最新變化
劍橋STEP 2021的官方更新
- 自2021年起取消STEP 1、僅保留STEP 2和STEP 3考試。
- 2021版STEP大綱內容與2020版一致，原STEP 1要求的知識點仍將在STEP 2和3中出現。
- STEP 2021考試相關日期已於3月8日公布。
- STEP 2021麵向所有考生開放，沒有offer的學生同樣可以參加考試。
- 劍橋官方保留變更STEP 2021考試政策的權利。
關(guan) 鍵日期
劍橋STEP 2021考試關(guan) 鍵日期
| 2021年3月15日 | 注冊報名開放 |
| 2021年5月07日 | 注冊報名截止 |
| 2021年6月14日 | STEP 2考試日 |
| 2021年6月17日 | STEP 3考試日 |
| 2021年8月10日 | STEP成績公布 |
注意：劍橋官方保留調整上述日期的權利。
特別提醒
STEP考試時間可能與(yu) A Level相關(guan) 考試時間衝(chong) 突，注意提前做好準備，如有時間衝(chong) 突請務必提前做好相關(guan) 考試科目的調整。報名方式劍橋STEP 2021考試如何報名？
考生需通過授權考試中心報名參加STEP考試：
- 考生所在學校是授權考試中心，則可通過學校報名和參加STEP考試。
- 學校不是授權考試中心，可以社會考生的身份登錄BC官網報名。
- 還有部分城市有校外機構，可以代社會考生報名並組織STEP考試。
報名時需提交給考試中心以下信息：
- 姓名、性別、出生日期及UCAS編號。
- 所申請大學的名稱、專業及專業代碼。
- 如因身體原因需要特殊照顧，需一並提交相關證明材料。
報名資格
哪些學生可以報考STEP 2021？
所有人均可報名參加STEP 2021考試。不再像2020年那樣要求必須是offer holder才能參加。
考試簡介
什麽(me) 是劍橋STEP數學考試？由劍橋大學招生考試委員會(hui) 組織的STEP考試是為(wei) 測試申請者數學能力而舉(ju) 行的筆試，其全稱為(wei) Sixth Term Examination Paper，直譯過來就是“第六學期考試”。STEP成績通常作為(wei) 英國幾所頂尖高等院校，包括劍橋大學、帝國理工學院、倫(lun) 敦大學學院、華威大學等院校的數學、計算機等相關(guan) 專(zhuan) 業(ye) 錄取條件之一。盡管牛津大學數學、計算機相關(guan) 專(zhuan) 業(ye) 不要求提供STEP成績，但牛津官網上也明確建議申請者參加STEP考試。
需考專(zhuan) 業(ye) 哪些大學和專(zhuan) 業(ye) 需要考劍橋STEP？
劍橋大學
| Courses | 專業名稱 | UCAS代碼 |
| Mathematics | 數學 | G100 |
| Mathematics with Physics | 數學物理 | G100 |
| Economics | 經濟 | L100 |
| Engineering | 工程 | H100 |
通常劍橋大學在條件錄取中要求考生的STEP成績達到等級1及以上。其中，數學專(zhuan) 業(ye) 通常要求STEP 2、3等級1、1甚至等級1、S的成績。自2018年起劍橋大學以下專(zhuan) 業(ye) 不再要求考STEP：
- 化學工程 Chemical Engineering via Engineering (H810)，但要求考ENGAA。
- 自然科學 Natural Sciences (BCF0)，但要求考NSAA。
自2019年起劍橋大學以下專(zhuan) 業(ye) 不再要求考STEP：
- 計算機 Computer Science with Mathematics (G400 BA/CS)，但要求考CTMUA。
華威大學
| Courses | 專業名稱 | UCAS代碼 |
| Mathematics | 數學 | G100 |
| Mathematics (Master of MATH) | 數學(四年) | G103 |
| Mathematics and Philosophy | 數學和哲學 | GV15 |
| Mathematics andStatistics | 數學與統計 | GG13 |
| Mathematics and Statistics (MMathStat) | 數學和統計(四年) | GGC3 |
| MORSE (Mathematics, Operational Research, Statistics and Economics) | 數學運籌學統計與經濟(四年) | G0L0 |
| MORSE(Mathematics, Operational Research, Statistics andEconomics) | 數學運籌學統計與經濟 | GLN0 |
| Data Science (Mathematics, Statistics and Computer Science) | 數據科學 | G103 |
一般華威大學要求STEP成績達到等級2及以上。盡管華威大學接受考生用MAT或TMUA代替STEP成績，但很多考生因為(wei) 各種原因錯過每年10月底或11月初的MAT和TMUA考試，不得不選擇參加次年6月的STEP考試。華威大學官方給出的TMUA最低要求為(wei) 6.5(滿分9.0分)，而MAT因為(wei) 每年成績會(hui) 有所變化，無法在考試成績統計結果出來以前給出MAT對應的最低分數。自2018年起華威大學以下專(zhuan) 業(ye) 不再要求考STEP：數學與(yu) 商學 Mathematics and Business Studies (G1NC)數學與(yu) 經濟學 Mathematics and Economics (GL11)
帝國理工
| Courses | 專業名稱 | UCAS代碼 |
| Computing | 計算 | G400 |
| Computing | 計算 | G401 |
| Computing(International Programme of Study） | 計算(國際項目) | G402 |
| Computing (Management and Finance） | 計算(管理和金融) | G501 |
| Computing(Software Engineering） | 計算(軟件工程) | G600 |
| Computing(Security and Reliability) | 計算(安全和可靠性) | G610 |
| Computing(Artificial Intelligence and Machine Learning） | 計算(人工智能和機器學習) | G700 |
| Computing(Visual Computing and Robotics) | 計算(視覺計算和機器人) | GG47 |
| Mathematics and Computer Science | 數學與計算機科學 | GG14 |
| Mathematics and Computer Science | 數學與計算機科學 | GG41 |
一般帝國理工在條件錄取中要求STEP 2或3達到等級2或1以上，或者STEP 2和3同時達到等級2甚至等級1以上。
其他大學其他要求STEP(或MAT、TMUA)的大學包括：
- 倫敦大學學院(UCL)
- 布裏斯托大學
- 巴斯大學
- 倫敦國王學院
上述大學的相關(guan) 專(zhuan) 業(ye) 會(hui) 在官網或錄取條件中明確提出具體(ti) STEP考試和成績等級要求。帝國理工的計算機專(zhuan) 業(ye) 必須要STEP，而數學專(zhuan) 業(ye) 通常要求MAT，如果沒有MAT成績則可以用STEP替代。通常帝國理工要求STEP 2或3等級2以上的成績。牛津大學的數學、計算機等相關(guan) 專(zhuan) 業(ye) 則要求考生必須參加自家組織的MAT(Mathematics Admissions Test數學入學考試）。盡管STEP成績不作為(wei) 牛津大學錄取的必要條件之一，但牛津也鼓勵考生參STEP考試並提供成績，以全麵評估考生的學術能力。
考試形式
劍橋STEP考試形式是怎樣的？
考試題型
自2021年起取消STEP 1考試後，STEP僅(jin) 提供STEP 2和STEP 3兩(liang) 種考試。
題型均為(wei) 計算題，不必做答所有題目，考生隻需從(cong) 試卷中選擇6道題作答。
自2019年改革以後，STEP 2和3試卷題量由13道減少為(wei) 12道，見下表。
| 考試 | Section A | Section B | Section C | 合計 |
| STEP 2 | 純數8道 | 力學2道 | 統計2道 | 12道 |
| STEP 3 | 純數8道 | 力學2道 | 統計2道 | 12道 |
答題方式筆試
考試時長3小時
公式表考試不提供公式表，大綱中涉及的公式要求學生全部掌握，如果有超過大綱給出的公式，試題中會(hui) 給出。
計算器不允許使用計算器詞典
允許使用紙質雙語詞典
試卷樣題
劍橋STEP試卷樣以下為(wei) 2020年STEP 2真題：Section A: Pure Mathematics
[STEP 2, 2020Q1]
(i) Use the substitution ，where , to find in terms of the integral
(where ).
(ii) Find in terms of the integral (where ).
(iii) Show that
[STEP 2, 2020Q2]
The curves and both satisfy the differential equation,
where .
All points on have positive and co-ordinates and passes through (1, 1). All points on have negative and co-ordinates and passes through (−1, −1).
(i) Show that the equation of can be written as .
Determine a similar result for curve .
Hence show that is a line of symmetry of each curve.
(ii) Sketch on the same axes the curves and , for . Hence show that lies between the lines and .
Sketch curve .
(iii) Sketch curve .
[STEP 2, 2020Q3]
A sequence of positive real numbers is said to be unimodal if there is a value such that
and
So the sequences ; ; and are all unimodal, but is not.
A sequence of positive real numbers is said to have property if for all with .
(i) Show that, in any sequence of positive real numbers with property L,
Prove that any sequence of positive real numbers with property is unimodal.
(ii) A sequence of real numbers satisfies for , where is a positive real constant. Prove that, for ,
and, for ,
Hence show that the sequence consists of positive terms and is unimodal, provided .
In the case and , prove by induction that . Let , where is an integer with .
In the case and , prove that ur is largest when .
[STEP 2, 2020Q4]
(i) Given that , and are the lengths of the sides of a triangle, explain why , and .
(ii) Use a diagram to show that the converse of the result in part (i) also holds: if , and are positive numbers such that , and then it is possible to construct a triangle with sides of length , and .
(iii) When , and are the lengths of the sides of a triangle, determine in each case whether the following sets of three lengths can
- always
- sometimes but not always
- never
form the sides of a triangle. Prove your claims.
(A) , , .
(B) , , .
(C) , , .
(D) , , .
(iv) Let f be a function defined on the positive real numbers and such that, whenever ,
but .
Show that, whenever , and are the lengths of the sides of a triangle, then , and can also be the lengths of the sides of a triangle.
[STEP 2, 2020Q5]
If is a positive integer, the value of the function is the sum of the digits of in base 10. For example, d(249) = 2 + 4 + 9 = 15.
An -digit positive integer is written in the form , where for all and .
(i) Prove that is non-negative and divisible by 9.
(ii) Prove that is a multiple of 9 if and only if is a multiple of 9.
Suppose that . Show that if has n digits, then and , and hence that .
Find a value of for which . Show that there are no further values of satisfying this equation.
(iii) Find a value of for which . Show that there are no further values of satisfying this equation.
[STEP 2, 2020Q6]
A matrix is real if it can be written as , where , , and are real.
In this case, the trace of matrix is defined to be tr and det() is the determinant of matrix . In this question, is a real 2 × 2 matrix.
(i) Prove that
tr() = tr − 2det().
(ii) Prove that
but and
and that
and
(iii) Use part (ii) to prove that
Find a necessary and sufficient condition on and so that .
(iv) Give an example of a matrix for which , but which does not represent a rotation or reflection. [Note that the matrices are both rotations.]
[STEP 2, 2020Q7]
In this question, .
(i) Let be the complex number , where . Show that is independent of . Hence show that, if is a complex number on the line in the Argand diagram, then lies on a circle in the Argand diagram with centre 1.
Let be the line , where is a real constant not equal to 2. Show that, if lies on , then lies on a circle whose centre and radius you should give in terms of . For which on is ?
(ii) Let be the line , where is a non-zero real constant. Show that, if lies on H, then lies on a circle whose centre and radius you should give in terms of . For which on is ?
[STEP 2, 2020Q8]
In this question, is a quartic polynomial where the coefficient of is equal to 1, and which has four real roots, 0, , and , where .
is defined by .
The area enclosed by the curve and the -axis between 0 and is equal to that between and , and half that between and .
(i) Sketch the curve , showing the co-ordinates of its turning points. Explain why must have the form , where . Find, in factorised form, an expression for in terms of , and .
(ii) If , explain why and why if . Hence show that or .
By considering also , show that and that .
(iii) Find an expression for in terms of and only. Show that the points of inflection on lie on the -axis.
Section B: Mechanics
[STEP 2, 2020Q9]
Point is a distance above ground level and point is directly below at ground level. Point is also at ground level, a distance horizontally from . The angle of elevation of from is . A particle is projected horizontally from , with initial speed . A second particle is projected from B with speed at an acute angle above the horizontal. The horizontal components of the velocities of the two particles are in opposite directions. The two particles are projected simultaneously, in the vertical plane through , and .
Given that the two particles collide, show that
and also that
(i) ;
(ii) ;
(iii) .
Show that the particles collide at a height greater than if and only if the particle projected from is moving upwards at the time of collision.
[STEP 2, 2020Q10]
A particle of mass m moves freely and without friction on a wire circle of radius , whose axis is horizontal. The highest point of the circle is , the lowest point of the circle is and angle . A light spring of modulus of elasticity λ is attached to and to . The natural length of the spring is , which is less than the diameter of the circle.
(i) Show that, if there is an equilibrium position of the particle at , where , then
Show also that there will only be such an equilibrium position if .
When the particle is at the lowest point of the circular wire, it has speed .
(ii) Show that, if the particle comes to rest before reaching , it does so when , where satisfies
where
Show also that this will only occur if .
Section C: Probability and Statistics
[STEP 2, 2020Q11]
A coin is tossed repeatedly. The probability that a head appears is and the probability that a tail appears is .
(i) A and B play a game. The game ends if two successive heads appear, in which case A wins, or if two successive tails appear, in which case B wins.
Show that the probability that the game never ends is 0.
Given that the first toss is a head, show that the probability that A wins is .
Find and simplify an expression for the probability that A wins.
(ii) A and B play another game. The game ends if three successive heads appear, in which case A wins, or if three successive tails appear, in which case B wins.
Show that
P(A wins | the first toss is a head) = P(A wins | the first toss is a tail) and give a similar result for P(A wins | the first toss is a tail).
Show that
P(A wins) =
(ii) A and B play a third game. The game ends if a successive heads appear, in which case A wins, or if successive tails appear, in which case B wins, where and are integers greater than 1.
Find the probability that A wins this game.
Verify that your result agrees with part (i) when .
[STEP 2, 2020Q12]
The score shown on a biased -sided die is represented by the random variable which has distribution for , where not all the are equal to 0.
(i) Find the probability that, when the die is rolled twice, the same score is shown on both rolls. Hence determine whether it is more likely for a fair die or a biased die to show the same score on two successive rolls.
(ii) Use part (i) to prove that, for any set of positive numbers ,
(ii) Determine, with justification, whether it is more likely for a fair die or a biased die to show the same score on three successive rolls.
計分方式
劍橋STEP考試計分方式是怎樣的？
STEP考試計分方式
考生作答6道題，每題均為(wei) 20分，全卷滿分120分。盡管隻需要做6道題，但不限製考生的答題數量。考生答題超過6道時，每道題都會(hui) 判分，但隻取得分最高的6道題計入總分。需要注意的是STEP計分采用鼓勵性原則：STEP強調考生能在解題過程中完整做答，如果使用的解題方法非常巧妙、且答題過程完整，考官會(hui) 酌情給出bonus mark。也即有考生一道題能得分超過20分！
STEP考試成績等級
| 等級 | 含義 | 占比 |
| S | Outstanding (優秀) | 約前5~15% |
| 1 | Very Good (非常好) | 約前15~30% |
| 2 | Good (好) | 約前30~50% |
| 3 | Satisfactory (合格) | 約前50~80% |
| U | Unclassified (不合格) | _ |
需要注意的是，盡管STEP 2、3的滿分、等級都一樣，但每種考試每年各個(ge) 等級對應的分數閾值都不一樣。
STEP考試成績等級劃分標準
以下為(wei) 2020年官方給出的STEP 2和3的等級劃分標準和統計圖。
STEP 2等級劃分標準和統計圖Grade boudaries (STEP 2, 2020)等級劃分標準
| Maximum Mark | S | 1 | 2 | 3 | U |
| 120 | 77 | 55 | 42 | 25 | 0 |
Cumulative percentage achieving each grade (STEP 2, 2020)達到各等級的累積百分比
| Maximum Mark | S | 1 | 2 | 3 | U |
| 120 | 9.3 | 30.8 | 48.4 | 81.3 | 100 |
Distribution of scores (STEP 2, 2020)分數分布
STEP 3等級劃分標準和統計圖Grade boudaries (STEP 3, 2020)等級劃分標準
| Maximum Mark | S | 1 | 2 | 3 | U |
| 120 | 80 | 67 | 53 | 30 | 0 |
Cumulative percentage achieving each grade (STEP 3, 2020)達到各等級的累積百分比
| Maximum Mark | S | 1 | 2 | 3 | U |
| 120 | 11.6 | 38.2 | 58.3 | 83.5 | 100 |
Distribution of scores (STEP 3, 2020)分數分布
考試範圍
劍橋STEP考試範圍有哪些變化？STEP考試的考查範圍見下表。
| 考試 | 考察範圍 |
| STEP 1(已取消，但仍作為STEP 2和3的知識點) | A Level數學的純數、力學、概率統計部分，附加2021大綱要求的內容 |
| STEP 2(同樣作為STEP 3的知識點) | AS進階數學 (高數) 的純數、力學、概率統計部分，附加2021大綱要求的內容 |
| STEP 3 | A level進階數學(高數)的純數、力學、概率統計部分，附加2021大綱要求的內容 |
為(wei) 了適應近幾年的A Level課程改革，STEP考試在2019年做了重大調整，最主要的變化是對STEP 2、3的考試範圍和試卷結構進行了調整，但題型沒有變化。主要變化簡述如下：
- 按照A Level數學和進階數學(高數)改革對應的修訂考試大綱；
- STEP 2和3試卷的題量由13道減少為12道；
- 出題風格不變，意味著往年的STEP真題可以用於備考；
- 考試的鼓勵性計分原則不變。
最新的2021年大綱已經出爐，與(yu) 2020年的大綱相比幾乎沒有變化。
考試難度
劍橋STEP考試有多難？一句話概括難度STEP有少量題比較簡單，但大多數題都是高考壓軸題難度，尤其是每道題的最後一問非常有挑戰性。
近幾年STEP考試越來越難嗎？
是的，經過牛劍課程教學和研發團隊的對比分析，發現最近幾年STEP試題難度有明顯提升。一個(ge) 明顯的變化是劃分等級S和等級1的分數線上有大幅下降的趨勢，另一方麵考慮到越來越多數學成績優(you) 秀的中國學生參加STEP考試，無疑會(hui) 在一定程度上提高各個(ge) 等級的分數線。
STEP考試難在哪兒(er) ？
- 強調邏輯推理的完備性；
- 計算量大且不能用計算器；
- 要求基本數學運算相當熟練；
- 注重數學基本知識和基本定理、公式的推導方法所蘊含的基本數學思想；
- 通過已知與未知知識的並行使用考查學生的領悟能力和知識遷移能力。
STEP和數學競賽相比哪個(ge) 難？
- 試題難度：總體來說STEP的難度沒有大多數數學競賽最後幾道題難；
- 側重考查點：STEP更強調對基本數學知識、思想方法的運用，而數學競賽強調巧思妙解、是對智商和數學技巧的雙重考驗；
- 答題策略：取決於題型，多數數學競賽的初賽以選擇題為主，相對於STEP所有的題都是解答題而言會更容易得分；
- 備考方式：STEP有考綱和明確的考試範圍，題型和解答套路相對比較固定，所以更容易備考，而競賽沒有考綱、也沒有明確的考試範圍，有些難題不能按常規套路求解、不容易準備。
參加數學競賽對考STEP有幫助嗎？
雖然STEP和數學競賽試題存在諸多差異，但在備考數學競賽過程中學到的數學知識、方法和思想對於(yu) 備考STEP考試也是非常有幫助的，建議學有餘(yu) 力的學生在備考STEP的同時參加數學競賽，比如美國數學競賽AMC 10/12、英國數學競賽BMO係列賽事等。
MAT跟STEP相比難度如何？
MAT比STEP簡單。MAT僅(jin) 考查純數知識，考查範圍比原STEP 1考試的純數部分考查的知識還要少。MAT有選擇題，且選擇題大部分都比較容易，通常隻有2-3道選擇題比較難或容易丟(diu) 分。MAT考試這樣設置的原因一方麵是要兼顧申請帝國理工和華威大學的考生數學水平，另一方麵也是為(wei) 了不讓考生的MAT成績太難看，鼓勵更多的學生參加MAT考試，並通過MAT考試測評其數學實力。
而STEP考試全是大題，且大題之間的難度差異比較大。所以STEP考試不全是考查數學水平，也不像考MAT的選擇題會(hui) 有一定的運氣成分，STEP考試中會(hui) 挑題比會(hui) 解題更重要！題目挑得好就會(hui) 更容易得高分。
STEP準備多久才能考到等級1以上？
不同學生的數學基礎差別較大，劍橋官方建議備考STEP時間不少於(yu) 6個(ge) 月。沒有學高數和數學競賽基礎的學生，建議備考時間9個(ge) 月以上。
學過高數且有數學競賽基礎的學生如果準備STEP 2和3，建議不少於(yu) 6個(ge) 月。
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