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# SAT Data Perfect Score

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)$$\ \ 18$$

Rectangle $$ABCD$$ has dimensions $$4\ \times \sqrt{3},$$ as shown in the above diagram. If you take a point $$P$$ in the rectangle and draw triangle $$PBC,$$ what is the probability that $$\triangle PBC$$ is an acute triangle?

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

The above graph shows the frequency distributions of the numbers of questions correctly answered by boys and girls in a class on a math test with $$10$$ questions. Which of the following statements is true?

$$\begin{array}{r r l} & \text{I.} & \text{The numbers of boys and girls who took the test are the same. }\\ & \text{II.} & \text{The areas under the blue and red lines are the same.}\\ & \text{III.} & \text{The girls performed better than the boys on average.}\\ \end{array}$$

(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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