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Castilla y León PAU Applied Mathematics for Social Sciences II study materials
Castilla y León PAU Applied Mathematics for Social Sciences II study resources

Castilla y León PAU Applied Mathematics for Social Sciences II practice questions

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Question 1Probabilidad, distribuciones e inferencia

If two events A and B are independent, what is P(A and B)?

  1. P(A) * P(B) Correct answer.
  2. P(A) + P(B)
  3. P(A) + P(B) - P(A and B)
  4. P(A) / P(B)

Explanation

By definition, two events A and B are independent if and only if P(A and B) = P(A) * P(B).

Question 2Probabilidad, distribuciones e inferencia

In a town, 60% of people read Newspaper A, 40% read Newspaper B, and 20% read both. What percentage of people read at least one of the two newspapers P(A or B)?

  1. 80% Correct answer.
  2. 100%
  3. 70%
  4. 50%

Explanation

Using inclusion-exclusion principle: P(A or B) = P(A) + P(B) - P(A and B) = 0.60 + 0.40 - 0.20 = 0.80 = 80%.

Question 3Probabilidad, distribuciones e inferencia

What is Bayes' Theorem formula for P(A|B)?

  1. [P(B|A) * P(A)] / P(B) Correct answer.
  2. [P(A|B) * P(B)] / P(A)
  3. P(A) * P(B)
  4. [P(A) + P(B)] / P(A and B)

Explanation

Bayes' Theorem states P(A|B) = [P(B|A) * P(A)] / P(B), relating conditional probability P(A|B) to its reverse P(B|A).

Question 4Probabilidad, distribuciones e inferencia

A random variable X follows a Binomial distribution B(n = 10, p = 0.3). What is the mean (expected value) E(X)?

  1. 3.0 Correct answer.
  2. 2.1
  3. 7.0
  4. 0.3

Explanation

For a binomial distribution B(n, p), the mean is mu = n * p = 10 * 0.3 = 3.0.

Question 5Probabilidad, distribuciones e inferencia

For a standard normal distribution Z ~ N(0, 1), if P(Z <= 1.96) = 0.975, what is the two-tailed 95% confidence interval Z-critical value Z_(alpha/2)?

  1. 1.96 Correct answer.
  2. 2.58
  3. 1.645
  4. 1.00

Explanation

For a 95% confidence level, alpha = 0.05, so alpha/2 = 0.025. P(Z <= Z_(alpha/2)) = 1 - 0.025 = 0.975, which corresponds to Z_(alpha/2) = 1.96.

Question 6Probabilidad, distribuciones e inferencia

How do you standardize a normal random variable X ~ N(mu, sigma) to a standard normal variable Z ~ N(0, 1)?

  1. Z = (X - mu) / sigma Correct answer.
  2. Z = (X + mu) / sigma
  3. Z = (X - sigma) / mu
  4. Z = (mu - X) / sigma^2

Explanation

Standardization subtracts the mean mu and divides by the standard deviation sigma: Z = (X - mu) / sigma.

Question 7Probabilidad, distribuciones e inferencia

A sample of n = 100 students has a sample mean score x_bar = 70 with known population standard deviation sigma = 10. What is the 95% confidence interval for the population mean mu (using Z = 1.96)?

  1. (68.04, 71.96) Correct answer.
  2. (60.00, 80.00)
  3. (69.02, 70.98)
  4. (65.00, 75.00)

Explanation

Margin of error E = Z * (sigma / sqrt(n)) = 1.96 * (10 / sqrt(100)) = 1.96 * (10 / 10) = 1.96. The 95% CI is (x_bar - E, x_bar + E) = (70 - 1.96, 70 + 1.96) = (68.04, 71.96).

Question 8Probabilidad, distribuciones e inferencia

To halve the margin of error E of a confidence interval for a population mean, by what factor must the sample size n be increased?

  1. Multiply sample size by 4 Correct answer.
  2. Multiply sample size by 2
  3. Multiply sample size by 8
  4. Halve the sample size

Explanation

Margin of error E is inversely proportional to sqrt(n). To cut E in half (E/2), sqrt(n) must double, which requires increasing n by a factor of 2^2 = 4.

Question 9Probabilidad, distribuciones e inferencia

In a survey of 400 voters, 240 state they favor Candidate A. What is the sample proportion p_hat?

  1. 0.60 (60%) Correct answer.
  2. 0.40 (40%)
  3. 0.24 (24%)
  4. 0.80 (80%)

Explanation

Sample proportion p_hat = x / n = 240 / 400 = 0.60 or 60%.

Question 10Probabilidad, distribuciones e inferencia

If events A and B are mutually exclusive (disjoint), what is P(A and B)?

  1. 0 Correct answer.
  2. 1
  3. P(A) * P(B)
  4. P(A) + P(B)

Explanation

Mutually exclusive events cannot occur at the same time, so their intersection is empty and P(A and B) = 0.

Question 11Probabilidad, distribuciones e inferencia

What is the complement rule for the probability of an event A?

  1. P(A') = 1 - P(A) Correct answer.
  2. P(A') = P(A)
  3. P(A') = 1 / P(A)
  4. P(A') = -P(A)

Explanation

The sum of probabilities of an event A and its complement A' equals 1, so P(A') = 1 - P(A).

Question 12Probabilidad, distribuciones e inferencia

A diagnostic test has a sensitivity P(Positive|Sick) = 0.95. If 5% of the population is sick, what is the joint probability P(Sick and Positive)?

  1. 0.0475 (4.75%) Correct answer.
  2. 0.95 (95%)
  3. 0.05 (5%)
  4. 0.50 (50%)

Explanation

By multiplication rule: P(Sick and Positive) = P(Sick) * P(Positive|Sick) = 0.05 * 0.95 = 0.0475.

Question 13Probabilidad, distribuciones e inferencia

What is the standard deviation of a Binomial distribution B(n, p)?

  1. sqrt(n * p * (1 - p)) Correct answer.
  2. n * p * (1 - p)
  3. n * p
  4. sqrt(n * p)

Explanation

The variance of B(n, p) is Var = n*p*(1-p). The standard deviation is the square root of the variance: sigma = sqrt(n*p*(1-p)).

Question 14Probabilidad, distribuciones e inferencia

In a 90% confidence interval for a population mean, what is the width of the interval if the margin of error is E = 3?

  1. 6 Correct answer.
  2. 3
  3. 1.5
  4. 9

Explanation

The confidence interval extends from (x_bar - E) to (x_bar + E). Its total width is 2 * E = 2 * 3 = 6.

Question 15Probabilidad, distribuciones e inferencia

What is the point estimate for a population mean mu?

  1. The sample mean x_bar Correct answer.
  2. The sample standard deviation s
  3. The confidence level 1 - alpha
  4. The margin of error E

Explanation

The sample mean x_bar is the unbiased single-value point estimate for the unknown population mean mu.

Question 16Probabilidad, distribuciones e inferencia

In rolling a fair 6-sided die, what is the probability of rolling a number greater than 4?

  1. 1/3 (33.3%) Correct answer.
  2. 1/2 (50%)
  3. 1/6 (16.7%)
  4. 2/3 (66.7%)

Explanation

Numbers greater than 4 on a 6-sided die are {5, 6} (2 favorable outcomes out of 6 possible). P = 2/6 = 1/3.

Question 17Probabilidad, distribuciones e inferencia

What is the margin of error E for a confidence interval for a population proportion p?

  1. Z_(alpha/2) * sqrt( (p_hat * (1 - p_hat)) / n ) Correct answer.
  2. Z_(alpha/2) * (sigma / n)
  3. p_hat / sqrt(n)
  4. Z_(alpha/2) * p_hat * (1 - p_hat)

Explanation

The standard error for sample proportion p_hat is sqrt(p_hat(1 - p_hat)/n). Multiplying by critical value Z_(alpha/2) yields margin of error E.

Question 18Probabilidad, distribuciones e inferencia

If a confidence level increases from 90% to 99% while sample size n remains constant, what happens to the width of the confidence interval?

  1. The interval width increases (becomes wider) Correct answer.
  2. The interval width decreases (becomes narrower)
  3. The interval width remains unchanged
  4. The interval width drops to zero

Explanation

Higher confidence requires a larger critical Z-value (Z_0.05 = 1.645 vs Z_0.005 = 2.58), which increases the margin of error E and widens the interval.

Question 19Probabilidad, distribuciones e inferencia

Three coins are flipped simultaneously. What is the probability of obtaining exactly two Heads?

  1. 3/8 (37.5%) Correct answer.
  2. 1/4 (25%)
  3. 1/2 (50%)
  4. 1/8 (12.5%)

Explanation

Sample space has 2^3 = 8 outcomes. Outcomes with exactly 2 Heads are {HHT, HTH, THH} (3 favorable). P = 3/8 = 0.375.

Question 20Probabilidad, distribuciones e inferencia

What is the total area under the probability density curve of a standard normal distribution N(0, 1)?

  1. 1 Correct answer.
  2. 0.5
  3. 100
  4. Infinity

Explanation

By definition of a probability density function, the total integral over (-infinity, +infinity) equals 1 (100% probability).

Question 21Probabilidad, distribuciones e inferencia

If P(A) = 0.4, P(B) = 0.5, and P(A|B) = 0.6, what is P(A and B)?

  1. 0.30 Correct answer.
  2. 0.20
  3. 0.24
  4. 0.90

Explanation

Using multiplication rule P(A and B) = P(A|B) * P(B) = 0.6 * 0.5 = 0.30.

Question 22Probabilidad, distribuciones e inferencia

A box contains 4 red balls and 6 blue balls. Two balls are drawn sequentially WITHOUT replacement. What is the probability that both balls are red?

  1. 2/15 (13.3%) Correct answer.
  2. 4/25 (16%)
  3. 2/5 (40%)
  4. 1/3 (33.3%)

Explanation

P(1st Red) = 4/10. P(2nd Red | 1st Red) = 3/9. P(Both Red) = (4/10) * (3/9) = 12/90 = 2/15.

Question 23Probabilidad, distribuciones e inferencia

In testing a hypothesis, what is a Type I error (error Tipo I)?

  1. Rejecting a true null hypothesis H0 Correct answer.
  2. Failing to reject a false null hypothesis H0
  3. Accepting a true alternative hypothesis H1
  4. Calculating the sample mean incorrectly

Explanation

A Type I error occurs when the researcher rejects the null hypothesis H0 when H0 is actually true (false positive rate alpha).

Question 24Probabilidad, distribuciones e inferencia

What is the expected value of a discrete random variable X with values x_i and probabilities p_i?

  1. Sum of x_i * p_i Correct answer.
  2. Sum of x_i / n
  3. Sum of p_i / x_i
  4. Max(x_i) - Min(x_i)

Explanation

The expected value mu = E(X) for a discrete random variable is the weighted sum E(X) = sum(x_i * p_i).

Question 25Probabilidad, distribuciones e inferencia

In a Normal distribution N(mu = 50, sigma = 5), what percentage of observations fall between 45 and 55 (within 1 standard deviation of the mean)?

  1. Approximately 68% Correct answer.
  2. Approximately 95%
  3. Approximately 99.7%
  4. 50%

Explanation

According to the 68-95-99.7 empirical rule for normal distributions, approximately 68.27% of data fall within mu +- 1*sigma (45 to 55).

Question 26Probabilidad, distribuciones e inferencia

What is the probability of obtaining at least one Head in 4 independent tosses of a fair coin?

  1. 15/16 (93.75%) Correct answer.
  2. 1/16 (6.25%)
  3. 1/2 (50%)
  4. 3/4 (75%)

Explanation

Using complement rule: P(at least 1 H) = 1 - P(all 4 Tails) = 1 - (1/2)^4 = 1 - 1/16 = 15/16.

Question 27Probabilidad, distribuciones e inferencia

In a binomial experiment B(n = 5, p = 0.5), what is the probability of getting all 5 successes?

  1. 1/32 (0.03125) Correct answer.
  2. 5/32 (0.15625)
  3. 1/16 (0.0625)
  4. 1/5 (0.20)

Explanation

P(X = 5) = C(5,5) * (0.5)^5 * (0.5)^0 = 1 * (1/32) * 1 = 1/32 = 0.03125.

Question 28Álgebra matricial y sistemas

What is the determinant of the 2x2 matrix A = [[3, 5], [1, 4]]?

  1. 7 Correct answer.
  2. 17
  3. 12
  4. -7

Explanation

For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here, det(A) = (3)(4) - (5)(1) = 12 - 5 = 7.

Question 29Álgebra matricial y sistemas

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

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

Explanation

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

Question 30Álgebra matricial y sistemas

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

  1. 2x4 Correct answer.
  2. 3x3
  3. 4x2
  4. The product AB is undefined.

Explanation

When multiplying an (m x n) matrix by an (n x p) matrix, the inner dimensions match (n=3) and the product matrix has dimension (m x p) = 2x4.

Question 31Álgebra matricial y sistemas

What condition must a square matrix A satisfy for its inverse matrix A^(-1) to exist?

  1. det(A) != 0 Correct answer.
  2. det(A) = 0
  3. det(A) > 1
  4. A must be a diagonal matrix

Explanation

A square matrix A is invertible (non-singular) if and only if its determinant is non-zero (det(A) != 0).

Question 32Álgebra matricial y sistemas

What is the transpose of the matrix M = [[1, 2, 3], [4, 5, 6]]?

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

Explanation

The transpose M^T converts rows into columns. Row 1 [1, 2, 3] becomes Column 1, and Row 2 [4, 5, 6] becomes Column 2, yielding [[1, 4], [2, 5], [3, 6]].

Question 33Álgebra matricial y sistemas

According to the Rouché-Frobenius theorem, a system of linear equations with coefficient matrix A and augmented matrix A|B is consistent and determined (has a unique solution) when:

  1. rank(A) = rank(A|B) = n (number of unknowns) Correct answer.
  2. rank(A) = rank(A|B) < n (number of unknowns)
  3. rank(A) != rank(A|B)
  4. rank(A) = 0

Explanation

By Rouché-Frobenius, if rank(A) = rank(A|B) = n, the system has a unique solution (sistema compatible determinado).

Question 34Álgebra matricial y sistemas

Solve the matrix equation AX = B for X, where A is an invertible square matrix.

  1. X = A^(-1)B Correct answer.
  2. X = BA^(-1)
  3. X = B / A
  4. X = A B^(-1)

Explanation

Multiplying both sides on the left by A^(-1) gives A^(-1)AX = A^(-1)B, which simplifies to IX = X = A^(-1)B. Matrix multiplication is non-commutative, so left-multiplication is required.

Question 35Álgebra matricial y sistemas

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

  1. 0 Correct answer.
  2. 4
  3. 8
  4. -4

Explanation

Expanding along the second column: det(A) = 1 * det([[1, 2], [2, 4]]) = 1 * (1*4 - 2*2) = 1 * (4 - 4) = 0. Alternatively, Row 3 is twice Row 1 (excluding middle element 0), making rows linearly dependent.

Question 36Álgebra matricial y sistemas

For what value of k is the system of equations { x + y = 3, 2x + ky = 6 } dependent with infinitely many solutions?

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

Explanation

Multiplying the first equation by 2 gives 2x + 2y = 6. For the second equation 2x + ky = 6 to be identical, we must have k = 2.

Question 37Álgebra matricial y sistemas

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

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

Explanation

For A = [[a, b], [c, d]], det(A) = ad - bc = 2*3 - 1*5 = 6 - 5 = 1. A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] = [[3, -1], [-5, 2]].

Question 38Álgebra matricial y sistemas

Using Cramer's Rule to solve for x in the system { 2x + y = 5, x - 3y = -1 }, what is the determinant of the coefficient matrix Det(A)?

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

Explanation

The coefficient matrix is A = [[2, 1], [1, -3]]. Det(A) = (2)(-3) - (1)(1) = -6 - 1 = -7.

Question 39Álgebra matricial y sistemas

A firm produces two types of goods. The raw material requirement matrix is R = [[2, 3], [1, 4]] and the cost vector per unit of raw material is C = [[10], [20]]. What is the total raw material cost per unit for each good?

  1. [[80], [90]] Correct answer.
  2. [[50], [70]]
  3. [[30], [80]]
  4. [[100], [60]]

Explanation

The cost per good is given by the matrix product R * C = [[2*10 + 3*20], [1*10 + 4*20]] = [[20 + 60], [10 + 80]] = [[80], [90]]. Good 1 costs 80 € and Good 2 costs 90 €.

Question 40Álgebra matricial y sistemas

If matrix A is symmetric, which of the following properties must hold?

  1. A = A^T Correct answer.
  2. A = -A^T
  3. det(A) = 0
  4. A * A^T = I

Explanation

By definition, a matrix A is symmetric if it equals its transpose: A = A^T.

Question 41Álgebra matricial y sistemas

What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [0, 0, 0]]?

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

Explanation

Row 2 is twice Row 1, and Row 3 contains all zeros. Therefore, there is only 1 linearly independent row, so rank(A) = 1.

Question 42Álgebra matricial y sistemas

Solve the system { 3x - 2y = 7, x + 4y = 7 } for x and y.

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

Explanation

From the second equation, x = 7 - 4y. Substituting into first equation: 3(7 - 4y) - 2y = 7 => 21 - 12y - 2y = 7 => 14y = 14 => y = 1. Then x = 7 - 4(1) = 3.

Question 43Álgebra matricial y sistemas

What is the trace of the matrix A = [[5, -2, 1], [3, 4, 0], [1, 7, -2]]?

  1. 7 Correct answer.
  2. 9
  3. 11
  4. 5

Explanation

The trace of a square matrix is the sum of its main diagonal elements: Trace(A) = 5 + 4 + (-2) = 7.

Question 44Álgebra matricial y sistemas

A homogenous linear system of equations AX = 0 always has at least one solution, which is:

  1. The trivial solution X = 0 Correct answer.
  2. No solution
  3. Infinitely many non-zero solutions
  4. X = 1

Explanation

A homogeneous system AX = 0 is always consistent because setting all variables to zero (X = 0) satisfies 0 = 0. This is called the trivial solution.

Question 45Álgebra matricial y sistemas

Calculate A^2 for the 2x2 matrix A = [[1, 2], [0, 1]].

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

Explanation

A^2 = [[1, 2], [0, 1]] * [[1, 2], [0, 1]] = [[1*1 + 2*0, 1*2 + 2*1], [0*1 + 1*0, 0*2 + 1*1]] = [[1, 4], [0, 1]].

Question 46Álgebra matricial y sistemas

If det(A) = 4 for a 3x3 matrix A, what is det(2A)?

  1. 32 Correct answer.
  2. 8
  3. 16
  4. 24

Explanation

For an n x n matrix A and scalar k, det(kA) = k^n * det(A). Here n = 3 and k = 2, so det(2A) = 2^3 * det(A) = 8 * 4 = 32.

Question 47Álgebra matricial y sistemas

Find the values of m for which the system { mx + y = 1, x + my = m } has infinitely many solutions.

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

Explanation

The determinant of the coefficient matrix is m^2 - 1 = 0 => m = 1 or m = -1. For m = 1, the equations become x + y = 1 and x + y = 1 (identical, infinitely many solutions). For m = -1, -x + y = 1 and x - y = -1 (identical), but let's check augmented matrix: for m = 1 both equations are x+y=1.

Question 48Álgebra matricial y sistemas

If A is an invertible matrix such that A^2 = A, what must A be?

  1. The identity matrix I Correct answer.
  2. The zero matrix 0
  3. Any diagonal matrix
  4. No such matrix exists

Explanation

Since A is invertible, we can multiply A^2 = A by A^(-1) on both sides: A^(-1) A^2 = A^(-1) A => A = I.

Question 49Álgebra matricial y sistemas

In an economy with two sectors, the input-output matrix is A = [[0.2, 0.3], [0.4, 0.1]]. What is the Leontief matrix (I - A)?

  1. [[0.8, -0.3], [-0.4, 0.9]] Correct answer.
  2. [[0.2, -0.3], [-0.4, 0.1]]
  3. [[-0.2, 0.3], [0.4, -0.1]]
  4. [[1.2, 0.3], [0.4, 1.1]]

Explanation

The identity matrix is I = [[1, 0], [0, 1]]. Subtracting A gives (I - A) = [[1 - 0.2, 0 - 0.3], [0 - 0.4, 1 - 0.1]] = [[0.8, -0.3], [-0.4, 0.9]].

Question 50Álgebra matricial y sistemas

What is the rank of the matrix A = [[1, 2], [3, 6], [5, 10]]?

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

Explanation

Row 2 = 3 * Row 1 and Row 3 = 5 * Row 1. All rows are scalar multiples of Row 1, so there is only 1 independent row. Hence rank(A) = 1.