25 Math Problems for 11th Graders with Answers and Explanations
Welcome to a collection of 25 challenging math problems designed specifically for 11th graders. These problems will push the boundaries of your mathematical knowledge and problem-solving skills. Each question is tailored to suit the 11th-grade level, covering topics such as algebra, trigonometry, calculus, statistics, and more. Get ready to expand your mathematical horizons and embrace the exhilarating world of advanced mathematics!
Problem 1: Quadratic Equations (Algebra) - Easy
Solve the equation: x² - 5x + 6 = 0.
Solution:
Step 1: Factor the quadratic equation: (x - 3)(x - 2) = 0.
Step 2: Set each factor equal to zero: x - 3 = 0 or x - 2 = 0.
Step 3: Solve for x: x = 3 or x = 2.
Answer: The solutions to the equation are x = 3 and x = 2.
Problem 2: Trigonometric Functions (Trigonometry) - Easy
Find the value of sin(π/6).
Solution:
Step 1: Recall the sine values of common angles: sin(π/6) = 1/2.
Answer: The value of sin(π/6) is 1/2.
Problem 3: Limits (Calculus) - Easy
Find the limit as x approaches 2 of (x² - 4) / (x - 2).
Solution:
Step 1: Simplify the expression: (x² - 4) / (x - 2) = (x + 2).
Step 2: Substitute the value of x: (x + 2) = (2 + 2) = 4.
Answer: The limit as x approaches 2 of (x² - 4) / (x - 2) is 4.
Problem 4: Linear Systems (Algebra) - Easy
Solve the system of equations:
2x + 3y = 7,
4x - 2y = 2.
Solution:
Step 1: Multiply the first equation by 2: 4x + 6y = 14.
Step 2: Add the two equations: (4x + 6y) + (4x - 2y) = 14 + 2.
Step 3: Simplify: 8x + 4y = 16.
Step 4: Divide by 4: 2x + y = 4.
Step 5: Substitute the value of y into the first equation: 2x + 3(4 - 2x) = 7.
Step 6: Solve for x: 2x + 12 - 6x = 7.
Step 7: Simplify: -4x + 12 = 7.
Step 8: Solve for x: -4x = -5.
Step 9: Divide by -4: x = 5/4.
Step 10: Substitute the value of x into the second equation: 4(5/4) - 2y = 2.
Step 11: Simplify: 5 - 2y = 2.
Step 12: Solve for y: -2y = -3.
Step 13: Divide by -2: y = 3/2.
Answer: The solution to the system of equations is x = 5/4 and y = 3/2.
Problem 5: Derivatives (Calculus) - Easy
Find the derivative of f(x) = 3x² + 2x - 1.
Solution:
Step 1: Apply the power rule: f'(x) = 6x + 2.
Answer: The derivative of f(x) = 3x² + 2x - 1 is f'(x) = 6x + 2.
Problem 6: Probability (Statistics) - Easy
A bag contains 5 red marbles and 3 blue marbles. What is the probability of selecting a red marble?
Solution:
Step 1: Determine the total number of marbles: 5 + 3 = 8.
Step 2: Determine the number of favorable outcomes: 5 (red marbles).
Step 3: Calculate the probability: P(red) = 5/8.
Answer: The probability of selecting a red marble is 5/8.
Problem 7: Exponential Functions (Algebra) - Medium
Solve the equation 2^x = 8.
Solution:
Step 1: Rewrite 8 as a power of 2: 2^3 = 8.
Step 2: Set the exponents equal to each other: x = 3.
Answer: The solution to the equation 2^x = 8 is x = 3.
Problem 8: Trigonometric Identities (Trigonometry) - Medium
Simplify the expression: tan²(x) + 1 = sec²(x).
Solution:
Step 1: Use the Pythagorean identity: sin²(x) + cos²(x) = 1.
Step 2: Divide both sides by cos²(x): (sin²(x) + cos²(x)) / cos²(x) = 1 / cos²(x).
Step 3: Simplify: tan²(x) + 1 = sec²(x).
Answer: The expression tan²(x) + 1 simplifies to sec²(x).
Problem 9: Integration (Calculus) - Medium
Find the integral of f(x) = 2x + 3 with respect to x.
Solution:
Step 1: Apply the power rule of integration: ∫(2x + 3) dx = x² + 3x + C.
Answer: The integral of f(x) = 2x + 3 with respect to x is x² + 3x + C.
Problem 10: Hypothesis Testing (Statistics) - Medium
A new drug claims to reduce headaches by 50%. To test this claim, a sample of 100 individuals is selected, and their headache intensity is measured before and after taking the drug. Determine whether the drug is effective at a significance level of 0.05.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The drug does not reduce headaches by 50%.
Alternative hypothesis (H₁): The drug reduces headaches by 50%.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.05), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.05, we can conclude that the drug is effective in reducing headaches by 50%.
Problem 11: Matrices (Algebra) - Medium
Solve the system of equations using matrices:
2x + 3y = 7,
4x - 2y = 2.
Solution:
Step 1: Write the system of equations in matrix form: AX = B, where
A = [[2, 3], [4, -2]],
X = [[x], [y]],
B = [[7], [2]].
Step 2: Calculate the inverse of matrix A: A⁻¹.
Step 3: Multiply A⁻¹ by B to obtain X: X = A⁻¹B.
Answer: The solution to the system of equations is x = ___ and y = ___.
Problem 12: Logarithmic Functions (Algebra) - Medium
Solve the equation log₂(x) = 3.
Solution:
Step 1: Rewrite the equation in exponential form: 2³ = x.
Step 2: Evaluate the exponent: 2³ = 8.
Answer: The solution to the equation log₂(x) = 3 is x = 8.
Problem 13: Vectors (Algebra) - Hard
Find the magnitude of the vector u = <3, -4, 5>.
Solution:
Step 1: Apply the magnitude formula: ||u|| = √(3² + (-4)² + 5²).
Step 2: Simplify: ||u|| = √(9 + 16 + 25) = √50 = 5√2.
Answer: The magnitude of the vector u = <3, -4, 5> is 5√2.
Problem 14: Trigonometric Equations (Trigonometry) - Hard
Solve the equation sin(x) + cos(x) = 1.
Solution:
Step 1: Rewrite the equation using the Pythagorean identity: sin(x) + √(1 - sin²(x)) = 1.
Step 2: Square both sides of the equation: sin²(x) + 2sin(x)√(1 - sin²(x)) + 1 - sin²(x) = 1.
Step 3: Simplify and combine like terms: 2sin(x)√(1 - sin²(x)) = 0.
Step 4: Solve for sin(x): sin(x) = 0 or √(1 - sin²(x)) = 0.
Step 5: Solve for x: x = 0 or sin(x) = 0.
Answer: The solutions to the equation sin(x) + cos(x) = 1 are x = 0 and x = π.
Problem 15: Differential Equations (Calculus) - Hard
Solve the differential equation dy/dx = 2x.
Solution:
Step 1: Integrate both sides of the equation with respect to x: ∫dy = ∫2x dx.
Step 2: Simplify: y = x² + C, where C is the constant of integration.
Answer: The solution to the differential equation dy/dx = 2x is y = x² + C.
Problem 16: Hypothesis Testing (Statistics) - Hard
A company claims that the average weight of their cereal boxes is 500 grams. To test this claim, a sample of 50 cereal boxes is selected, and their weights are measured. Determine whether the company's claim is statistically supported at a significance level of 0.01.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The average weight of the cereal boxes is 500 grams.
Alternative hypothesis (H₁): The average weight of the cereal boxes is not 500 grams.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.01), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.01, we can conclude that there is sufficient evidence to reject the company's claim about the average weight of the cereal boxes.
Problem 17: Complex Numbers (Algebra) - Hard
Solve the equation x² + 4 = 0 for complex values of x.
Solution:
Step 1: Rewrite the equation: x² = -4.
Step 2: Take the square root of both sides: x = ±√(-4) = ±2i.
Answer: The solutions to the equation x² + 4 = 0 are x = 2i and x = -2i.
Problem 18: Definite Integrals (Calculus) - Hard
Evaluate the definite integral ∫(0 to 3) (2x + 1) dx.
Solution:
Step 1: Apply the power rule of integration: ∫(2x + 1) dx = x² + x.
Step 2: Evaluate the integral at the upper and lower limits: [x² + x] from 0 to 3.
Step 3: Substitute the upper and lower limits: (3² + 3) - (0² + 0) = 12.
Answer: The value of the definite integral ∫(0 to 3) (2x + 1) dx is 12.
Problem 19: Probability Distributions (Statistics) - Hard
A fair six-sided die is rolled 100 times. Find the probability of getting exactly three 6s.
Solution:
Step 1: Determine the number of favorable outcomes: 1 (getting a 6).
Step 2: Determine the total number of possible outcomes: 6.
Step 3: Calculate the probability of getting a 6 on one roll: 1/6.
Step 4: Apply the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n = 100, k = 3, and p = 1/6.
Step 5: Calculate the probability: P(X = 3) = (100 choose 3) * (1/6)^3 * (5/6)^(100-3).
Answer: The probability of getting exactly three 6s in 100 rolls of a fair six-sided die is P(X = 3) = ___.
Problem 20: Logarithmic Differentiation (Calculus) - Hard
Differentiate y = (ln x)².
Solution:
Step 1: Apply the logarithmic differentiation rule: dy/dx = (2(ln x) / x) * (ln x)'.
Step 2: Differentiate ln x with respect to x: (ln x)' = 1/x.
Step 3: Substitute the value: dy/dx = (2(ln x) / x) * (1/x).
Answer: The derivative of y = (ln x)² is dy/dx = (2(ln x) / x²).
Problem 21: Matrices and Transformations (Algebra) - Hard
Given the matrix A = [[3, -1], [2, 4]], perform a 90-degree counterclockwise rotation transformation.
Solution:
Step 1: Set up the rotation matrix: R = [[0, -1], [1, 0]].
Step 2: Multiply A by R: A' = R * A.
Answer: The result of performing a 90-degree counterclockwise rotation transformation on matrix A is A' = [[1, 3], [-4, 2]].
Problem 22: Complex Analysis (Algebra) - Hard
Find the principal value of the complex logarithm log(i).
Solution:
Step 1: Convert the complex number to polar form: i = 1∠(π/2).
Step 2: Apply the logarithm rule for complex numbers in polar form: log(z) = ln|r| + i(arg(z) + 2kπ), where k is an integer.
Step 3: Calculate the principal value: log(i) = ln|1| + i(π/2 + 2kπ).
Answer: The principal value of the complex logarithm log(i) is ln(1) + i(π/2 + 2kπ).
Problem 23: Vector Calculus (Calculus) - Hard
Find the curl of the vector field F = ⟨3xy, x² + z, 2yz⟩.
Solution:
Step 1: Calculate the partial derivatives: ∂F/∂x = ⟨3y, 2x, 0⟩, ∂F/∂y = ⟨3x, 0, 2z⟩, ∂F/∂z = ⟨0, 1, 2y⟩.
Step 2: Evaluate the curl: curl(F) = ∇ × F = ∂F/∂y - ∂F/∂x + ∂F/∂z = ⟨3x - 3y, 2z - 2x, 2y - 0⟩.
Answer: The curl of the vector field F = ⟨3xy, x² + z, 2yz⟩ is curl(F) = ⟨3x - 3y, 2z - 2x, 2y⟩.
Problem 24: Hypothesis Testing (Statistics) - Hard
A researcher wants to determine if the mean height of a population is significantly different from 170 cm. A sample of 50 individuals is taken, and their heights are recorded. Conduct a two-sample t-test at a significance level of 0.05.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The mean height of the population is 170 cm.
Alternative hypothesis (H₁): The mean height of the population is not 170 cm.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., two-sample t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.05), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.05, we can conclude that the mean height of the population is significantly different from 170 cm.
Problem 25: Differential Equations (Calculus) - Hard
Solve the differential equation dy/dx = y³ - y.
Solution:
Step 1: Rewrite the equation as dy/(y³ - y) = dx.
Step 2: Perform partial fraction decomposition: 1/(y³ - y) = A/y + B/(y - 1).
Step 3: Integrate both sides of the equation: ∫(1/y) dy + ∫(1/(y - 1)) dy = ∫dx.
Step 4: Simplify: ln|y| - ln|y - 1| = x + C.
Step 5: Combine logarithms: ln|y/(y - 1)| = x + C.
Step 6: Exponentiate both sides of the equation: y/(y - 1) = e^(x + C).
Step 7: Solve for y: y = (e^C) * (y - 1).
Step 8: Simplify: y - (e^C) * y = (e^C).
Step 9: Factor out y: y(1 - e^C) = (e^C).
Step 10: Solve for y: y = (e^C) / (1 - e^C).
Answer: The solution to the differential equation dy/dx = y³ - y is y = (e^C) / (1 - e^C).
Problem 1: Quadratic Equations (Algebra) - Easy
Solve the equation: x² - 5x + 6 = 0.
Solution:
Step 1: Factor the quadratic equation: (x - 3)(x - 2) = 0.
Step 2: Set each factor equal to zero: x - 3 = 0 or x - 2 = 0.
Step 3: Solve for x: x = 3 or x = 2.
Answer: The solutions to the equation are x = 3 and x = 2.
Problem 2: Trigonometric Functions (Trigonometry) - Easy
Find the value of sin(π/6).
Solution:
Step 1: Recall the sine values of common angles: sin(π/6) = 1/2.
Answer: The value of sin(π/6) is 1/2.
Problem 3: Limits (Calculus) - Easy
Find the limit as x approaches 2 of (x² - 4) / (x - 2).
Solution:
Step 1: Simplify the expression: (x² - 4) / (x - 2) = (x + 2).
Step 2: Substitute the value of x: (x + 2) = (2 + 2) = 4.
Answer: The limit as x approaches 2 of (x² - 4) / (x - 2) is 4.
Problem 4: Linear Systems (Algebra) - Easy
Solve the system of equations:
2x + 3y = 7,
4x - 2y = 2.
Solution:
Step 1: Multiply the first equation by 2: 4x + 6y = 14.
Step 2: Add the two equations: (4x + 6y) + (4x - 2y) = 14 + 2.
Step 3: Simplify: 8x + 4y = 16.
Step 4: Divide by 4: 2x + y = 4.
Step 5: Substitute the value of y into the first equation: 2x + 3(4 - 2x) = 7.
Step 6: Solve for x: 2x + 12 - 6x = 7.
Step 7: Simplify: -4x + 12 = 7.
Step 8: Solve for x: -4x = -5.
Step 9: Divide by -4: x = 5/4.
Step 10: Substitute the value of x into the second equation: 4(5/4) - 2y = 2.
Step 11: Simplify: 5 - 2y = 2.
Step 12: Solve for y: -2y = -3.
Step 13: Divide by -2: y = 3/2.
Answer: The solution to the system of equations is x = 5/4 and y = 3/2.
Problem 5: Derivatives (Calculus) - Easy
Find the derivative of f(x) = 3x² + 2x - 1.
Solution:
Step 1: Apply the power rule: f'(x) = 6x + 2.
Answer: The derivative of f(x) = 3x² + 2x - 1 is f'(x) = 6x + 2.
Problem 6: Probability (Statistics) - Easy
A bag contains 5 red marbles and 3 blue marbles. What is the probability of selecting a red marble?
Solution:
Step 1: Determine the total number of marbles: 5 + 3 = 8.
Step 2: Determine the number of favorable outcomes: 5 (red marbles).
Step 3: Calculate the probability: P(red) = 5/8.
Answer: The probability of selecting a red marble is 5/8.
Problem 7: Exponential Functions (Algebra) - Medium
Solve the equation 2^x = 8.
Solution:
Step 1: Rewrite 8 as a power of 2: 2^3 = 8.
Step 2: Set the exponents equal to each other: x = 3.
Answer: The solution to the equation 2^x = 8 is x = 3.
Problem 8: Trigonometric Identities (Trigonometry) - Medium
Simplify the expression: tan²(x) + 1 = sec²(x).
Solution:
Step 1: Use the Pythagorean identity: sin²(x) + cos²(x) = 1.
Step 2: Divide both sides by cos²(x): (sin²(x) + cos²(x)) / cos²(x) = 1 / cos²(x).
Step 3: Simplify: tan²(x) + 1 = sec²(x).
Answer: The expression tan²(x) + 1 simplifies to sec²(x).
Problem 9: Integration (Calculus) - Medium
Find the integral of f(x) = 2x + 3 with respect to x.
Solution:
Step 1: Apply the power rule of integration: ∫(2x + 3) dx = x² + 3x + C.
Answer: The integral of f(x) = 2x + 3 with respect to x is x² + 3x + C.
Problem 10: Hypothesis Testing (Statistics) - Medium
A new drug claims to reduce headaches by 50%. To test this claim, a sample of 100 individuals is selected, and their headache intensity is measured before and after taking the drug. Determine whether the drug is effective at a significance level of 0.05.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The drug does not reduce headaches by 50%.
Alternative hypothesis (H₁): The drug reduces headaches by 50%.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.05), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.05, we can conclude that the drug is effective in reducing headaches by 50%.
Problem 11: Matrices (Algebra) - Medium
Solve the system of equations using matrices:
2x + 3y = 7,
4x - 2y = 2.
Solution:
Step 1: Write the system of equations in matrix form: AX = B, where
A = [[2, 3], [4, -2]],
X = [[x], [y]],
B = [[7], [2]].
Step 2: Calculate the inverse of matrix A: A⁻¹.
Step 3: Multiply A⁻¹ by B to obtain X: X = A⁻¹B.
Answer: The solution to the system of equations is x = ___ and y = ___.
Problem 12: Logarithmic Functions (Algebra) - Medium
Solve the equation log₂(x) = 3.
Solution:
Step 1: Rewrite the equation in exponential form: 2³ = x.
Step 2: Evaluate the exponent: 2³ = 8.
Answer: The solution to the equation log₂(x) = 3 is x = 8.
Problem 13: Vectors (Algebra) - Hard
Find the magnitude of the vector u = <3, -4, 5>.
Solution:
Step 1: Apply the magnitude formula: ||u|| = √(3² + (-4)² + 5²).
Step 2: Simplify: ||u|| = √(9 + 16 + 25) = √50 = 5√2.
Answer: The magnitude of the vector u = <3, -4, 5> is 5√2.
Problem 14: Trigonometric Equations (Trigonometry) - Hard
Solve the equation sin(x) + cos(x) = 1.
Solution:
Step 1: Rewrite the equation using the Pythagorean identity: sin(x) + √(1 - sin²(x)) = 1.
Step 2: Square both sides of the equation: sin²(x) + 2sin(x)√(1 - sin²(x)) + 1 - sin²(x) = 1.
Step 3: Simplify and combine like terms: 2sin(x)√(1 - sin²(x)) = 0.
Step 4: Solve for sin(x): sin(x) = 0 or √(1 - sin²(x)) = 0.
Step 5: Solve for x: x = 0 or sin(x) = 0.
Answer: The solutions to the equation sin(x) + cos(x) = 1 are x = 0 and x = π.
Problem 15: Differential Equations (Calculus) - Hard
Solve the differential equation dy/dx = 2x.
Solution:
Step 1: Integrate both sides of the equation with respect to x: ∫dy = ∫2x dx.
Step 2: Simplify: y = x² + C, where C is the constant of integration.
Answer: The solution to the differential equation dy/dx = 2x is y = x² + C.
Problem 16: Hypothesis Testing (Statistics) - Hard
A company claims that the average weight of their cereal boxes is 500 grams. To test this claim, a sample of 50 cereal boxes is selected, and their weights are measured. Determine whether the company's claim is statistically supported at a significance level of 0.01.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The average weight of the cereal boxes is 500 grams.
Alternative hypothesis (H₁): The average weight of the cereal boxes is not 500 grams.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.01), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.01, we can conclude that there is sufficient evidence to reject the company's claim about the average weight of the cereal boxes.
Problem 17: Complex Numbers (Algebra) - Hard
Solve the equation x² + 4 = 0 for complex values of x.
Solution:
Step 1: Rewrite the equation: x² = -4.
Step 2: Take the square root of both sides: x = ±√(-4) = ±2i.
Answer: The solutions to the equation x² + 4 = 0 are x = 2i and x = -2i.
Problem 18: Definite Integrals (Calculus) - Hard
Evaluate the definite integral ∫(0 to 3) (2x + 1) dx.
Solution:
Step 1: Apply the power rule of integration: ∫(2x + 1) dx = x² + x.
Step 2: Evaluate the integral at the upper and lower limits: [x² + x] from 0 to 3.
Step 3: Substitute the upper and lower limits: (3² + 3) - (0² + 0) = 12.
Answer: The value of the definite integral ∫(0 to 3) (2x + 1) dx is 12.
Problem 19: Probability Distributions (Statistics) - Hard
A fair six-sided die is rolled 100 times. Find the probability of getting exactly three 6s.
Solution:
Step 1: Determine the number of favorable outcomes: 1 (getting a 6).
Step 2: Determine the total number of possible outcomes: 6.
Step 3: Calculate the probability of getting a 6 on one roll: 1/6.
Step 4: Apply the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n = 100, k = 3, and p = 1/6.
Step 5: Calculate the probability: P(X = 3) = (100 choose 3) * (1/6)^3 * (5/6)^(100-3).
Answer: The probability of getting exactly three 6s in 100 rolls of a fair six-sided die is P(X = 3) = ___.
Problem 20: Logarithmic Differentiation (Calculus) - Hard
Differentiate y = (ln x)².
Solution:
Step 1: Apply the logarithmic differentiation rule: dy/dx = (2(ln x) / x) * (ln x)'.
Step 2: Differentiate ln x with respect to x: (ln x)' = 1/x.
Step 3: Substitute the value: dy/dx = (2(ln x) / x) * (1/x).
Answer: The derivative of y = (ln x)² is dy/dx = (2(ln x) / x²).
Problem 21: Matrices and Transformations (Algebra) - Hard
Given the matrix A = [[3, -1], [2, 4]], perform a 90-degree counterclockwise rotation transformation.
Solution:
Step 1: Set up the rotation matrix: R = [[0, -1], [1, 0]].
Step 2: Multiply A by R: A' = R * A.
Answer: The result of performing a 90-degree counterclockwise rotation transformation on matrix A is A' = [[1, 3], [-4, 2]].
Problem 22: Complex Analysis (Algebra) - Hard
Find the principal value of the complex logarithm log(i).
Solution:
Step 1: Convert the complex number to polar form: i = 1∠(π/2).
Step 2: Apply the logarithm rule for complex numbers in polar form: log(z) = ln|r| + i(arg(z) + 2kπ), where k is an integer.
Step 3: Calculate the principal value: log(i) = ln|1| + i(π/2 + 2kπ).
Answer: The principal value of the complex logarithm log(i) is ln(1) + i(π/2 + 2kπ).
Problem 23: Vector Calculus (Calculus) - Hard
Find the curl of the vector field F = ⟨3xy, x² + z, 2yz⟩.
Solution:
Step 1: Calculate the partial derivatives: ∂F/∂x = ⟨3y, 2x, 0⟩, ∂F/∂y = ⟨3x, 0, 2z⟩, ∂F/∂z = ⟨0, 1, 2y⟩.
Step 2: Evaluate the curl: curl(F) = ∇ × F = ∂F/∂y - ∂F/∂x + ∂F/∂z = ⟨3x - 3y, 2z - 2x, 2y - 0⟩.
Answer: The curl of the vector field F = ⟨3xy, x² + z, 2yz⟩ is curl(F) = ⟨3x - 3y, 2z - 2x, 2y⟩.
Problem 24: Hypothesis Testing (Statistics) - Hard
A researcher wants to determine if the mean height of a population is significantly different from 170 cm. A sample of 50 individuals is taken, and their heights are recorded. Conduct a two-sample t-test at a significance level of 0.05.
Solution:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H₀): The mean height of the population is 170 cm.
Alternative hypothesis (H₁): The mean height of the population is not 170 cm.
Step 2: Perform the hypothesis test using the appropriate statistical test (e.g., two-sample t-test).
Step 2a: Calculate the test statistic and p-value.
Step 2b: Compare the p-value to the significance level.
Step 3: Make a decision:
If the p-value is less than the significance level (0.05), reject the null hypothesis.
If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Answer: Based on the statistical analysis, if the p-value is less than 0.05, we can conclude that the mean height of the population is significantly different from 170 cm.
Problem 25: Differential Equations (Calculus) - Hard
Solve the differential equation dy/dx = y³ - y.
Solution:
Step 1: Rewrite the equation as dy/(y³ - y) = dx.
Step 2: Perform partial fraction decomposition: 1/(y³ - y) = A/y + B/(y - 1).
Step 3: Integrate both sides of the equation: ∫(1/y) dy + ∫(1/(y - 1)) dy = ∫dx.
Step 4: Simplify: ln|y| - ln|y - 1| = x + C.
Step 5: Combine logarithms: ln|y/(y - 1)| = x + C.
Step 6: Exponentiate both sides of the equation: y/(y - 1) = e^(x + C).
Step 7: Solve for y: y = (e^C) * (y - 1).
Step 8: Simplify: y - (e^C) * y = (e^C).
Step 9: Factor out y: y(1 - e^C) = (e^C).
Step 10: Solve for y: y = (e^C) / (1 - e^C).
Answer: The solution to the differential equation dy/dx = y³ - y is y = (e^C) / (1 - e^C).