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Andy T.

Student at Swarthmore College

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SAT II Mathematics Level 2

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Question:

What is the distance in space between the points with coordinates (-3, 6, 7) and (2, -1 , 4)? (A) 4.36 (B) 5.92 (C) 7.91 (D) 9.11 (E) 22.25

Andy T.

Answer:

First we realize that the question is asking for a distance between points so we would have to recall the distance formula for points in 3 dimensions. The distance is the square root of the sums of the squares of the difference of each direction. Therefore in this question we would first find the differences of each direction: (-3 - 2) = -5, (6 - (-1)) = 7, (7 - 4) = 3. Then we square each difference: (-5)^2 = 25, 7^2 = 49, 3^2 = 9. Afterwards we add them all up: 25 + 49 + 9 = 83. Finally take the square root of the sum: sqrt(83) = 9.11 (D)

SAT

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Question:

2^30 + 2^30 + 2^30 + 2^30 = A. 8^120 B. 8^30 C. 2^32 D. 2^30 E. 2^26

Andy T.

Answer:

First we should realize that if we have adding 4 of the same numbers, the addition should be the same as multiplying 4 to one of the numbers, so 2^30 + 2^30 + 2^30 + 2^30 = 4 (2^30). Then we can convert 4 to a power of 2, so 4 = 2^2. After we change the 4 into a power a two, then we can use an exponent rule which states that if you multiply numbers with the same base, then we may add the exponents instead. 4 (2^30) = 2^2(2^30) = 2^32 (C)

ACT

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Question:

What is the product of the two real solutions of the equation: 2x = 3 - x^2 A) 3 B) 2 C) -2 D) 6 E) -3

Andy T.

Answer:

First we should start out by realizing that we need to find "two real solutions" from the given equation, so by solving for x we would find the numbers we need. 2x = 3 - x^2 => x^2 + 2x - 3 = 0 => (x + 3) * ( x - 1) = 0. Through rearranging the equation and factoring we see that (x + 3) or (x - 1) equals 0. Therefore solutions to the equation must be either x = -3 or 1. Since the question asks for the product of the solutions then the answer must be -3 * 1 = -3 (E)

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