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Solve for x in a determinant problem involving trigonometric functions. The solution involves calculating the determinant of a 3x3 matrix, using trigonometric identities, and solving a trigonometric equation.

Find the values of sine, cosine, tangent, cosecant, secant, and cotangent for the given right triangles. The solution involves applying trigonometric ratios and the Pythagorean theorem.

Solve a matrix problem involving motor production and component requirements. The solution involves setting up production and component matrices and performing matrix multiplication to find the total components needed.

Solve matrix equations to find the unknown matrix X using matrix inverse and transpose properties.

Find the solution to a system of differential equations given its general solution and an initial condition. The solution involves finding the constants and then substituting them back into the general solution to find the specific solution for x1(t), x2(t), and x3(t).

Find the general solution to a system of differential equations X' = AX, given the eigenvalues and eigenvectors of the matrix A. The solution involves finding linearly independent solutions based on the eigenvalues and eigenvectors, including complex conjugate pairs.

Solve the initial value problem (IVP) for a system of linear differential equations X' = AX, finding the particular solution given the matrix A and initial condition X(0). The solution involves finding eigenvalues, eigenvectors, and applying the initial condition.

Find the solution y(t) to the second-order linear non-homogeneous differential equation y" - 3y' - 10y = 1 with initial conditions y(0) = -1 and y'(0) = 2. The solution involves finding the complementary and particular solutions and applying initial conditions.

Find the Laplace Transform of the solution for a mass-spring-damper system with an applied force, given initial conditions. The solution involves setting up the differential equation, applying the Laplace Transform, and solving for X(s).

Find the general solution and the value at pi for a mass-spring-damper system with an applied force, using differential equations and initial conditions.

Find the solution to a differential equation given its Laplace transform. The solution involves inverse Laplace transforms and the time-shifting property.

Solve probability and input-output model problems. Includes fox hunting probabilities and production analysis for interconnected factories.

This calculus problem involves evaluating the integral of (e^x + 1) / (e^x - 1) dx. The solution uses substitution and partial fraction decomposition to find the antiderivative.

Find the maximum value of an algebraic expression given a constraint on the sum of three positive variables. The solution involves applying algebraic manipulations and inequalities to determine the upper bound of the expression.

Solve the second-order linear differential equation y'' - y = 0 using the Method of Undetermined Power Series Coefficients, resulting in a solution involving hyperbolic functions.

Solve the algebraic equation A + AA + AAA = 738 to find the value of the digit A. The solution involves simplifying the equation and solving for A.

Solve the algebraic equation (a+2)(a+3)(a+4)(a+5) / (a-2)(a-3)(a-4)(a-5) = 1 to find the value of a. The solution involves simplifying the equation and solving for a.

Solve the division problem involving fractions: (55 / 5) / (5 / 55). The solution involves simplifying the numerator and denominator, then dividing the resulting fractions.

Solve the linear equation 2x + 5 = 12 to find the value of x. The solution involves isolating x by subtracting and dividing.

Solve for X in the equation X = √(100² - 96²). This problem involves simplifying the expression using the difference of squares formula and calculating the square root.

The equation (X-1)^2 = X-1 has two solutions, X = 1 and X = 2.

Solve the given system of linear equations by using the elimination method to find the values of x and y.

Solve the equation involving division of fractions and find the value of X.

The solutions to the equation \sqrt{5x^2-6x+8} - \sqrt{5x^2-6x-7} = 1 are x = 4 and x = -14/5.

The provided text contains a detailed solution to various integration problems, including trigonometric substitutions, partial fraction decomposition, and integration by parts. It also demonstrates the use of properties of definite integrals to simplify calculations.

The area of a square with side length 3 is 9.

The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It determines the dataset is negatively skewed.

The solution calculates the range, arithmetic mean, population variance, and standard deviation of the annual incomes of five vice presidents.

The mixed strategy Nash equilibria are (Pass, Pass) and (Two, Two). For (Pass, Pass), the expected utility for each player is 1. For (Two, Two), the expected utility for each player is 0.

The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It demonstrates the relationships between these measures and concludes that the distribution is slightly negatively skewed.