Let \(A=\{1, 2, 3, 4, 5\},\) and \(U\) the set of all the subsets of \(A.\) If you select one element of \(U,\) what is the probability that all of the elements of the selected set are either prime numbers or non-prime numbers?

(A)\(\ \ \frac{5}{16}\)

(B)\(\ \ \frac{3}{8}\)

(C)\(\ \ \frac{1}{2}\)

(D)\(\ \ \frac{9}{16}\)

(E)\(\ \ \frac{31}{32}\)

The mean, median and mode of seven positive integers are \(4, 5\) and \(6,\) respectively. If \(a_1, a_2, \ldots, a_6, a_7\) are the seven numbers arranged in ascending order, what can be \(a_3?\)

(A)\(\ \ 1 \text{ or } 2\)

(B)\(\ \ 1 \text{ or } 3\)

(C)\(\ \ \text{only } 2\)

(D)\(\ \ 2 \text{ or } 3\)

(E)\(\ \ \text{only } 3\)

The twelve integers \(1\) through \(12\) are the elements of three sets \(A, B\) and \(C,\) and each integer is included in one or more of the three sets. The numbers of elements of \(A, B\) and \(C,\) are \(6, 5\) and \(7,\) respectively. The integers exclusively included in \(A\) are \(1\) and \(2,\) the integers included in both \(A\) and \(C\) but not \(B\) are \(8\) and \(12,\) and all the elements of \(B\) are odd. If the integer included in all of \(A, B\) and \(C\) is \(9,\) and the sum of all the elements of \(C\) is \(52,\) what is the minimum possible sum of the elements exclusively included in \(B?\)

(A)\(\ \ 10\)

(B)\(\ \ 12\)

(C)\(\ \ 14\)

(D)\(\ \ 16\)

(E)\(\ \ 18\)

(A)\(\ \ \frac{6-\sqrt{3}\pi}{6}\)

(B)\(\ \ \frac{27-4\sqrt{3}\pi}{36}\)

(C)\(\ \ \frac{1}{4}\)

(D)\(\ \ \frac{\sqrt{3}}{4}\)

(E)\(\ \ \frac{1}{2}\)

\(\begin{array}{r r l}
& \text{I.} & \text{The numbers of boys and girls who took the test are the same. }\\

& \text{II.} & \text{The areas under the blue and red lines are the same.}\\

& \text{III.} & \text{The girls performed better than the boys on average.}\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) II only

(C)\(\ \ \) I and III only

(D)\(\ \ \) II and III only

(E)\(\ \ \) I, II and III

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