#### Unknown Soldier

##### Senior Member

I. Lets set A = {1, 2, 3, 4, 5, 6, 7} and set B = {4, 5, 6, 7, 8, 9, 10}. Find the symmetric difference A ∆ B.

II. You poll ten people who drink cola asking each if they like Pepsi and dislike Coke, like Coke and dislike Pepsi, and possibly like both Pepsi and Coke. If 3 people like Pepsi and dislike Coke, and 4 people like Coke and dislike Pepsi, then how many of the ten people like Coke and Pepsi?

III. If x is any real number and a, b, and c are real numbers where f(x) = ax

^{2}+ bx + c and f(1) = 1, f(2) = 1, and f(4) = 3, then find the values of a, b, and c.

IV. If x is any real number, and g(x) = 2sin(3x), then find the area under the curve of g(x) and above the x-axis over the interval 0 ≤ x ≤ π. Also, find the equation of the tangent line to the curve of g(x) when x = π/6.

V. Let A be the square matrix

y 2

3 x

What real values of x and y will result in matrix A having a determinant equal to 4?

VI. If a and b are both positive real numbers, then prove that a/b + b/a ≥ 2.

VII. Prove if arbitrary n ∈ N = {0, 1, 2, ... }, then 2 | n

^{2}+ n.