SAT数学Problem Solving练习题三.

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 = 12. a/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 (3^{x + 1} - 3^x) / (3^x - 3^{x - 1})?

(a) 6

(b) 3^x

(c) 3^{x + 1}

(d) 3^{x - 1}

(e) 3

• Answer: The numerator of the fraction is: 3^{x + 1} - 3^x = 3^x·(3 - 1) = 2 · 3^x The denominator of the fraction is: 3^x - 3^{x - 1} = 3^{x - 1}·(3 - 1) = 2 · 3^{x - 1} We can write the fraction as (2 · 3^x) / (2 · 3^{x - 1}) = 3^x / 3^{x - 1} = 3

Question #5: Two diameters of a circle create an angle AOB of 45^o 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 45^o angle and 360^o. In conclusion, AB = 20 · 45^o/360^o = 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:

(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

(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) = 12 m + 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.