solutions exist iff det(A - k I) = 0 (eigenvalues). det([[3-k, 1], [2, 2-k]]) = (3-k)(2-k) - 2 = k^2 - 5k + 6 - 2 = k^2 - 5k + 4 = 0. Factoring gives (k - 1)(k - 4) = 0 -> k = 1 or k = 4.

Question 17Álgebra matricial y sistemas lineales

Determine the inverse of the 3x3 matrix A = [[1, 0, 1], [0, 1, 0], [2, 0, 1]].

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

Explanation

det(A) = 1(1) - 0 + 1(-2) = 1 - 2 = -1. Matrix of cofactors C = [[1, 0, -2], [0, -1, 0], [-1, 0, 1]]. Transpose adj(A) = [[1, 0, -1], [0, -1, 0], [-2, 0, 1]]. A^-1 = (1/-1) adj(A) = [[-1, 0, 1], [0, 1, 0], [2, 0, -1]].

Question 18Álgebra matricial y sistemas lineales

If A is a 3x3 invertible matrix with det(A) = 3, compute the determinant of its adjugate matrix det(adj(A)).

  1. 9 Correct answer.
  2. 3
  3. 27
  4. 1/3

Explanation

For any n x n matrix, A * adj(A) = det(A) I -> det(A) det(adj(A)) = (det(A))^n -> det(adj(A)) = (det(A))^(n-1). For n = 3 and det(A) = 3: det(adj(A)) = 3^(3-1) = 3^2 = 9.

Question 19Álgebra matricial y sistemas lineales

According to the Rouché-Capelli theorem, a system of 3 linear equations in 3 unknowns is compatible determined (unique solution) if and only if:

  1. rank(A) = rank(A|B) = 3 Correct answer.
  2. rank(A) = rank(A|B) = 2
  3. rank(A) < rank(A|B)
  4. rank(A) = 3 and rank(A|B) = 4

Explanation

By Rouché-Capelli, the system is compatible if rank(A) = rank(A|B). If this common rank equals the number of unknowns (n = 3), the system has a unique solution (compatible determined).

Question 20Álgebra matricial y sistemas lineales

For what values of parameter m does the system x + y + z = 1, x + 2y + 3z = 2, 2x + 3y + mz = 3 have a unique solution?

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

Explanation

Compute det(A) = det([[1, 1, 1], [1, 2, 3], [2, 3, m]]). Row operations R3 -> R3 - R1 - R2 gives det = det([[1, 1, 1], [1, 2, 3], [0, 0, m-4]]) = (m-4)(2 - 1) = m - 4. A unique solution exists if det(A) != 0, so m != 4.

Question 21Álgebra matricial y sistemas lineales

Solve for y using Cramer's Rule in the system: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.

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

Explanation

det(A) = 1(1-2) - 1(-2-1) + 1(4+1) = -1 + 3 + 5 = 7. For det(A_y), replace col 2 with B: det([[1, 6, 1], [2, 3, 1], [1, 2, -1]]) = 1(-3-2) - 6(-2-1) + 1(4-3) = -5 + 18 + 1 = 14. Thus y = 14 / 7 = 2.

Question 22Álgebra matricial y sistemas lineales

Find the value of k such that the system x + 2y = 3, 3x + 6y = k is compatible.

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

Explanation

The second equation's LHS (3x + 6y) is 3 times the first equation's LHS (x + 2y). For the system to be compatible, the RHS must also be multiplied by 3: k = 3 * 3 = 9.

Question 23Álgebra matricial y sistemas lineales

For a homogeneous linear system Ax = 0 with 3 equations and 3 unknowns, non-trivial solutions exist if and only if:

  1. det(A) = 0 Correct answer.
  2. det(A) != 0
  3. rank(A) = 3
  4. A is invertible

Explanation

A homogeneous system always has the trivial solution x = 0. Non-trivial solutions exist iff rank(A) < 3, which is equivalent to det(A) = 0.

Question 24Álgebra matricial y sistemas lineales

For the parametric system { ax + y + z = 1, x + ay + z = a, x + y + az = a^2 }, for which value of 'a' is the system compatible undetermined (infinitely many solutions)?

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

Explanation

det(A) = (a - 1)^2 (a + 2). Setting det(A) = 0 gives a = 1 or a = -2. For a = 1: all three equations become x + y + z = 1, so rank(A) = 1 = rank(A|B) < 3 (compatible undetermined). For a = -2: rank(A) = 2 but rank(A|B) = 3 (incompatible).

Question 25Álgebra matricial y sistemas lineales

Find the general solution in terms of parameter lambda for the system { x + y - z = 0, 2x - y + z = 0 }.

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

Explanation

Adding the two equations: (x + y - z) + (2x - y + z) = 0 -> 3x = 0 -> x = 0. Substituting x = 0 into equation 1 gives y - z = 0 -> y = z. Setting z = lambda yields x = 0, y = lambda, z = lambda.

Question 26Geometría analítica tridimensional

What is the dot product of vectors u = (2, -1, 4) and v = (3, 5, -2)?

  1. -7 Correct answer.
  2. 19
  3. -3
  4. 7

Explanation

The dot product is u . v = u1 v1 + u2 v2 + u3 v3 = (2)(3) + (-1)(5) + (4)(-2) = 6 - 5 - 8 = -7.

Question 27Geometría analítica tridimensional

Two non-zero vectors u and v are orthogonal (perpendicular) if and only if:

  1. u . v = 0 Correct answer.
  2. u x v = 0
  3. u = k v
  4. |u| = |v|

Explanation

Since u . v = |u||v| cos(theta), u . v = 0 when theta = 90 degrees (cos 90 = 0).

Question 28Geometría analítica tridimensional

What is a normal vector to the plane with equation 3x - 2y + 5z - 7 = 0?

  1. n = (3, -2, 5) Correct answer.
  2. n = (3, 2, 5)
  3. n = (7, 7, 7)
  4. n = (-3, 2, 5)

Explanation

For a plane Ax + By + Cz + D = 0, the coefficients vector n = (A, B, C) is normal to the plane. Here n = (3, -2, 5).

Question 29Geometría analítica tridimensional

Find the point of intersection of the line (x-1)/2 = (y+3)/1 = (z-2)/(-1) with the plane z = 0.

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

Explanation

Set the line equations equal to t: x = 1 + 2t, y = -3 + t, z = 2 - t. Setting z = 0 gives 2 - t = 0 -> t = 2. Then x = 1 + 2(2) = 5, y = -3 + 2 = -1, z = 0.

Question 30Geometría analítica tridimensional

What is the distance from the point P(0, 0, 0) to the plane 2x - 3y + 6z - 14 = 0?

  1. 2 Correct answer.
  2. 14
  3. 7
  4. 1

Explanation

Distance d = |A x0 + B y0 + C z0 + D| / sqrt(A^2 + B^2 + C^2) = |0 + 0 + 0 - 14| / sqrt(4 + 9 + 36) = 14 / sqrt(49) = 14 / 7 = 2.

Question 31Geometría analítica tridimensional

Compute the cross product u x v of vectors u = (1, 0, 2) and v = (0, 3, 1).

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

Explanation

u x v = det([[i, j, k], [1, 0, 2], [0, 3, 1]]) = i(0*1 - 2*3) - j(1*1 - 2*0) + k(1*3 - 0*0) = -6i - 1j + 3k = (-6, -1, 3).

Question 32Geometría analítica tridimensional

Find the general equation of the plane passing through point P(1, 2, 3) with normal vector n = (2, -1, 4).

  1. 2x - y + 4z - 12 = 0 Correct answer.
  2. 2x - y + 4z + 12 = 0
  3. x + 2y + 3z - 12 = 0
  4. 2x + y + 4z - 14 = 0

Explanation

Plane equation: A(x - x0) + B(y - y0) + C(z - z0) = 0 -> 2(x - 1) - 1(y - 2) + 4(z - 3) = 0 -> 2x - 2 - y + 2 + 4z - 12 = 0 -> 2x - y + 4z - 12 = 0.

Question 33Geometría analítica tridimensional

Determine the relative position of the two planes pi1: 2x - y + 3z = 4 and pi2: 4x - 2y + 6z = 8.

  1. Coincident planes (same plane) Correct answer.
  2. Parallel and distinct planes
  3. Intersecting along a line
  4. Perpendicular planes

Explanation

Comparing coefficients: 4/2 = -2/(-1) = 6/3 = 8/4 = 2. Since all ratios including constants are equal, the planes are coincident.

Question 34Geometría analítica tridimensional

What is the angle between the two planes pi1: x + y = 0 and pi2: y + z = 0?

  1. 60 degrees (pi/3 rad) Correct answer.
  2. 45 degrees (pi/4 rad)
  3. 90 degrees (pi/2 rad)
  4. 30 degrees (pi/6 rad)

Explanation

Normal vectors: n1 = (1, 1, 0) and n2 = (0, 1, 1). cos(theta) = |n1 . n2| / (|n1||n2|) = |0 + 1 + 0| / (sqrt(2) * sqrt(2)) = 1 / 2. theta = arccos(1/2) = 60 degrees.

Question 35Geometría analítica tridimensional

What is the distance between the parallel planes pi1: 2x - 2y + z - 3 = 0 and pi2: 2x - 2y + z + 6 = 0?

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

Explanation

Formula for distance between parallel planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0 is d = |D2 - D1| / sqrt(A^2 + B^2 + C^2) = |6 - (-3)| / sqrt(4 + 4 + 1) = 9 / sqrt(9) = 9 / 3 = 3.

Question 36Geometría analítica tridimensional

Determine the value of the scalar triple product [u, v, w] for u = (1, 0, 0), v = (0, 1, 0), w = (0, 0, 1).

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

Explanation

[u, v, w] is the determinant of the 3x3 matrix with rows u, v, w. det([[1,0,0],[0,1,0],[0,0,1]]) = 1. It represents the volume of the unit cube.

Question 37Geometría analítica tridimensional

Find the shortest distance between the two skew lines r: (x-1)/1 = y/2 = (z+1)/1 and s: x/2 = (y-1)/(-1) = z/1.

  1. sqrt(35) / 5 Correct answer.
  2. 3 / sqrt(35)
  3. 1 / sqrt(6)
  4. 2 / sqrt(14)

Explanation

For points P(1, 0, -1) on r and Q(0, 1, 0) on s, vector PQ is (-1, 1, 1). Direction vectors are u = (1, 2, 1) and v = (2, -1, 1). Their cross product u x v is (3, 1, -5) with magnitude sqrt(35). The scalar triple product |PQ . (u x v)| = |-3 + 1 - 5| = 7. The distance d = 7 / sqrt(35) = sqrt(35) / 5.

Question 38Geometría analítica tridimensional

Find the symmetric point P' of the point P(1, 2, 3) relative to the plane pi: x + y + z - 12 = 0.

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

Explanation

Line perpendicular to plane through P: x = 1 + t, y = 2 + t, z = 3 + t. Substitute into plane equation: (1+t) + (2+t) + (3+t) - 12 = 0 -> 3t + 6 - 12 = 0 -> 3t = 6 -> t = 2. Midpoint M (projection of P onto plane): M(1+2, 2+2, 3+2) = (3, 4, 5). Since M is midpoint of PP': P' = 2M - P = 2(3, 4, 5) - (1, 2, 3) = (6-1, 8-2, 10-3) = (5, 6, 7).

Question 39Geometría analítica tridimensional

Determine the relative position of line r: (x-1)/1 = (y-2)/1 = (z-3)/1 and plane pi: x + y - 2z = 5.

  1. Line is parallel to the plane (does not intersect) Correct answer.
  2. Line intersects the plane at a single point
  3. Line is contained within the plane
  4. Line is perpendicular to the plane

Explanation

Line direction vector v = (1, 1, 1), plane normal vector n = (1, 1, -2). v . n = 1(1) + 1(1) + 1(-2) = 1 + 1 - 2 = 0. So line is parallel to or contained in plane. Check point P(1, 2, 3) on line: 1 + 2 - 2(3) = 3 - 6 = -3 != 5. Point P is not in plane, so line r is strictly parallel to plane pi.

Question 40Cálculo diferencial y límites

What is the derivative of f(x) = x^3 - 4x^2 + 7x - 12?

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

Explanation

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

Question 41Cálculo diferencial y límites

What is the derivative of g(x) = e^(2x)?

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

Explanation

By the chain rule, d/dx(e^(u(x))) = e^(u(x)) * u'(x). Here u(x) = 2x, so u'(x) = 2, giving g'(x) = 2 e^(2x).

Question 42Cálculo diferencial y límites

What is the limit lim (x -> 0) [sin(3x) / x]?

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

Explanation

Using the standard limit lim (u->0) [sin(u)/u] = 1, rewrite as 3 * [sin(3x) / (3x)]. As x -> 0, 3x -> 0, so the limit is 3 * 1 = 3 (or apply L'Hôpital: lim cos(3x)*3 / 1 = 3).

Question 43Cálculo diferencial y límites

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

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

Explanation

The slope of the tangent line at x = 2 is given by f'(2). Since f'(x) = 2x - 3, evaluating at x = 2 gives f'(2) = 2(2) - 3 = 4 - 3 = 1.

Question 44Cálculo diferencial y límites

What are the critical points of f(x) = x^3 - 3x?

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

Explanation

Critical points occur where f'(x) = 0. Here f'(x) = 3x^2 - 3. Setting 3(x^2 - 1) = 0 yields x^2 = 1, so x = 1 and x = -1.

Question 45Cálculo diferencial y límites

What is the vertical asymptote of the rational function f(x) = (2x + 1) / (x - 4)?

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

Explanation

A vertical asymptote occurs where the denominator is zero and the numerator is non-zero. Setting x - 4 = 0 gives x = 4.

Question 46Cálculo diferencial y límites

If f''(x) > 0 on an interval (a, b), the graph of f(x) on that interval is:

  1. Concave upward (convex) Correct answer.
  2. Concave downward
  3. Decreasing strictly
  4. Linear

Explanation

A positive second derivative f''(x) > 0 implies that the first derivative f'(x) is increasing, meaning the curve bends upward (concave upward / convex).

Question 47Cálculo diferencial y límites

What does Bolzano's Theorem state for a continuous function f on [a, b] if f(a) * f(b) < 0?

  1. There exists at least one c in (a, b) such that f(c) = 0. Correct answer.
  2. The function has a local maximum in (a, b).
  3. f'(c) = 0 for some c in (a, b).
  4. f(x) is monotonic on [a, b].

Explanation

Bolzano's theorem (Intermediate Value Theorem for zeros) guarantees that if a continuous function takes opposite signs at the endpoints, it must cross zero at least once in (a, b).

Question 48Cálculo diferencial y límites

Evaluate lim (x -> 0) [(e^x - 1 - x) / x^2].

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

Explanation

This gives indeterminate form 0/0. Applying L'Hôpital once: lim (e^x - 1) / (2x) [still 0/0]. Applying L'Hôpital a second time: lim (e^x) / 2 = e^0 / 2 = 1/2.

Question 49Cálculo diferencial y límites

Find the equation of the tangent line to the curve f(x) = ln(x) at x = e.

  1. y = (1/e)x Correct answer.
  2. y = (1/e)x + 1
  3. y = x - e + 1
  4. y = (1/e)x - 1

Explanation

Point: (e, f(e)) = (e, 1). Derivative: f'(x) = 1/x, so slope m = f'(e) = 1/e. Tangent line equation: y - 1 = (1/e)(x - e) -> y - 1 = (1/e)x - 1 -> y = (1/e)x.

Question 50Cálculo diferencial y límites

What is the derivative of h(x) = x^2 * sin(x)?

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

Explanation

Using the product rule d/dx(u v) = u' v + u v': h'(x) = d/dx(x^2) * sin(x) + x^2 * d/dx(sin(x)) = 2x sin(x) + x^2 cos(x).