代数英文练习2
美国高中代数2 Square Roots
1-3
Square Roots
Example 3B: Rationalizing the Denominator
Simplify the expression.
Multiply by a form of 1.
Leona Algebra 2
1-3
Square Roots
Check It Out! Example 3a
Simplify by rationalizing the denominator.
Multiply by a form of 1.
Leona Algebra 2
1-3
Square Roots
Check It Out! Example 3b
Simplify by rationalizing the denominator.
1-3
Square Roots
Check It Out! Example 4b
Add or subtract.
Simplify radical terms.
Combine like radical terms.Βιβλιοθήκη Leona Algebra 2
1-3
Square Roots
Lesson Quiz: Part I
D.
Quotient Property of Square Roots
Leona Algebra 2
1-3
Square Roots
Check It Out! Example 2 Simplify each expression. A.
Find a perfect square factor of 48. Product Property of Square Roots
抽象代数Chapter2习题答案
HW 2 Solutions, (October 5th, 2011)
All problems from Chapter 2 of Artin’s Algebra. 1.3 If map r : N → N was a right inverse for the shift map s, then the composition sr would send 1 to 1. However, the number 1 is not in the image of s so such a right inverse is impossible. For any n ∈ N, define the map ℓn by ℓn (i) = i − 1 if i ≥ 2 and ℓn (1) = n. Then the composition ℓn s is the identity on N. Thus we have exhibited an infinite number of left inverses. 2.1 Group multiplication table for S3 = {e, (12), (13), (23), (123), (132)} with first row (resp. column) corree (12) (23) (13) (123) (132) (12) e (123) (132) (23) (13) (23) (132) e (123) (13) (12) sponding to left (resp. right-) multiplication by identity e: . (13) (123) (132) e (12) (23) (123) (13) (12) (23) (132) e (132) (23) (13) (12) e (123) 2.2 Let S ′ be the subset of S consisting of invertible elements. We must show that the associative law of composition, ◦, on S restricts to a law of composition on S ′ . In other words, we need to show closure: if s1 and s2 are invertible (i.e. in S ′ ), then s1 ◦ s2 is also invertible (i.e. in S ′ ). But this is clearly true since 1 −1 ′ s− 2 ◦ s1 is the inverse of s1 ◦ s2 . This law of composition on S is associative since it is associative on S . To complete the proof that subset S ′ is a group, we need to check that identity and inverses are in S ′ , and these follow quickly. 2.4 a) Yes, GLn (R) is a subgroup of GLn (C), clearly the product of two invertible matrices with real entries is an invertible matrix with real entries (implies closure ). The identity matrix has real entries, and the inverse of a matrix with real entries also has real entries. b) Yes, {−1, 1} is a subgroup of R× . (Similar technique as in part (a).) c) No, the inverse of a positive integer (under addition) is not a positive integer. d) Yes, {positive reals} is a subgroup of R× . (Similar technique as in part (a).) e) No, matrix a 0 is not invertible. 0 0 1
数学英语练习题2
Math Problems in English (2)Directions: Each of the questions has four answer choices. For each of these questions, select the best of the answer choices given.41. A necklace is made by stringing N individual beads together in the repeating pattern red bead, greenbead, white bead, blue bead, and yellow bead. If the necklace design begins with a red bead and ends with a white bead, then N could equalA. 54B. 68C. 76D. 8242. John was assigned some math exercises for homework. He answered half of them in study hall. Afterschool he answered 7 more exercises. If he still has 11 exercises to do, how many exercises were assigned?A. 36B. 24C. 12D. 843. The average of 3 different positive integers is 100 and the largest of these integers is 120, what is theleast possible value of the smallest of these integers?A. 1B. 10C. 61D. 7144. If when a certain integer x is divided by 5 the remainder is 2, then each of the following could also be aninteger EXCEPTA. x/17B. x/11C. x/10D. x/645. Over the last three years a scientist had an average yearly income of $45,000. The scientist earned 1.5times as much the second year as the first year and 2.5 times as much the third year as the first year.What was the scientist’s income the second year?A. $9,000B. $13,500C. $27,000D. $40,50046. Salesperson A’s compensation for any week is $360 plus 6% of the proportion of A’s total sales above$1,000 for that week. Salesperson B’s compensation for any week is 8% of B’s total sales for that week.For what amount of total weekly sales would both salespeople earn the same compensation?A. $21,000B. $18,000C. $15,000D. $4,50047. An instructor scored a student’s test of 50 questions by subtracting 2 times the number of incorrectanswers from the number of correct answers. If the student answered all the questions and received a sore of 38, how many questions did that student answer correctly?A. 46B. 47C. 48D. 4948. When you add 5 to a certain number, then subtract -10, multiply by 4, and divide by 6, you get 12. What isthe number?A. -33B. 1/3C. 3D. 13 2/349. If k is an integer greater than 44 and less than 51, then which of the following could be the product of 11and k?A. 572B. 550C. 484D. 44050. If x and y are prime integers, which of the following CANNOT be the sum of x and y?A. 23B. 16C. 13D. 951. To fill a number of vacancies, an employer must hire 3 programmers from among 6 applicants, and 2managers from among 4 applicants. What is the total number of ways in which she can make her decision?A. 1,940B. 132C. 120D. 6052. A $500 investment and a $1,500 investment have a combined yearly return of 8.5% of the total of the twoinvestments. If the $500 investment has a yearly return of 7%, what percent yearly return does the $1,500 investment have?A. 9%B. 10%C. 10.5%D. 11%53. All of the following have the same number of distinct prime factors EXCEPTA. 20B. 21C. 24D. 3054. A haberdasher sells neckties for $7 each and shirts for $12 each. If he sells $95 worth of ties and shirts,what is the least amount of ties he could have sold?A. 3B. 4C. 5D. 655. A machine costs x dollars per day to maintain and y cents for each unit it produces. If the machine isoperated 7 days a week and produces n units in a week, which of the following is the total cost, in dollars, of operating the machine for a week?A. 7x+100ynB. 7x+ynC. (700x+yn)/100D. (7x+100yn)/10056. Coins are to be put into 7 pockets so that each pocket contains at least one coin. At most 3 of the pocketsare to contain the same number of coins, and no two of the remaining pockets are to contain an equal number of coins. What is the least possible number of coins needed for the pocket?A. 7B. 13C. 17D. 2257. Jack is standing 30 yards due north of point P. Sue is standing 72yards due east of point P. What is the shortest distance betweenJack and Sue?A. 60 yardsB. 78 yardsC. 90 yardsD. 100yards58. If n is a prime number greater than 3, what is the remainder when n2 is divided by 12?A. 0B. 1C. 2D. 359. In each production lot for a certain toy, 25 % of the toys are red and 75% of the toys are blue. Half the toysare size A and half are size B. If 10 out of a lot of 100 toys are red and size A, how many of the toys are blue and size B?A. 15B. 25C. 30D. 3560. A certain copy machine produces 13 copies every 10 seconds. If the machine operates withoutinterruption, how many copies would it produce in an hour?A. 4,680B. 4,690C. 4,710D. 4,73261. What is the twenty-first term of the sequence given by x n = 4n - 3 ?A. 6B. 72C. 81D. 8762. In a marketing survey for products A, B, and C, 1000 people were askedwhich of the products, if any, they use. The three circular regions in thediagram represent the numbers of people who use products A, B, and C,according to the survey results. Of the people surveyed, a total of 400 use A, a total of 400 use B, and a total of 450 use C. How many of the people surveyed use exactly one of the products?A. 325B. 250C. 150D. 10075100125125B AC63. The contents of a certain box consists of 14 apples and 23 oranges. How many oranges must be removedfrom the box so that 70 percent of the pieces of fruit in the box will be apples?A. 3B. 6C. 14D. 1764. A rectangle with dimensions 24 inches by 42 inches is to be divided into squares of equal size. Which ofthe following could be a length of the squares?A. 4 inchesB. 6 inchesC. 7 inchesD. 8 inchesaverage of the student’s two lowest scores ?A. 129-2aB. 258-2aC. 129-aD. 258+2a67. The lengths of the three sides of a right triangle are given by three consecutive even integers. Find thelengths of the three sides.A. 4, 6, 8B. 6, 8, 10C. 8, 10, 12D. 10, 12, 1468. A prize of $240 is divided between two persons. If one person receives $180, then what is the differencebetween the amounts received by the persons ?A. $30B. $60C. $120D.$21069. A salesman makes a profit of 25% on all sales. How many fax machines will he sell for $375 each to makea total commission of at least $500 ?A. 4B. 5C. 6D. 1570. Joe works two part-time jobs. One week Joe worked 8 hours at one job, earning $150, and 4.5 hours atother job, earning $90. What were his average hourly earnings for the week ?A. $8.00B. $9.60C. $16.00D. $19.20A. 10B. 13C. 16D. 1873. If a three-digit integer is selected at random from the integers 100 through 199, inclusive, what is theprobability that the first digit and the last digit of the integer are each equal to one more than the middle digit ?A. 2/225B. 1/111C. 1/110D. 1/10074. The width of a rectangle is 6 cm less than the length. If the perimeter of the rectangle is 48 cm, what is thelength of the rectangle in centimeters ?A. 15B. 12C. 9D. 675. In a certain company, the ratio of the number of women employees to the number of men employees is 3to 2. If the total number of employees is 240, then how many of the employees are men ?A. 40B. 48C. 96D. 14476. If x, y, and z are positive integers and 3x=4y=7z, then the least possible value of x+y+z isA. 33B. 40C. 49D. 6177. A number of bricks were purchased to build a fireplace, at a cost of 40 cents each, but only 3/4 of themwere needed. If the unused 190 bricks were returned and their cost refunded, what was the cost of the bricks used to make the fireplace ?A. $228B. $304C. $414D. $57078. If the width of a rectangle is increased by 25% while the length remains constant, the resulting area iswhat percent of the original area ?A. 25%B. 75%C. 125%D. 225%79. A four-character password consists of one letter of the alphabet and three different digits between 0 and 9,inclusive. The letter must appear as the second or third character of the password. How many different passwords are possible ?A. 5,040B. 18,720C. 26, 000D. 37,44080. In the figure, PQRS is a square and each of the four circles has aradius of r. What fractional part of the area of the square isunshaded ?A. (π-4)/2B. (4-π)/4C. π/4D. 4/π数学术语angle 角area 面积average 平均数arithmetic mean 算术平均数base 底面circle 圆形circumference 圆周长cubic foot 立方英尺consecutive even integers连续偶整数consecutive positive integers连续正整数cylinder 圆柱体denominator 分母difference 差digit 数字even integer 偶整数fraction 分数height 高度hexagon 六边形integer 整数length 长度median 中数multiple 倍数numerator 分子odd number 奇数parallelogram 平行四边形perimeter 周长point 点positive integer 正整数prime factor 质因子prime number 质数probability 概率product 乘积quadrilateral 四边形radius 半径reciprocal 倒数rectangle 长方形rectangular solid 长方体remainder 余数right triangle 直角三角形sequence 数列side 边square 正方形square inch 平方英寸standard deviation 标准差sum 和term 项triangle 三角形vertex 顶点volume 体积width 宽度其他生词和词组due north 正北duo 二重唱necktie 领带string 使连成一串tie 平局trio 三重唱。
做一些数学练习题英文
Math Practice ExercisesI. Arithmetic1. Calculate: 345 + 6782. Subtract: 1000 4323. Multiply: 56 × 784. Divide: 144 ÷ 125. Find the square root of 646. Calculate: (8 + 3) × 6 57. Simplify: 18 ÷ (2 + 1)8. Evaluate: 7^39. Find the sum of the first 10 even numbers.10. Calculate the product of the first 5 odd numbers. II. Algebra1. Solve for x: 3x + 7 = 252. Simplify the expression: 4(x 3) + 2x3. Expand and simplify: (2x + 5)(x 3)4. Solve the equation: 5x 3 = 2x + 125. Find the value of x: 2(x 4) + 3x = 216. Solve the system of equations:2x + 3y = 8x y = 37. Factorize: x^2 5x + 68. Solve the inequality: 3x 7 > 2x + 49. Simplify the expression: (3x^2)^2 ÷ (9x)10. Solve for x: |x 4| = 5III. Geometry1. Find the area of a rectangle with length 8 units and width 5 units.2. Calculate the perimeter of a square with side length 7 units.3. Find the volume of a cube with side length 6 units.4. Determine the circumference of a circle with radius 10 units.5. Calculate the surface area of a sphere with radius 5 units.6. Find the slope of the line passing through points (2, 3) and (5, 7).7. Write the equation of a line with slope 2 and yintercept 5.8. Find the midpoint of the line segment with endpoints (4, 3) and (2, 5).9. Determine the angle between the lines y = 2x + 1 and y = 0.5x + 3.10. Calculate the distance between the points (3, 8) and (5, 2).IV. Trigonometry1. Find the sine of 45°.2. Calculate the cosine of 30°.3. Determine the tangent of 60°.4. Solve for x: sin(x) = 0.55. Find the value of x: cos(x) = 0.66. Calculate the value of sin(2θ) if sin(θ) = 0.3.7. Determine the length of the hypotenuse in a right triangle with legs of lengths 6 units and 8 units.8. Find the angle θ in a right triangle if the op posite side is 5 units and the hypotenuse is 10 units.9. Solve for x: 2sin(x) 1 = 010. Calculate the value of cos^2(x) + sin^2(x).V. Probability and Statistics1. A coin is tossed 3 times. What is the probability of getting exactly 2 heads?2. A standard sixsided die is rolled. What is the probability of rolling a 4 or a 6?3. In a class of 30 students, 18 are girls. What is the probability that a randomly selected student is a boy?4. Calculate the mean of the following numbers: 4, 7, 10, 12, 15.5. Find the median of the data set: 3, 8, 5, 12, 7, 9.6. Determine the mode of the following numbers: 2, 5, 7, 5, 8, 9, 5, 11.7. Calculate the variance of the data set: 6, 8, 10, 12, 14.8. Find the standard deviation of the following numbers: 9, 12, 15, 18, 21.9. A survey of 200 people found that 120 preferred coffee to tea. What is the probability that a randomly selected person prefers coffee?10. In a binomial experiment with n = 10 and p = 0.4,find the probability of getting exactly 6 successes.VI. Precalculus1. Evaluate the expression: (3^2)^3 ÷ 3^43. Write the expression as a single logarithm: log(5x) + log(3) log(2x)4. Solve the equation: 4^x = 165. Expand the binomial expression: (a + b)^46. Find the domain of the function f(x) = 1 / (x^2 9)7. Determine the range of the function g(x) = 2x^2 + 4x + 58. Solve the system of inequalities:2x + 3y ≤ 6x y > 29. Graph the function h(x) = |x 2| 310. Find the inverse function of f(x) = (x + 3) / 2.VII. Calculus1. Find the derivative of f(x) = x^3 2x^2 + 4x 72. Calculate the second derivative of g(x) = sin(x)3. Evaluate the definite integral: ∫(from 0 to π) sin(x) dx4. Find the limit as x approaches 0 of (sin(x) / x)5. Determine the critical points of the function h(x) =x^4 8x^3 + 18x^26. Find the area under the curve y = e^x from x = 0 to x= 27. Calculate the volume of the solid obtained rotatingthe region bounded y = x^2, y = 4, and x = 0 about the xaxis.8. Solve the differential equation: dy/dx = 3x^2 2x9. Find the antiderivative of f(x) = 1 / (x^2 + 1)10. Determine the arc length of the curve y = ln(x) fromx = 1 to x = e.VIII. Logic and Reasoning1. If all cats are mammals and some mammals are not cats, which of the following must be true?All mammals are cats.No mammals are cats.Some mammals are cats.None of the above.2. Given the statement "If it rains, then the ground is wet," which of the following is a contrapositive?If the ground is wet, then it rains.If it does not rain, then the ground is not wet.If the ground is not wet, then it does not rain.None of the above.3. Determine whether the following argument is valid:All birds have wings.Penguins are birds.Therefore, penguins have wings.4. If p and q are statements, which of the following represents the negation of "p and q"?p or qnot p and not qnot p or not qp and not q5. Which of the following is an example of a conditional statement?All squares are rectangles.I will go to the store if I need milk.The sun is shining.None of the above.6. If A is a subset of B and B is a subset of C, which of the following must be true?A is a subset of C.C is a subset of A.A is equal to C.None of the above.7. Given the statement "All students who study hard pass their exams," which of the following is a converse?All students who pass their exams study hard.No students who study hard fail their exams.Some students who pass their exams do not study hard.None of the above.8. Which of the following is an example of a tautology?p or not pp and not pp or qp and q9. If the statement "All triangles have three sides" is true, which of the following must also be true?All shapes with three sides are triangles.No shapes with more than three sides are triangles.Some shapes with three sides are not triangles.None of the above.10. Determine whether the following biconditional statement is true or false: "A number is even if and only if it is divisible 2."答案I. Arithmetic1. 10232. 5683. 43684. 125. 86. 397. 68. 3439. 9010. 945II. Algebra1. x = 62. 10x 63. 2x^2 11x 154. x = 35. x = 36. x = 2, y = 17. (x 2)(x 3)8. x > 49. 4x^310. x = 1, x = 6 III. Geometry1. 40 square units2. 28 units3. 216 cubic units4. 62.83 units5. 314.16 square units6. 17. y = 2x + 58. (1, 2)9. 90°10. 8.06 unitsIV. Trigonometry1. 0.7072. 0.63. 1.7324. 30°, 150°5. 53.13°6. 0.67. 10 units8. 30°9. x = 30°, 150°10. 1V. Probability and Statistics1. 3/82. 1/33. 6/10 or 0.64. 95. 7.56. 57. 48. 2.2369. 3/5 or 0.610. 0.2304VI. Precalculus1. 272. 5/63. log(15x)4. x = 25. a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^46. x ≠ 3 and x ≠ 37. All real numbers8. No solution9. arctan(x) + C10. 1VII. Calculus1. f'(x) = 3x^2 4x + 42. g''(x) = sin(x)3. 24. 15. x = 1, x = 3.56. 6.3897. 50.2658. y = x^3 x^2 + C9. arctan(x) + C10. 1.299VIII. Logic and Reasoning1. Some mammals are cats.2. If the ground is not wet, then it does not rain.3. Valid4. Not p or not q5. I will go to the store if I need milk.6. A is a subset of C.7. All students who pass their exams study hard.8. p or not p9. No shapes with more than three sides are triangles.10. True。
代数拓扑学习题(英文)
2. (a) Suppose a CW X is the union of a finite number of subcomplexes Xi and that a subcomplex A of X is the union of subcomplexes Ai ⊂ Xi . Show that if each Xi deformation retracts onto Ai and each intersection of a subcollection of the Xi ’s deformation retracts onto the corresponding intersection of Ai ’s, then X deformation retracts onto A . [By an induction argument the problem reduces to the case of two
plexes f : X→Y is a homotopy equivalence if it restricts to homotopy equivalences → Xi1 ∩ ··· ∩ Xik Yi1 ∩ ··· ∩ Yik for decompositions of X and Y as finite unions of
Section 1.2. 1. Rederive the calculation π1(RP2) ≈ Z2 using the CW structure on RP2 obtained by identifying antipodal vertices, edges, and faces of a cube. 2. Let K be the graph with six vertices and nine edges shown at the right, and let X be obtained from K by attaching a 2 cell along each loop formed by a cycle of four edges in K . Show that π1(X) = 0 .
初二英语代数运算单选题20题
初二英语代数运算单选题20题1. What is the result of 2x + 3x? A. 5x. B. 6x. C. 1x. Answer: A. When we combine like terms, 2x + 3x = 5x.2. If y = 4, what is the value of 2y? A. 6. B. 8. C. 10. Answer: B. When y = 4, 2y = 2×4 = 8.3. What is the sum of 3a + 2a - a? A. 4a. B. 5a. C. 6a. Answer: A. 3a + 2a = 5a, then 5a - a = 4a.4. If x = 3 and y = 2, what is the value of x + y? A.5. B.6. C.7. Answer: A. x = 3 and y = 2, so x + y = 3 + 2 = 5.5. What is the result of 4b - 2b? A. 2b. B. 6b. C. 8b. Answer: A. 4b - 2b = 2b.6. What is the result of 5 + 3? A. 8. B. 2. C. 15. Answer: A. 5 plus 3 is equal to 8.7. If a = 7 and b = 4, what is a - b? A. 3. B. 11. C. 28. Answer: A. 7 minus 4 is 3.8. The sum of two numbers is 12. One number is 5. What is the other number? A. 7. B. 17. C. 3. Answer: A. 12 minus 5 is 7.9. What is 8 + (-3)? A. 5. B. 11. C. -11. Answer: A. 8 plus negative 3 is 5.10. If x = 10 and y = 6, what is x - y + 2? A. 6. B. 8. C. 4. Answer: A.10 minus 6 is 4, then add 2 is 6.11. What is the result of 3 times 4? A. 7 B. 12 C. 10 D. 11. Answer: B.3 times4 is 12.12. If a number is multiplied by 5 and the result is 40. What is the number? A. 6 B. 8 C. 10 D. 12. Answer: B. Let the number be x. Then 5x = 40. So x = 8.13. How many times does 6 go into 36? A. 5 B. 6 C. 7 D. 8. Answer:B. 36 divided by 6 is 6.14. The product of two numbers is 48. If one number is 6, what is the other number? A. 8 B. 10 C. 12 D. 14. Answer: A. Let the other number be x. Then 6x = 48. So x = 8.15. If 12 is divided by a number and the result is 3. What is the number?A. 4B. 5C. 6D. 7. Answer: A. Let the number be x. Then 12 divided by x is 3. So x = 4.16. If x + y = 7 and x - y = 3, what is the value of x?Solution: By adding the two equations, we get 2x = 10, so x = 5.17. If 2x + 3y = 12 and 3x - 2y = 5, what is the value of x + y?Solution: Multiply the first equation by 2 and the second equation by 3, then add them. We get 13x = 46, so x = 46/13. Substitute x into one of the equations to find y. Then calculate x + y.18. If 3(x + y) = 24 and 2(x - y) = 8, what is the value of x * y?Solution: Solve the two equations to find x and y, then multiply them to get x * y.19. If 4x - 3y = 10 and x + 2y = 7, what is the value of 2x + y?Solution: Multiply the second equation by 2 and subtract the first equation from it to find y. Then substitute y into one of the equations to find x. Finally, calculate 2x + y.20. If 2(x + 3y) = 18 and 3(2x - y) = 21, what is the value of x/y?Solution: Solve the two equations to find x and y, then calculate x/y.。
初二英语代数运算单选题20题
初二英语代数运算单选题20题1.There are 20 students in the class. 5 students are absent. How many students are present?A.15B.25C.10D.30答案：A。
20 个学生，5 个缺席，用减法，20 - 5 = 15。
B 选项是20 + 5 的结果；C 选项是20 - 10 的结果；D 选项是20 + 10 的结果。
本题考查整数的减法运算规则。
2.A book costs 15 yuan. Tom has 20 yuan. How much money will he have left after buying the book?A.5 yuanB.35 yuanC.10 yuanD.25 yuan答案：A。
书15 元，有20 元，用减法，20 - 15 = 5。
B 选项是20 + 15 的结果；C 选项是20 - 10 的结果；D 选项是20 + 5 的结果。
本题考查整数的减法运算规则。
3.Lily has 3.5 yuan. She wants to buy a pencil that costs 1.5 yuan. How much money will she have left?A.2 yuanB.5 yuanC.1 yuanD.4 yuan答案：A。
3.5 元减去1.5 元等于2 元。
B 选项是3.5 + 1.5 的结果；C 选项是3.5 - 2.5 的结果；D 选项是3.5 + 0.5 的结果。
本题考查小数的减法运算规则。
4.There are 12 apples. Each apple costs 2.5 yuan. How much do all the apples cost?A.25 yuanB.30 yuanC.14.5 yuanD.10 yuan答案：B。
12 个苹果，每个2.5 元，用乘法，12×2.5 = 30。
初二英语代数运算单选题20题
初二英语代数运算单选题20题1. In the school store, the pen costs 5 yuan each and the notebook costs8 yuan each. If you buy 3 pens and 2 notebooks, the total cost is _____.A. 29 yuanB. 31 yuanC. 34 yuanD. 37 yuan答案：A。
买3 支笔，每支5 元，花费3×5 = 15 元；买2 个笔记本，每个8 元，花费2×8 = 16 元。
总共花费15 + 16 = 29 元。
选项B，3×5 + 2×8 = 15 + 16 = 31 元，计算错误。
选项C，3×5 + 2×9 = 15 + 18 = 33 元，笔记本价格计算错误。
选项D，3×6 + 2×8 = 18 + 16 = 34 元，笔的价格计算错误。
2. The school library has x books. If 20 new books are added and 15 books are borrowed, the number of books left is _____.A. x + 5B. x - 5C. x + 35D. x - 35答案：B。
原本有x 本书，新增20 本后有x + 20 本，借出去15 本后剩下x + 20 - 15 = x + 5 本。
选项A，x + 20 - 15 = x + 5，不是x + 5。
选项C，x + 20 + 15 = x + 35，计算错误。
选项D，x - 20 + 15 = x - 5，不是x - 35。
3. Each student in Class 2 needs to pay 10 yuan for the school trip. If there are y students in the class, the total amount collected is _____.A. 10y yuanB. 10 + y yuanC. 10 - y yuanD. y / 10 yuan答案：A。
03B数学双语单句训练40句
代数Algebra与几何Geometry双语表达40句3×(4+5)+6÷3 Do the operations within the paretheses first.3×9+6÷3 Multify and divide from left to right.11. A coordinate plane坐标平面is formed by two intersecting number lines. Thehorizontal line is called the x-axis. The vertical line is called y-axis. The two-axissystem is called the coordinate system座标系.12. The coordinates of a point are presented by the ordered pair (x, y). This shows thedistance the point is from the origin (0, 0), in the domain (the set of x coordinates) and the range (the set of y coordinates).13. Integers整数are the set of whole numbers and their opposites. Positive integers正整数are greater than zero. Negative integers负整数are less than zero. A negative integer is less than a positve integer. The smaller of two integers is always the zero to the left on th number of the line.14. The opposite of 4 is -4. They are both 4 spaces from 0.15. The sum of two positive integers is positive. (negative?) 4+3=7, -4+(-3)=-716. The Addition and Subtraction Properties of Equality state that when the same numberis added to both sides of an equation, the two sides remain equal: 4+17+5=21+5 17. The Multiplication and Division Properties of Equality state that when each side of theequation is multiplied by the same number, the two sides remain equal. (3+4) ×5=7×5 18. A fraction分数is simplified, or reduced to lowest terms, when its numberator分子anddenominator分母have no common factors公约数other than 1. To simplify a fraction, divide the numberator and denominator by their greatest common factor.19. Use exponents 指数to show how numbers with the same base are multiplied anddivided:20. If a town with population of 3,000 grows by 2% per year, how large will th populationbein 10 years? (Hint: to find the population after 1 year, multiply by 1.02)-- The town’s population will be __________ after 10 years.❖①❖①❖①❖①❖①❖①❖①❖①❖①21. Parallel lines 平行线do not cross or intersect 交叉and are equidistant距离相等的from each other.22. 2:1 - A two to one ratio; two to one;23. The sum of the angles in a rectangle长方形add up to 360 degrees.24. A point, in the mathematical sense, is a location in space that does not have size orshape.25. A line is set of connected points without endpoints终点.26. A line segment局部, in the mathematical sense, is a set of connected points with twoendpoints.27. A ray, in the mathematical sense, is a line that has one endpoint and continuesindefinitely in one direction.28. An angle is a geometric几何学的figure formed by two rays that connect at a singlepoint or vertex角的顶点.29. The vertex is the point of intersection of the lines or rays that form an angle.30. The ruler measured 8 centimeters (cm).31. An acute angle锐角is an angle that is less than 90 degrees.32. A right angle直角is an angle that is exactly 90 degrees.33. An obtuse angle钝角is an angle that is greater than 90 degrees but less than 180degrees.34. The sum of the three angles of a triangle is equal to 180 degrees.35. A quadrilateral四边形is a closed figure with four sides and four vertices.36. A parallelogram平行四边形is a quadrilateral with opposite sides which are paralleland congruent.37. A trapezoid梯形/ 不等边四边形is a quadrilateral with one pair of opposite sides thatare parallel. (An octagon八边形has eight sides.)38. Congruent全等的means exactly the same or equal.39. In geometry, similar can mean that two figures have the same shape but not the samesize.40. The circumference圆周is the measure of the distance around the outside edge of acircle.。
七年级英语代数基础单选题50题
七年级英语代数基础单选题50题1.There are many _____ in our school library.A.bookB.booksC.a bookD.bookes答案：B。
本题考查名词复数形式。
选项 A 是单数形式，题干中说有很多，所以要用复数形式books，选项B 正确。
选项C 有a 表示一本，不符合many 的要求。
选项D 的复数形式错误。
2.The teacher has two _____ on the desk.A.penB.pensC.a penD.penes答案：B。
考查名词复数。
选项 A 是单数，题干中two 表明要用复数形式pens，选项B 正确。
选项C 有a 是单数，不符合要求。
选项D 复数形式错误。
3.We can see some _____ in the classroom.A.studentB.studentsC.a studentD.studentes答案：B。
考查名词复数。
some 后面接可数名词复数或不可数名词，student 是可数名词，要用复数students，选项B 正确。
选项A 是单数，选项C 有a 是单数，选项D 复数形式错误。
4.There are a few _____ in the playground.A.girlB.girlsC.a girlD.girles答案：B。
考查名词复数。
a few 表示一些，后面接可数名词复数，girl 的复数是girls，选项B 正确。
选项A 是单数，选项C 有a 是单数，选项D 复数形式错误。
5.The school has many _____ and teachers.A.studentB.studentsC.a studentD.studentes答案：B。
考查名词复数。
many 后面接可数名词复数，student 的复数是students，选项B 正确。
选项A 是单数，选项C 有a 是单数，选项D 复数形式错误。
高二英语代数基础单选题50题
高二英语代数基础单选题50题1.This is ___ useful book. I read it every day.A.aB.anC.theD./答案：A。
“useful”是以辅音音素开头的单词，所以用“a”。
“a”用于辅音音素开头的单词前，表示泛指；“an”用于元音音素开头的单词前；“the”表示特指；“/”即零冠词，这里不是零冠词的用法。
2.I have ___ apple and ___ orange.A.an,aB.a,anC.the,theD./,/答案：B。
“apple”是以元音音素开头的单词，所以用“an”；“orange”是以元音音素开头的单词，所以用“a”。
3.He is ___ honest boy.A.aB.anC.theD./答案：B。
“honest”是以元音音素开头的单词，所以用“an”。
4.___ sun rises in the east.A.AB.AnC.TheD./答案：C。
“sun”是独一无二的事物，要用“the”来特指。
5.She plays ___ piano very well.A.aB.anC.theD./答案：C。
“piano”是乐器，乐器前面要用“the”。
6.She speaks English very _____.A.goodB.wellC.niceD.beautiful答案：B。
本题考查副词的用法。
“speak”是动词，要用副词“well”来修饰。
“good”“nice”“beautiful”都是形容词，不能修饰动词。
7.The book is _____ interesting.A.realB.reallyC.trueD.truly答案：B。
“interesting”是形容词，要用副词“really”来修饰。
“real”“true”是形容词，“truly”虽然是副词，但通常不修饰“interesting”这个词。
8.He runs _____ fast.A.quiteB.veryC.tooD.much答案：A。
初一英语代数方程求解单选题30题
初一英语代数方程求解单选题30题1.There are 5 pencils in a box. If there are x boxes, the total number of pencils is 20. How many boxes are there? A.3 B.4 C.5 D.6答案：B。
本题考查简单的代数方程求解。
设盒子的数量为x，则5x = 20，解得x = 4。
A 选项3，代入方程5×3 = 15 不等于20；C 选项5，代入方程5×5 = 25 不等于20；D 选项6，代入方程5×6 = 30 不等于20。
2.A book costs $5. If you buy x books and pay $30, how many books can you buy? A.5 B.6 C.7 D.8答案：B。
设可以买的书的数量为x，则5x = 30，解得x = 6。
A 选项5，代入方程5×5 = 25 不等于30；C 选项7，代入方程5×7 = 35 不等于30；D 选项8，代入方程5×8 = 40 不等于30。
3.There are some apples in a basket. If you take out 3 apples and there are 12 left. How many apples were there in the basket at first? A.14 B.15 C.16 D.17答案：B。
设原来篮子里有x 个苹果，则x - 3 = 12，解得x = 15。
A 选项14，14 - 3 = 11 不等于12；C 选项16，16 - 3 = 13 不等于12；D 选项17，17 - 3 = 14 不等于12。
4.You have some stickers. If you give away 4 stickers and you have 10 left. How many stickers did you have at first? A.13 B.14 C.15 D.16答案：B。
初二英语代数运算练习题20题
初二英语代数运算练习题20题1.There are 5 apples on the table. Tom puts 3 more apples on the table. How many apples are there on the table now?A.6B.7C.8D.9答案：C。
一开始有5 个苹果，又放了3 个，5+3=8。
A 选项是计算错误；B 选项也是计算错误；D 选项是把加法算成了乘法。
2.Lily has 10 pencils. She gives 4 pencils to her friend. How many pencils does Lily have left?A.4B.5C.6D.7答案：C。
有10 支铅笔，给出去4 支，10-4=6。
A 选项是只减了一部分；B 选项计算错误；D 选项也是计算错误。
3.There are 8 students in a classroom. Each student has 2 books. How many books are there in total?A.14B.16C.18答案：B。
8 个学生，每人2 本书，8×2=16。
A 选项计算错误；C 选项是把乘法算成了加法；D 选项也是计算错误。
4.Jack has 12 marbles. He divides them equally among 3 friends. How many marbles does each friend get?A.3B.4C.5D.6答案：B。
12 个弹珠平均分给3 个朋友，12÷3=4。
A 选项是计算错误；C 选项是把除法算成了减法；D 选项也是计算错误。
5.Mary has 7 stickers. She buys 5 more stickers. Then she gives 3 stickers to her sister. How many stickers does Mary have now?A.7B.8C.9D.10答案：C。
高等代数与解析几何英文习题
《高等代数与解析几何》英文习题主讲老师：林 磊1. (Feb. 28)Is ⎭⎬⎫⎩⎨⎧⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛0101,0110,1112,1011 a basis for the linear space of all 22⨯matrices?2. (Mar. 1)Let k j i u 32++=. Find vectors v and w that are bothorthogonal to u and to each other.3. (Mar. 4)Let },...,,{21n v v v S = be a basis for a linear space V and let U bea subspace of V . Is it necessarily true that a basis for U is a subset of S ? Why?4. (Mar. 7)In (1)-(2) determine which of the given functions are inner products on 3R where⎪⎪⎪⎭⎫ ⎝⎛=321u u u α and ⎪⎪⎪⎭⎫ ⎝⎛=321v v v β(1) 332211432),(v u v u v u ++=βα;(2) 132231),(v u v u v u ++=βα.5. (Mar. 8)In Exercises (1)-(2) determine whether the given set of vectors is orthogonal, orthonormal, or neither with respect to the Euclidean inner product.(1) {})3,0(),2,1(;(2) {})1,0,1(),0,1,0(),1,0,1(-.6. (Mar. 11)Compute the area of the triangle with vertices )7,2,0(, )3,5,2(-, and )1,1,1(.7. (Mar. 14)Show that ),(4||||22βαβαβα=--+.8. (Mar. 15)In Exercises (1) and (2) find an equation for the plane that passes through the point P and that is parallel to the plane whose general equation is given.(1) )5,3,2(-=P ; 01273=++-z y x .(2) )1,4,6(-=P ; 06352=+++-z y x .9. (Mar. 18)Let T : 32R R → be a linear transformation such that.15211,34011⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡T T (a) Find ⎪⎪⎭⎫⎝⎛⎥⎦⎤⎢⎣⎡67T ;(b) Find ⎪⎪⎭⎫⎝⎛⎥⎦⎤⎢⎣⎡y x T ; (c) Find a matrix A such that⎥⎦⎤⎢⎣⎡=⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡y x A y x T . 10. (Mar. 21)If },,,{21n v v v spans a linear space V , is it possible for },,,{32n v v v to span V ? Explain your answer.11. (Mar. 22)In Exercises (1) and (2) find parametric equations for the line of intersection between the planes whose general equations are given.(1) 04532=+++z y x ; 02652=-+-z y x .(2) 02453=-+-z y x ; 01274=+++z y x .12. (Mar. 25)In Exercises (1)-(2) name the surface determined by the given equation and give its equation in a coordinate system in which the surface is in standard position.(1) 014824222=--++-y x z y x .(2) 048161863222=-----+-z y x z y x13. (Mar. 28)In Exercises (1)-(2) find the matrix representation of the given linear transformation T : 33][x R R →with respect to the ordered bases{})1,0,0(),0,1,0(),0,0,1(=B for 3R and },,1{'2x x B =for 3][x R .(1) c bx ax c b a T ++=2)),,((.(2) c x b a c b a T -+=)()),,((.14. (Mar. 29)In Exercises (1)-(2) find bases for the kernel and range of the given linear transformation T : 33][][x x R R →.(1) b ax c bx ax T +=++2)(2.(2) x c b c bx ax T )()(2+=++.。
线性代数英文试卷（习题）
线性代数英文试卷（习题）第一篇：线性代数英文试卷(习题)ZheJiang University Of Science And Technology Civil Engineering 14 Final TestLinear Algebra Final Test(15.06)Cautions:[1]You are allowed to finish this test within 60 minutes.[2]Fill the answer in the question paper in part.1 1.Filling Blanks(45 points)姓名：x[1]figure out the value of the following determinant A=xyx+yx+yxy=______________x+yy学号：班级：⎛32⎫T ⎪102⎛⎫[2]Here are two matrix,A=01⎪,B=010⎪⎪⎝⎭14⎪⎝⎭T,please find AB=____________,BA=______________.[3]A known matrix B satisfies the following equation,B2-B-2E=0,if B,B+2E are nonsingular,thus(B+2E)-1=___________ [4]Pick up the vectors which are linearly independent__________ ①(-2,1)T,(1,3)T,(2,4)T②2,x2,x,2x+3③x+2,x+1,x2-1④(1,2)T,(-1,1)T⎛12-2⎫⎪[5]A=4t3⎪.if B is a nonzero 3x3 matrix,AB=0,thus t=________ 3-11⎪⎝⎭O-A[6]|A|=|B|=|C|=2,and they’re all 3*3 matrix,find the value of D=2(B)-1C=____________3[7]Judge whether ⎛200⎫⎛253⎫⎪⎪A=052⎪，B=050⎪ have the same eigenvalues______(‘Y’OR’N’),if yes,please find 004⎪004⎪⎝⎭⎝⎭ them=_____________[8]find matrix X,which satisfies the equation⎛1-1⎫⎛21⎫,X=___________X=⎪⎪⎝24⎭⎝12⎭⎛50⎫⎪⎪[9]find the eigenvectors of A=⎝18⎭,______________________2.Solve problems(55 points)[1]A,Bare 3x3 matrix,and they satisfy the equationAB=2A+B，and⎛002⎫⎪B=040⎪，findA-E. 200⎪⎝⎭[2]If ⎛12⎫A=⎪，find a matrix U⎝21⎭，making U-1AU=Λexist.(Tip:Λis a diagonal matrix).ZheJiang University Of Science And Technology Civil Engineering 14 Final Test⎧2x1+x2-x3+x4=1⎡3⎪[3]The equation is⎨3x1-2x2+x3-3x4=4,find its general solution;[4]A=⎢4⎢⎪x+4x-3x+5x=-2⎢0234⎩1⎢⎣04-30000100⎤find0⎥⎥4⎥⎥-1⎦A-1 and A*.⎛1⎫⎛111⎫⎪⎪[5]α=0is one of eigenvectors of matrixA=m-1-1， ⎪⎪1⎪1-1n⎪⎝⎭⎝⎭（1）find the eigenvalue α,then figure out the value of m,n；（2）Judge whether matrix A can be diagonalizable;if yes,please find a matrix diagonal matrix).U,making U-1AU=Λexist.(Tip:Λis a⎛1⎫⎛1⎫⎛1⎫⎛1⎫⎛1⎫⎪⎪⎪⎪⎪[6]The array of vectors β,α are given,β1=0,β2=m,β3=1 ,α1=1,α2=2.They have the same rank of ⎪⎪⎪⎪⎪1⎪3⎪2⎪0⎪n⎪⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭matrix,meanwhile,β3is linearly independent withα1,α2，find the value ofm,n.----------------Draft paper Area-------------ZheJiang University Of Science And Technology Civil Engineering 14 Final TestAnswer of Liner Algebra(2015.06)3.-0.25(A-3E)4.[4]5.t=-3 1.Filling blanks(each question with 5 marks)1.2xy(x+y)⎛301⎫⎪50⎛⎫⎪2.AB=⎪,BA=214⎪101⎝⎭602⎪⎝⎭6.27/87.Y 2,5,48.X=-0.50⎪⎪⎝⎭⎛1.51⎫9.(-3,1)T,(0,1)T 2.Solve questions(each question with 8-10 marks)1.A-E=2(B-2E)-1(FIND THE EXACT ANSWER BY YOURSELF) ⎛1⎫⎛6⎫⎛0⎫0⎛⎫⎪⎪⎪⎪015 ⎪7⎪-⎪⎪513.X=α ⎪+0⎪+7⎪+β ⎪7⎪⎪7⎪0⎪0 4⎪⎪9⎪⎪⎪⎪⎝0⎭⎝0⎭-⎪⎝7⎭⎝7⎭2.U= ⎛-0.50⎫⎪⎪11⎝⎭4.⎛C-1A=O⎝-1O⎫-1⎪A*=|A|xA -1⎪B⎭5.[1]Eigenvalue is 2,m=n=16.M=2,n=1[2]U consists of 3 eigenvectors whose eigenvalue are 1.2.-2 第二篇：线性代数试卷厦门理工学院继续教育学院20 第学期期末试卷线性代数（考试时间：120分钟）专业姓名层次形式成绩一、选择题（每小题4分，共16分）1.A，B为三阶方阵，矩阵X满足AXA-BXB=BXA-AXB+E则().22-1-1-1(A)X=(A-B);(B)X=(A-B)(A+B)(C)X=(A+B)(A-B)(D)以上答案都不对.2.-1-1；A、B、C为n阶方阵，且AB=C,A、B、C的列向量组分别为α1,α2,⋅⋅⋅,αn；β1,β2,⋅⋅⋅,βn(A)；γ1,γ2,⋅⋅⋅,γn.若γ1,γ2,⋅⋅⋅,γn线性相关，则().α1,α2,⋅⋅⋅,αn线性相关;(B)β1,β2,⋅⋅⋅,βn线性相关；(C)（A）与(B)都成立；(D)(A)或(B)成立.3.设A,B为三阶矩阵，且r(A+3A+2E)=3，若r(B)=2则r(AB+B)=().(A)1 ；(B)2；(C)3；(D)无法判断．⎛αA=2γ2 3γ⎝34.设三阶矩阵⎫⎛β⎫⎪⎪B=γ2⎪⎪⎪2γ⎪⎭，⎝3⎭，其中α,β,γ2,γ3均为三维行向量，已知A=18，2B=2，则A-B=().(A)1 ；(B)2；(C)3；(D)4.二、填空题（每小题4分，共16分）⎡En-1⎢0AB=OB为n阶非零矩阵，5.设A、，且A的阶梯形为⎣1D=a1111b1111c1111n0⎤⎥0⎦，则矩阵B的秩=.6.已知，则此行列式的所有代数余子式之和i,j=1∑Aij=.1⎛1A=0Tx=(1,1)⎝7.已知是1⎫⎪a⎪⎭的一个特征向量，则a=.8.为已知A是3阶方阵，α1,α2,α3是三维线性无关的向量.若Aα1=α1+α2，Aα2=α2+α3，Aα3=α1+α3，则A的行列式等于.三、计算下列各题（每小题7分，共28分）01D=1Λ1101Λ11110Λ11ΛΛΛOΛΛ111Λ01111Λ10.9.计算n阶行列式10.若二次型1f(x1,x2,x3)=2x1+8x2+x3+2ax1x2222正定，求a的取值范围.411.已知α=(1,1,1),β=(1,0,1)，且A=αβ.求A.TTT⎛2 A=0 2⎝ 0301⎫⎛1⎪0B=0⎪⎪02⎭⎝0-100⎫⎪0⎪0⎪⎭12.已知矩阵X满足AX+2B=BA+2X，求X．四、解答下列各题（每小题14分，共28分）⎧2x1+3x2+3x3=a⎧x1+x2+x3=1⎨⎨3x+4x2+(a+2)x3=a+1x+2x+ax= 12313.求a使方程组⎩1与⎩1有公共解，并求公共解.14.已知二次型f(x1,x2,x3)=XAX=x1+x3+2ax1x2+2x1x3+2bx2x3T22的秩为2，Tf(x1,x2,x3)α=(1,1,1)是A的特征向量.（1）求a,b的值；（2）求经正交变换所得的标准型，并写出相应的正交矩阵.3五.解答下列各题（每小题4分，共12分）15.设α1,α2,⋅⋅⋅,αt是线性方程组Ax=O的基础解系，向量β满足Aβ=b≠O.证明α1,α2,⋅⋅⋅,αt,β线性无关.16.已知A是n阶方阵且可对角化，问B=A+A+E可否对角化？证明你的结论.2 T17.已知A为n阶矩阵.证明方程组Ax=O与AAx=O的解相同.第三篇：线性代数试卷线性代数试题请考生按规定用笔将所有试题的答案涂、写在答题纸上。
小学数学 第九章 代数表达式与恒等式 Algebraic Expressions 英语双语练习题
Maths Chapter 9 Algebraic Expressions and Identitieswith Answers数学第九章代数表达式与恒等式Algebraic Expressions and Identities Class 8 Questions with Answers代数表达式与恒等式Question 1. The expression x + 3 is in (a) one variable (b) two variables (c) no variable (d) none of these.问题1.表达式x+3有（a）一个变量（b）两个变量（c）没有变量（d）以上都没有。
Answer: (a) one variableHint: The only variable is x.答：（a）一个变量。
提示：唯一的变量是x。
Question 2. The expression 4xy + 7 is in (a) one variable (b) two variables (c) no variable (d) none of these.问题2.表达式4xy+7有（a）一个变量（b）两个变量（c）没有变量（d）以上都没有。
Answer: (b) two variables. Hint: There are two variables x and y.答：（b）两个变量。
提示：有两个变量x和y。
Question 3. The expression x + y + z is in (a) one variable (b) no variable (c) three variables (d) two variables.问题3.表达式x+y+z有（a）一个变量（b）没有变量（c）三个变量（d）两个变量。
Answer: (c) three variables Hint: There are three variables x, y and z.答：（c）三个变量。
国际学校八年级数学测试卷英文版(代数)
国际学校八年级数学测试卷英文版(代数) Chapter Two: XXXName: _________ Score: __________1.Choose XXX describes the graph.A) X < -2B) X ≥ -2C) X ≤ -2D) X。
-22.Solve: 3(2+m) ≥ 15.A) m ≥ 5B) m ≥ 3C) m ≤ 5D) m ≤ 33.Which one is the n of x+1≤2x+3<5?A)B)C)D)4.If a and b are negative numbers。
and a < b。
then:A)B)C)D) ab < 15.In the following n intervals。
which one is greater than or equal to 5?A) (5,+∞)B) (-∞,5)C) [5,+∞)D) (-∞,5]6.How many integers are there in the n of 3x+1.6?A) 1B) 2C) 3D) 47.Identify the type of interval (0,2].A) Left-open。
right-closedB) OpenC) ClosedD) Left-closed。
right-open8.The set given in set builder n as {x:-3<x≤7} can also be expressed as which of the following?A) (-3,7)B) [-3,7]C) (-3,7]D) [-3,7)9.Which of the following represents a closed interval?A) [-5,4]B) (-3,7)C) [7,12)10.If a。
b。
which one of the following is wrong?A) a+m。
小学数学思维拓展 代数表达式简化 Expressions and Equations 英语双语练习题
Worksheet on Simplification of Algebraic Expressions代数表达式简化练习题Problem 1:问题1：After striking a floor, a certain ball rebounds 4/5th of the height from which it has fallen. What is the total distance that it travels before coming to rest if it is gently dropped from a height of 120 meters?一个特定的球在击中地板后，反弹到它所落高度的五分之四。
如果它从120米的高度轻轻落下，在静止前的总距离是多少？Solution: As given in the question,解：如问题所示，The height at which the ball rebounds = 4/5球反弹的高度=4/5The height at which the ball drops = 120 meters球落下的高度=120米让落地总次数为nThe total distance it travels to come to rest position = 120 + 2*[96 * （1-4/5n） / (1-4/5)]=120+2*96 /(1-4/5)到达静止位置的总距离=120+2*96 /(1-4/5)= 120 + 2 *96 *5= 1080 kmsTherefore, the total distance is 1080 kms.因此，总距离为1080公里。
拓展：等比数列求和公式推导：Problem 2:问题2：8 years hence the sum of A’s and B’s age is 70 years. 4 years ago, the ratio between the sum of ages of A and B together and C’s age was 2:1. What is the present age of C?因此，A和B的年龄之和为70岁。
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WorkSHEET 1.2 Financial decisions
Name: ___________________________ 1 Fiona’s fortnightly income is $962.38. What is
her annual salary?
2 Inge earns $80.41 for 2
15 hours work. What is her hourly rate?
3 If the hourly rate for nursing is $12.39, how
much is paid for 3 hours overtime at
time-and-a-half?
4 The standard hourly rate for a 1
5 year old is
$7.56. Casual workers receive a 25% loading.
What is the hourly rate for a 15-year-old casual
worker?
5 Jason earns $11.25 per hour for a 38-hour
working week. Any overtime that was worked
is paid at the rate of time-and-a-half. Calculate
Jason’s wage in a week where he works
45 hours.
6Darian works as a postal worker. Darian’s normal wage is $10.90 per hour. Over the
Christmas period, to deliver all the extra mail,
Darian works Saturday and Sunday. Postal
workers are paid time-and-a-half for working
on Saturday and double-time on Sunday.
Calculate Darian’s wage for the week where
she works 8 hours on Monday to Friday as well
as 6 hours on Saturday and Sunday.
7Sven receives $5.50 for each box of fruit he picks. How many boxes does he need to fill to
earn $87.00?
8Alison assembles wooden toys. She is paid $1.75 for the first 50 toys she assembles, then
$2.25 thereafter. How much would she receive
if she assembled 117 toys?
9Peter works as a real estate agent. He is paid a retainer of $350 per week plus a commission
based on his sales as follows:
4.5% on the first $80 000
2.5% on the remainder.
What would Peter earn in a week when he sold
a property valued at $350 000?
10Clarissa is on a salary of $39 250 per annum.
She pays 6.5% of her salary into a
superannuation fund. She is paid fortnightly
and has $312.80 tax deducted from this pay.
What is Clarissa’s net pay per fortnight after
her superannuation and tax deductions?。