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The sum of three angles around a point, 2x, x, and x, equals 360 degrees, leading to the solution x = 90 degrees.

To find the unknown interior angle x, the exterior angle theorem was used, setting up and solving an equation to find x = 40 degrees.

The provided text contains a series of math problems, including algebraic expressions, exponents, fractions, and polynomial operations. The solutions demonstrate various techniques for simplifying and manipulating these expressions, such as factoring, expanding, and applying the rules of exponents and fractions.

The function f(x,y) = -2x^2 + 3xy^2 - 3y^2 + 4 has a local maximum value of 4 at (0,0).

The solution determines the Fourier coefficients for a periodic square pulse with a width of 3 and period of 12, finding the values of α, β, and λ in the given formula for ak.

The derivative of tan(θ) with respect to θ is sec^2(θ), derived using right-angled triangles and the definition of the derivative.

The provided examples demonstrate the continuity of functions at specific points. Polynomials are continuous everywhere, functions with undefined points are not continuous at those points, and absolute value functions are continuous everywhere, including at points where the expression inside the absolute value is zero.

The double integral, using polar coordinates and integration by parts, evaluates to π(e^2 + 1).

Roulette outcomes are random and independent, making future predictions impossible based on past results. Each number has an equal probability of appearing.

The solution to the system of linear equations is x = 1, y = -2, z = 3, and w = 2.

Given a^2 - 4 = 0, the possible values for a are 2 and -2. Substituting these values into a^3 - 4, we find that a^3 - 4 can be either 4 or -12.

The solution attempts to derive a reduction formula for the integral \(\int_0^\infty \frac{x}{\cosh^n(x)} dx\) using integration by parts and Feynman's trick, but ultimately finds that expressing the integral in terms of hypergeometric functions is a more suitable approach.

The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).

To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.

The eigenvalues of the matrix [[3, 1], [1, 3]] are 4 and 2.

The eigenvalues of the matrix [[0, 1], [-1, 0]] are i and -i.

The eigenvalues of the given 3x3 matrix are 1, 2, and 2.

The problem involves finding the equation of a circle given a point on the circle, the radius, and the fact that the center of the circle lies on a given line. Two possible circles satisfy the conditions, each with a different center, and the equations for both circles are derived.

The solution finds the trace lines of a plane by setting one coordinate to zero and solving for the other two coordinates in terms of the remaining parameter. This results in parametric equations for the trace lines in the xy, xz, and yz planes.

The equation represents an ellipse centered at (-2, 0) with a vertical major axis of length 4 and a horizontal minor axis of length 2.

The solution to the equation \sqrt{(X - 4)(X + 4)} = 3 is X = 5 and X = -5.

The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.

The solution involves finding the values of a and b, given a*a = 8 and a*b = 24. Then, b*b is calculated, resulting in a final answer of 72.

Cramer's rule is used to solve a system of four linear equations with four unknowns, resulting in the values of x1, x2, x3, and x4.

The equation of the line equidistant from the parallel lines y = (1/3)x - 4 and y = (1/3)x + 1 is y = (1/3)x - 5/3.

The solution finds the equation of a line g' that is the reflection of line g across line s. It involves finding the intersection point of g and s, projecting a direction vector of g onto a normal vector of s, and then calculating the direction vector of g'. Finally, the equation of g' is determined using the intersection point and the direction vector of g'.

The square root of 19 squared plus 19 plus 20 is calculated to be 20.

To find (a+b)^2, given a^2 + b^2 = 29 and ab = 10, substitute the given values into the equation (a+b)^2 = a^2 + 2ab + b^2, resulting in (a+b)^2 = 49.

The expression 3³ ÷ 3 ÷ 3 simplifies to 1.

The only real solution to the cubic equation a^3 + a^2 - 36 = 0 is a = 3.