首页>SAT考试>SAT 数 学>SAT 数学Problem Solving训练题三 含答案解读
SAT 数学Problem Solving训练题三 含答案解读
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导读:SAT数学Problem Solving练习题三 含答案解析 Question #1: In the x,y plane, which of the following statements are true? I. Line y + x = 5 is perpendicular to line y - x = 5. II. Lines y + x = 5 and y - x ...

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SAT数学Problem Solving练习题三 含答案解析

Question #1: In the x,y plane, which of the following statements are true? I. Line y + x = 5 is perpendicular to line y - x = 5. II. Lines y + x = 5 and y - x = 5 intersect each other on the y axis. III. Lines y + x = 5 and y - x = 5 intersect each other on the x axis. (a) I and III are both true. (b) I is the only true statement. (c) II is the only true statement. (d) I and II are both true. Answer: y + x = 5 can be written as y = -x + 5. The slope of this equation is m1 = -1.y - x = 5 can be written as y = x + 5. The slope of this equation is m2 = 1. m2 = -1/m1 so the 2 lines are perpendicular. We also need to find where the 2 lines intersect. If we add the 2 equations, 2?y = 10, y = 5. From the first equation, x = 5 - y = 5 - 5 = 0. In conclusion the lines intersect at (0, 5) and this point is on the y axis.In conclusion I and II statements are correct. ------------------------------------------------------- Question #2: If a is an integer chosen randomly from the set {3, 5, 6, 9} and b is an integer chosen randomly from the set {2, 3, 4}, what is the probability that a/b is an integer? (a) .125 (b) .250 (c) .333 (d) .5 (e) .55 Answer: We have 4 possible integers for a and 3 for b, so the number of possible combinations for a/b is 4 ? 3 = 12a/b is an integer only for 4 combinations: 1. a = 3 and b = 3 2. a = 6 and b = 2 3. a = 6 and b = 3 4. a = 9 and b = 3 The probability that a/b is an integer is 4/12 = 1/3 = .333. ------------------------------------------------------- Question #3: What is the value of integer a, if x = 2 is a solution of the equation √(a + x) = 2?x? (a) a = 10 (b) a = 12 (c) a = 14 (d) a = 16 (e) a = 18 Answer: If we square the equation we get a + x = 4?x2 By replacing x with 2, a + 2 = 4?22, so a + 2 = 16. In conclusion, a = 14. ------------------------------------------------------- Question #4: What is the value of (3x + 1 - 3x) / (3x - 3x - 1)? (a) 6 (b) 3x (c) 3x + 1 (d) 3x - 1 (e) 3 Answer: The numerator of the fraction is: 3x + 1 - 3x = 3x?(3 - 1) = 2 ? 3x The denominator of the fraction is: 3x - 3x - 1 = 3x - 1?(3 - 1) = 2 ? 3x - 1 We can write the fraction as (2 ? 3x) / (2 ? 3x - 1) = 3x / 3x - 1 = 3 ------------------------------------------------------- Question #5: Two diameters of a circle create an angle AOB of 45o between them. What is the length of arc AB if the radius of the circle is 10/?? (a) 5/2 (b) 3/2 (c) 2 (d) 4 (e) 6 Answer: The circumference of the circle is 2???r = 2???10/? = 20. The ratio between the length of arc AB and the circumference of the circle is equal between the ratio between the 45o angle and 360o. In conclusion, AB = 20 ? 45o/360o = 20/8 = 5/2. ------------------------------------------------------- Question #6: A bus travels from town A to town B for 2 hours at a speed of 60 miles/hour. The bus stops in town B for 2 hours and then travels from town B to town C for 1 hour, at a speed of 50 miles/hour. What is the average speed of the bus? (a) 30miles/hour (b) 31miles/hour (c) 32miles/hour (d) 34miles/hour (e) 40miles/hour Answer: The distance the bus travels from A to B is 60 miles/hour ? 2 hours = 120 miles. Then, the bus travels from B to C: 50 miles/hour ? 1 hour = 50 miles. The total distance traveled is 120 + 50 = 170 miles and the total time is 2 hours + 2 hours stop + 1 hour = 5 hours. In conclusion the average speed was 170 miles / 5 hours = 34 miles/hour. ------------------------------------------------------- Question #7: If a?b + b?c + c?a = 0, what is (a + b)2 + (b + c)2 + (c + a)2? (a) a2 + b2 + c2 (b) 2?(a2 + b2 + c2) (c) (a2 + b2 + c2)/2 (d) a2 + a + b2 + b + c2 +c (e) (a + b + c)/2 Answer: (a + b)2 + (b + c)2 + (c + a)2 = 2?(a2 + b2 + c2) + 2?(a?b + b?c + c?a) Since a?b + b?c + c?a = 0, the correct result is 2?(a2 + b2 + c2). ------------------------------------------------------- Question #8: Column A Column B x2 + 1 x + 1 (a) The quantity in Column A is greater then the quantity in Column B. (b) The quantity in Column B is greater then the quantity in Column A. (c) The two quantities are equal. (d) The relationship cannot be determined from the information given. Answer: We need to compare x2 + 1 with x + 1. This results in a comparison between x2 and x. For some x, x2 will be greater than x, e.g. for x = 2. For others, e.g. x = 1/2, x2 will be lower so the relationship cannot be determined from the information given. ------------------------------------------------------- Question #9: 2?m - n = 4 m + 2?n = 12 Column A Column B (m + n)2 61 (a) The quantity in Column A is greater then the quantity in Column B. (b) The quantity in Column B is greater then the quantity in Column A. (c) The two quantities are equal. (d) The relationship cannot be determined from the information given. Answer: From the first equation, n = 2?m - 4. Then, the first equation will be m + 2?(2?m - 4) = 1m + 4?m - 8 = 12 so 5?m = 20 and m = 4 From the first equation, n = 2?m - 4 = 2?4 - 4 = 4 Column A expression will be (m + n)2 = (4 + 4)2 = (8)2 = 64 The quantity in Column A is greater than the quantity in Column B. ------------------------------------------------------- Question #7: If a and b are positive integers and a?b = 200, which of the following can be the sum a + b? (a) 40 (b) 46 (c) 33 (d) 55 (e) 50 Answer: 200 = 2?2?2?5?5. If a and b are positive integers, the 2 numbers and their sum can be: 2 + 100 = 102 4 + 50 = 54 5 + 40 = 45 8 + 25 = 33 10 + 20 = 30 (c) is the correct answer.
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