SAT数学练习题四（附答案）

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　　1. If f(x) = │(x² – 50)│, what is the value of f( -5) ?

　　A. 75

　　B. 25

　　C. 0

　　D. -25

　　E. -75

　　2. ( √2 - √3 )² =

　　A. 5 - 2√6

　　B. 5 - √6

　　C. 1 - 2√6

　　D. 1 - √2

　　E. 1

　　3. 230 + 230 + 230 + 230 =

　　A. 8120

　　B. 830

　　C. 232

　　D. 230

　　E. 226

　　4. Amy has to visit towns B and C in any order. The roads connecting these towns with her home are shown on the diagram. How many different routes can she take starting from A and returning to A, going through both B and C (but not more than once through each) and not travelling any road twice on the same trip?

　　A. 10

　　B. 8

　　C. 6

　　D. 4

　　E. 2

　　5. In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ?

　　A. 18

　　B. 13.5

　　C. 9

　　D. 4.5

　　E. 3

　　答案：

　　1.Correct Answer: B

　　Explanation:

　　If x = -5, then (x² – 50) = 25 – 50 = -25

　　But the sign │x│ means the absolute value of x (the distance between the number and zero on the number line). Absolute values are always positive.

　　│-25 │ = 25

　　2.Correct Answer: A

　　Explanation:

　　Expand as for (a + b)2.

　　(√2 - √3)(√2 - √3) = 2 - 2(√2 + √3) + 3 = 5 - 2 √6

　　3.Correct Answer: C

　　Explanation:

　　All four terms are identical therefore we have 4 (230).

　　But 4 = 22, and so we can write 22. 230

　　Which is equivalent to 232

　　4. Correct Answer: B

　　Explanation:

　　Amy can travel clockwise or anticlockwise on the diagram.

　　Clockwise, she has no choice of route from A to B, a choice of one out of two routes from B to C, and a choice of one out of two routes from C back to A. This gives four possible routes. Similarly, anticlockwise she has four different routes.

　　Total routes = 8

　　5.Correct Answer: D

　　Explanation:

　　If we take AE as the base of triangle AEC, then the height is CD.

　　The height of the triangle is therefore, 9 (given).

　　To find the base we need to see that triangles AEB and CDE are similar. The ratio AB: CD, is therefore equal to the ratio AE: ED. The given information shows that the ratio is 3:9, or 1:3. Now dividing AD (4) in this ratio gives us AE as 1.

　　The area of AEC = ½ base x height

　　=1/2 x 9 = 4.5

　　SAT 考试数学练习题 3

　　1. Which of the following could be a value of x, in the diagram above?

　　A. 10

　　B. 20

　　C. 40

　　D. 50

　　E. any of the above

　　2. Helpers are needed to prepare for the fete. Each helper can make either 2

　　large cakes or 35 small cakes per hour. The kitchen is available for 3 hours and

　　20 large cakes and 700 small cakes are needed. How many helpers are required?

　　A. 10

　　B. 15

　　C. 20

　　D. 25

　　E. 30

　　3. Jo's collection contains US, Indian and British stamps. If the ratio of US

　　to Indian stamps is 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what

　　is the ratio of US to British stamps?

　　A. 5 : 1

　　B. 10 : 5

　　C. 15 : 2

　　D. 20 : 2

　　E. 25 : 2

　　4. A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle?

　　A. 2.5π

　　B. 3π

　　C. 5π

　　D. 4π

　　E. 10π

　　5. Two sets of 4 consecutive positive integers have exactly one integer in common. The sum of the integers in the set with greater numbers is how much greater than the sum of the integers in the other set?

　　A. 4

　　B. 7

　　C. 8

　　D. 12

　　E. it cannot be determined from the information given.

　　答案：

　　1.Correct Answer: B

　　Explanation:

　　The marked angle, ABC must be more than 90 degrees because it is the external angle of triangle BDC, and must be equal to the sum of angles BDC (90) and DCB. Also ABC is not a straight line and must be less than 180. Therefore 90 < 5x < 180 The only value of x which satisfies this relation is 20.

　　2.Correct Answer: A

　　Explanation:

　　20 large cakes will require the equivalent of 10 helpers working for one hour.

　　700 small cakes will require the equivalent of 20 helpers working for one hour. This means if only one hour were available we would need 30 helpers. But since three hours are available we can use 10 helpers.

　　3.Correct Answer: E

　　Explanation:

　　Indian stamps are common to both ratios. Multiply both ratios by factors such

　　that the Indian stamps are represented by the same number. US : Indian = 5 : 2,

　　n(n)dian : British = 5 : 1. Multiply the first by 5, and the second by 2.

　　Now US : Indian = 25 : 10, and Indian : British = 10 : 2 Hence the two ratios can

　　be combined and US : British = 25 : 2

　　4.Correct Answer: C

　　Explanation:

　　Draw the diagram. The diagonal of the rectangle is the diameter of the circle.

　　The diagonal is the hypotenuse of a 3,4,5 triangle, and is therefore, 5.

　　Circumference = π.diameter = 5π

　　5.Correct Answer: D

　　Explanation:

　　If two sets of four consecutive integers have one integer in common, the total

　　in the combined set is 7., and we can write the sets as n + (n + 1) + (n + 2) + (n

　　+ 3 ) and (n + 3) + (n + 4) + (n + 5) + (n + 6) Note that each term in the second

　　set is 3 more than the equivalent term in the first set. Since there are four terms

　　the total of the differences will be 4 x 3 = 12

　　SAT 考试数学练习题 4

　　1. If f(x) = (x + 2) / (x-2) for all integers except x=2, which of the following has the greatest value?

　　A. f(-1)

　　B. f(0)

　　C. f(1)

　　D. f(3)

　　E. f(4)

　　2. ABCD is a square of side 3, and E and F are the mid points of sides AB and BC

　　respectively. What is the area of the quadrilateral EBFD ?

　　A. 2.25

　　B. 3

　　C. 4

　　D. 4.5

　　E. 6

　　3. If n ≠ 0, which of the following mus t be greater than n?

　　I 2n

　　II n²

　　III 2 - n

　　A. I only

　　B. II only

　　C. I and II only

　　D. II and III only

　　E. None

　　4. After being dropped a certain ball always bounces back to 2/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (in inches) will it reach after its fourth bounce?

　　A. 20

　　B. 15

　　C. 8

　　D. 5

　　E. 3.2

　　5. n and p are integers greater than 1 5n is the square of a number

　　75np is the cube of a number.

　　The smallest value for n + p is

　　A. 14

　　B. 18

　　C. 20

　　D. 30

　　E. 50

　　答案：

　　1.Correct Answer: D

　　Explanation:

　　You can solve this by back solving – substitute the answer choices in the expression and see which gives the greatest value.sat

　　A (-1 + 2) / (-1-2) = -2 / 2 = -1;

　　B (0 + 2) / (0-2) = 2/ -2 = -1;

　　C (1 + 2) / (1-2) = 3/-1 = -3;

　　D (3 + 2) / (3-2) = 5/1 = 5 ;

　　E (4+ 2) / (4-2) = 6/2 = 3

　　If you had just chosen the largest value for x you would have been wrong. So although it

　　looks a long method, it is actually quick and accurate since the numbers are really simple and you can do the math in your head.

　　2.Correct Answer: D

　　Explanation:

　　(Total area of square - sum of the areas of triangles ADE and DCF) will give the area of the quadrilateral 9 - (2 x ½ x 3 x 1.5) = 4.5

　　3.Correct Answer: E

　　Explanation:

　　Remember that n could be positive negative or a fraction. Try out a few cases: In case I, if n is -1, then 2n is less than n. In case II, if n is a fraction such as ½ then n2 will be less than n. In

　　case III, if n is 2, then 2-n = 0, which is less than n. Therefore, none of the choices must be greater

　　than n

　　4.Correct Answer: C

　　Explanation:

　　If after each bounce it reaches 2/5 of the previous height, then after the second bounce it will reach 2/5 x 125. After the third it will reach 2/5 x 2/5 x 125. After the fourth it will reach 2/5 x 2/5 x 2/5 x 125. This cancels down to 2 x 2 x 2 = 8

　　5.Correct Answer: A

　　Explanation:

　　The smallest value for n such that 5n is a square is 5. 75np can now be written as 75 x 5 x p. This gives prime factors.... 3 x 5 x 5 x 5 x p To make the expression a perfect cube, p will have to have factors 3 x 3 , and hence p =9 n + p = 5 + 9 = 14

　　SAT 考试数学练习题 5

　　1. If f(x) = x² – 3, where x is an integer, which of the following could be a value of f(x)? I 6

　　II 0

　　III -6

　　A. I only

　　B. I and II only

　　C. II and III only

　　D. I and III only

　　E. I, II and III Correct Answer: A 解析:

　　Choice I is correct because f(x) = 6 when x=3. Choice II is incorrect because to make f(x) =

　　0, x² would have to be 3. But 3 is not the square of an integer. Choice III is incorrect because to make f(x) = 0, x² would have to be –3 but squares cannot be negative. (The minimum value for x2 is zero; hence, the minimum value for f(x) = -3)

　　2. For how many integer values of n will the value of the expression 4n + 7 be an integer greater than 1 and less than 200?

　　A. 48

　　B. 49

　　C. 50

　　D. 51

　　E. 52

　　Correct Answer: C

　　解析:

　　1 < 4n + 7 < 200. n can be 0, or -1. n cannot be -2 or any other negative integer or the expression 4n + 7 will be less than1. The largest value for n will be an integer < (200 - 7) /4. 193/4 = 48.25, hence 48. The number of integers between -1 and 48 inclusive is 50

　　3. In the following correctly worked addition sum, A,B,C and D represent different digits, and all the digits in the sum are different. What is the sum of A,B,C and D?

　　A. 23

　　B. 22

　　C. 18

　　D. 16

　　E. 14

　　Correct Answer: B

　　解析:

　　First you must realize that the sum of two 2-digit numbers cannot be more that 198 (99 + 99). Therefore in the given problem D must be 1. Now use trial and error to satisfy the sum 5A + BC = 143. A + C must give 3 in the units place, but neither can be 1 since all the digits have to be different. Therefore A + C = 13. With one to carry over into the tens column, 1 + 5 + B = 14, and B = 8. A + C + B + D = 13 + 8 + 1 = 22

　　4. 12 litres of water a poured into an aquarium of dimensions 50cm length , 30cm breadth, and 40 cm height. How high (in cm) will the water rise?

　　(1 litre = 1000cm³)

　　A. 6

　　B. 8

　　C. 10

　　D. 20

　　E. 40

　　Correct Answer: B

　　解析:

　　Total volume of water = 12 liters = 12 x 1000 cm3. The base of the aquarium is 50 x 30 = 1500cm3. Base of tank x height of water = volume of water. 1500 x height = 12000; height = 12000 / 1500 = 8

　　5. Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ?

　　A. 11/P + 6

　　B. P/11 +6

　　C. 17 - P/6

　　D. 17/P

　　E. 11.5P

　　Correct Answer: A

　　解析:

　　Let Ben’s age now be B. Anita’s age now is A. (A - 6) = P(B - 6)

　　But A is 17 and therefore 11 = P(B - 6). 11/P = B-6

　　(11/P) + 6 = B

　　SAT 考试数学练习题 6

　　1. The distance from town A to town B is five miles. C is six miles from B. Which of the following could be the distance from A to C?

　　I 11

　　II 1

　　III 7

　　A. I only

　　B. II only

　　C. I and II only

　　D. II and III only

　　E. I, II, or III. Correct Answer: E 解析:

　　Do not assume that AB and C are on a straight line. Make a diagram with A and B marked 5 miles apart. Draw a circle centered on B, with radius 6. C could be anywhere on this circle. The minimum distance will be 1, and maximum 11, but anywhere in between is possible.

　　2. √5 percent of 5√5 =

　　A. 0.05

　　B. 0.25

　　C. 0.5

　　D. 2.5

　　E. 25

　　Correct Answer: B

　　解析:

　　We can write the statement mathematically, using x to mean ‘of’ and /100 for ‘per cent’ . S o ( √5/100) x 5√5 = 5 x 5 /100 = 0.25

　　3. If pqr = 1 , rst = 0 , and spr = 0, which of the following must be zero?

　　A. P

　　B. Q

　　C. R

　　D. S

　　E. T

　　Correct Answer: D

　　解析:

　　If pqr = 1, none of these variable can be zero. Since spr = 0 , and since p and r are not zero, s must be zero. (Note that although rst = 0, and so either s or t must be zero, this is not sufficient to state which must be zero)

　　4.

　　A. 1/5

　　B. 6/5

　　C. 6³

　　D. 64 / 5

　　E. 64

　　Correct Answer: E

　　解析:

　　65 = 64x 6

　　(64 x 6) - 64 = 64(6 - 1) = 64 x 5 Now, dividing by 5 will give us 64

　　5. -20 , - 16 , - 12 , -8 ....

　　In the sequence above, each term after the first is 4 greater than the preceding term. Which of the following could not be a term in the sequence?

　　A. 0

　　B. 200

　　C. 440

　　D. 668

　　E. 762

　　Correct Answer: E

　　解析:

　　All terms in the sequence will be multiples of 4. 762 is not a multiple of 4

　　1. If a² = 12, then a4 =

　　A. 144

　　B. 72

　　C. 36

　　D. 24

　　E. 16

　　Correct Answer: A

　　解析:

　　a4 = a2 x a2 = 12 x 12 = 144

　　2. If n is even, which of the following cannot be odd?

　　I n + 3

　　II 3n

　　III n² - 1

　　A. I only

　　B. II only

　　C. III only

　　D. I and II only

　　E. I, II and III Correct Answer: B 解析:

　　In case I , even plus odd will give odd. In case II, odd times even will give even. In case III even squared is even, and even minus odd is odd. (You can check this by using an easy even

　　number like 2 in place of n). Only case II cannot be odd.

　　3. One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?

　　I 24

　　II 20

　　III 5

　　A. I only

　　B. II only

　　C. III only

　　D. II and III only

　　E. I, II and III Correct Answer: D 解析:

　　The maximum area of the triangle will come when the given sides are placed at right angles. If we take 8 as the base and 5 as the height the area = ½ x 8 x 5 = 20. We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20.

　　4. A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?

　　A. 12

　　B. 11

　　C. 10

　　D. 9

　　E. 8

　　Correct Answer: E

　　解析:

　　Food consumed per day = 39/6. In the remaining days it will consume 91 - 39 pounds = 52 pounds. Now divide the food by the daily consumption to find the number of days. 52 / (39/6) = 52 x (6 / 39) = 8

　　5. A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?

　　A. 8p

　　B. pq

　　C. pq + 27

　　D. -p

　　E. (p - q)6

　　Correct Answer: C

　　解析:

　　A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5. Consider the answer choices in turn. 8 is the cube of 2, and p is a cube, and so the product will also be a cube. pq will also be a cube as shown above.pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube. -p will be a cube. Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).

　　SAT 考试数学练习题 8

　　6. What is the length of the line segment in the x-y plane with end points at (-2,-2) and (2,3)?

　　A. 3

　　B. √31

　　C. √41

　　D. 7

　　E. 9

　　Correct Answer: C

　　解析:

　　Sketch a diagram and calculate the distance (hypotenuse of a right triangle) using Pythagoras theorem.

　　Vertical height of triangle = 5 ; horizontal side = 4 ; hypotenuse = √(25 + 16) = √41

　　7. n is an integer chosen at random from the set

　　{5, 7, 9, 11 }

　　p is chosen at random from the set

　　{2, 6, 10, 14, 18}

　　What is the probability that n + p = 23 ?

　　A. 0.1

　　B. 0.2

　　C. 0.25

　　D. 0.3

　　E. 0.4

　　Correct Answer: A

　　解析:

　　Each of the integers in the first set could be combined with any from the second set, giving a total of 4 x 5 = 20 possible pairs. Of these the combinations that could give a sum of 23 are (5 + 18), and (9 + 14). This means that the probability of getting a sum of 23 is 2/20 = 1/10

　　8. A dress on sale in a shop is marked at $D. During the discount sale its price is reduced by 15%. Staff are allowed a further 10% reduction on the discounted price. If a staff member buys the dress what will she have to pay in terms of D ?

　　A. 0.75D

　　B. 0.76D

　　C. 0.765D

　　D. 0.775D

　　E. 0.805D

　　Correct Answer: C

　　解析:

　　If the price is reduced by 15 %, then the new price will be 0.85D. If this new price is further reduced by 10%, the discounted price will be 0.9 x 0.85D = 0.765D

　　9. All the dots in the array are 2 units apart vertically and horizontally. What is the length of the longest line segment that can be drawn joining any two points in the array without passing through any other point ?

　　A. 2

　　B. 2√2

　　C. 3

　　D. √ 10

　　E. √20

　　Correct Answer: E

　　解析:

　　The longest line segment that can be drawn without passing through any dots other than those at the beginning and end of the segment, such a line could go from the middle dot in the top row to either the bottom left or right dot. In any case the segment will be the hypotenuse of a right triangle with sides 2 and 4. Using Pythagoras theorem the hypotenuse will be √(2 ² + 4 ² ) = √20

　　10. If the radius of the circle with centre O is 7 and the measure of angle AOB is 100, what is

　　the best approximation to the length of arc AB ?

　　A. 9

　　B. 10

　　C. 11

　　D. 12

　　E. 13

　　Correct Answer: D

　　解析:

　　If the radius is 7, the circumference = 14π . The length of the arc is 100/360 of the circumference. Taking π as 22/7 we get. (100 x 14 x 22) / (360 x 7) which reduces to 440/ 36 = 12.22 (i.e. approx. 12)

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