Solve the equation: 2^(3x-1) = 5^(x+2).
Find the inverse of the function f(x) = (3x - 2) / (2x + 1).
Solve the system of inequalities: 2x - y > 3, x + 3y <= 5.
Determine the complex roots of the equation x^3 - 2x^2 + 4x - 8 = 0.
Find the area of a regular octagon inscribed in a circle with radius 5 cm.
Determine the volume of a frustum of a cone with top radius 3 cm, bottom radius 5 cm, and height 4 cm.
Calculate the surface area of a sphere with a volume of 36π cubic units.
Find the equation of the ellipse with foci at (-3, 0) and (3, 0) and vertices at (-5, 0) and (5, 0).
Prove the identity: (tan(x) + cot(x))^2 = sec^2(x) + csc^2(x).
Solve the trigonometric equation: 2cos^2(2x) - 3cos(2x) + 1 = 0 for 0 <= x <= π.
Find the exact value of sin(75 degrees) without using a calculator.
Determine the equation of the sinusoid with amplitude 3, period π, phase shift π/4, and vertical shift 2.
Find the derivative of f(x) = x^sin(x).
Determine the integral of (x^2 + 1) / (x^3 + 3x) dx.
Find the arc length of the curve y = ln(x) from x = 1 to x = e.
Determine the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 - x^2 about the line x = 2.
Solve the differential equation: dy/dx = xy, y(0) = 1.
Calculate the coefficient of determination for two variables with a correlation coefficient of 0.8.
Conduct a chi-square test for independence to determine if there is a relationship between two categorical variables.
Find the confidence interval for a population proportion given a sample proportion, sample size, and confidence level.
Calculate the expected value and variance of a binomial distribution with n trials and probability of success p.
Perform a one-way ANOVA to compare the means of three or more groups.
Prove or disprove: The set of all 2x2 matrices with real entries forms a group under matrix multiplication.
Find all the subgroups of the symmetric group S3.
Determine whether the polynomial ring Z[x] is a principal ideal domain.
Estimate the round-off error in computing e^3.14 using a Taylor series approximation with 10 terms.
Numerical integration: Approximate the definite integral of sin(x^2) from 0 to 1 using Simpson's rule with 10 subintervals.
Root finding: Use Newton's method to find the root of x^3 - 2x - 5 = 0 accurate to 6 decimal places, starting with an initial guess of x = 2.
Estimate the round-off error in computing e^3.14 using a Taylor series approximation with 10 terms.
Approximate the definite integral of sin(x^2) from 0 to 1 using Simpson's rule with 10 subintervals.
Use Newton's method to find the root of x^3 - 2x - 5 = 0 accurate to 6 decimal places, starting with an initial guess of x = 2.
Encrypt the message "HELLO" using the RSA algorithm with public key (n = 143, e = 7).
Minimize the function f(x) = x^4 - 3x^2 + 2 using the gradient descent method with a learning rate of 0.1, starting at x = 1.
A sample of 100 people has a mean height of 172 cm and a standard deviation of 5 cm. Test the hypothesis that the population mean height is 170 cm at a 0.05 significance level.
Calculate a 95% confidence interval for the population mean weight based on a sample of 50 people with a mean weight of 70 kg and a standard deviation of 8 kg.
Implement the Runge-Kutta method of order 4 to solve a system of first-order ordinary differential equations.
Determine the convergence or divergence of the series Σ(n=1 to infinity) (-1)^n / sqrt(n).
Show that the function f(x) = x^3 is uniformly continuous on the interval [0, 1].
Find the eigenvalues and eigenvectors of the matrix A = [[2, -1, 0], [5, 2, -1], [6, -2, 1]].
Determine the rank and nullity of the linear transformation T: R^4 -> R^3 defined by T(x, y, z, w) = (x + y - z, 2x - y + w, x + z + w).
Evaluate the contour integral ∮(C) (z^2 + 1) / (z - i) dz, where C is the circle |z| = 2.
Find the Laurent series expansion of f(z) = 1 / (z^2 - 1) centered at z = 1.
Solve the second-order linear homogeneous differential equation y'' - 4y' + 3y = 0.
Find the general solution of the nonhomogeneous differential equation y'' + y = sec(x).
Use the method of Laplace transforms to solve the initial value problem y'' + 4y = δ(t - π), y(0) = 0, y'(0) = 1.
Derive the moment generating function of the Poisson distribution.
Implement the Newton-Raphson method to find the root of the equation x^3 - 2x - 5 = 0.
Use the Euler method to approximate the solution to the initial value problem y' = xy, y(0) = 1, on the interval [0, 1] with step size h = 0.1.
Prove the identity: (tan(x) + cot(x))^2 = sec^2(x) + csc^2(x).
Solve the trigonometric equation: 2cos^2(2x) - 3cos(2x) + 1 = 0 for 0 <= x <= π.
Find the exact value of sin(75 degrees) without using a calculator.
Determine the equation of the sinusoid with amplitude 3, period π, phase shift π/4, and vertical shift 2.
Find the derivative of f(x) = x^sin(x).
Determine the integral of (x^2 + 1) / (x^3 + 3x) dx.
Find the arc length of the curve y = ln(x) from x = 1 to x = e.
Determine the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 - x^2 about the line x = 2.
Solve the differential equation: dy/dx = xy, y(0) = 1.
Calculate the z-score for a value of 85 in a normal distribution with mean 75 and standard deviation 5.
Find the probability of getting exactly 3 heads in 5 coin flips.
Calculate the compound interest on $5000 invested for 3 years at an annual interest rate of 4.5% compounded quarterly.
Calculate the probability of drawing 3 aces from a standard deck of cards without replacement.
Solve the system of differential equations: dx/dt = 2x - y, dy/dt = x + 3y.
Implement the Euler method to approximate the solution to a first-order ODE.
Conduct a hypothesis test to compare the means of two independent samples with unknown and unequal variances.
Prove or disprove: The set of even integers is a subgroup of the integers under addition.
Determine the convergence of the series Σ(n=1 to infinity) (-1)^n / n^2.
Evaluate the contour integral ∮(C) dz / (z^2 + 4) where C is the circle |z| = 3.
Determine if the interval (0, 1) is compact.
Find the volume of the solid bounded by the surfaces z = x^2 + y^2 and z = 4.
A coin is flipped 8 times. What is the probability of getting exactly 5 heads?
Solve the initial value problem: y'' + 4y = cos(2x), y(0) = 1, y'(0) = 0.
Use the trapezoidal rule with n = 4 to approximate the integral of x^2 from 0 to 2.
Calculate the sample mean and standard deviation for the data set: 2, 4, 5, 7, 9.
Determine the order of the element (123)(45) in the symmetric group S5.
Determine the convergence of the series Σ(n=1 to infinity) 1/n^3.
Find the residues of the function f(z) = 1 / (z^2 + 1) at its poles.
Determine if the space R^2 with the usual topology is compact.
Find the inverse of the matrix [[2, -1], [3, 2]].
Find the gradient of the function f(x, y) = x^2y + y^3 at the point (1, 2).
A box contains 4 red balls, 3 green balls, and 2 blue balls. What is the probability of drawing a red or green ball?
Solve the initial value problem: y' + 2y = e^(-x), y(0) = 1.
Calculate the sample variance for the data set: 3, 5, 7, 9, 11.
Determine if the group of integers under addition is cyclic.
Determine the uniform continuity of the function f(x) = x^2 on the interval [0, 1].
Find the Taylor series expansion of f(z) = 1 / (1 - z) centered at z = 0.
Determine if the subspace X = {(x, y) ∈ R^2 | x^2 + y^2 <= 1} of R^2 is open.
Solve the system of linear equations: x + 2y - z = 3, 2x - y + 3z = 7, x + y + z = 4.
Find the surface area of the part of the plane 2x + 3y + z = 6 that lies in the first octant.
Evaluate the line integral of the vector field F(x, y) = <x^2, xy> along the curve y = x^2 from (0, 0) to (1, 1).
A card is drawn from a standard deck of 52 cards. What is the probability that it is a face card or a heart?
Solve the initial value problem: y'' - 2y' + y = e^x, y(0) = 1, y'(0) = 0.
Use Simpson's rule with n = 6 to approximate the integral of e^x from 0 to 1.
Calculate the sample standard deviation for the data set: 10, 12, 14, 16, 18, 20.
Find all the subgroups of the group Z12.
Determine the convergence of the series Σ(n=1 to infinity) (n + 1) / (n^2 + 1).
Find the image of the unit circle under the mapping w = z^2.
Calculate the covariance between the random variables X and Y, where X is the number of heads in 3 coin flips and Y is the number of tails.
Solve the Laplace's equation uxx + uyy = 0 on the unit square with boundary conditions u(0, y) = y, u(1, y) = 1 - y, u(x, 0) = x, and u(x, 1) = 1 - x.
Implement the Gauss-Seidel method to solve the system of linear equations: 4x - y = 2, x + 3y = 1.
Determine if the set of 2x2 matrices with determinant 1 forms a group under matrix multiplication.
