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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'.