2022年劍橋大學STEP數學考試全麵解讀

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。

劍橋STEP 2022考試關(guan) 鍵日期

2022年3月1日	注冊報名開放
2022年5月5日	注冊報名截止
2022年6月9日	STEP 2考試日
2022年6月20日	STEP 3考試日
2022年8月18日	STEP成績公布
2022年8月25日	STEP成績問詢截止

注：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，請提前做好相關(guan) 考試安排。

劍橋STEP 2022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British Council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。

劍橋STEP 2022考試政策有哪些變化？

2022年劍橋STEP考試延續2021年各項政策，包括：

取消STEP 1、僅保留STEP 2和3考試。
仍采用2021版大綱，延續2019年的重要改革。
原STEP 1要求的知識點仍將在STEP 2和3中出現。
繼續麵向所有考生開放（沒有offer的學生同樣可以參加考試）。

全麵解讀｜劍橋STEP數學考試【2022】

注：劍橋官方保留變更STEP 2022考試政策的權利。

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TMUA。

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 and Statistics	數學與(yu) 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(yu) 經濟(四年)	G0L0
MORSE(Mathematics, Operational Research, Statistics andEconomics)	數學運籌學統計與(yu) 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：

數學與商學 Mathematics and Business Studies (G1NC)
數學與經濟學 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(xi) )	G700
Computing (Visual Computing and Robotics)	計算 (視覺計算和機器人)	GG47
Mathematics and Computer Science	數學與(yu) 計算機科學	GG14
Mathematics and Computer Science	數學與(yu) 計算機科學	GG41

通常帝國理工計算機專(zhuan) 業(ye) 的條件錄取中會(hui) 帶有STEP成績要求。而數學專(zhuan) 業(ye) 則通常要求MAT，如果沒有MAT成績可以用STEP成績替代。

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。

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數學考試（Mathematics Admissions Test）。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加STEP考試並提供成績，以全麵評估考生的學術能力。

劍橋STEP考試形式是怎樣的？

基本信息

答題方式：線下，紙質試卷筆試

考試時長：3小時

公式表：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

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

劍橋STEP試卷樣題

以下為(wei) 2021年STEP 2真題：

[STEP 2, 2021Q1]

Prove, from the identities for , that

Find a similar identity for .
(i) Solve the equation

for .
(ii) Prove that if

then or .
Hence determine the solutions of equation with .

[STEP 2, 2021Q2]

In this question, the numbers , and may be complex.
(i) Let , and be real numbers. Given that there are numbers and such that

show that .
(ii) Conversely, you are given that the real numbers , and satisfy . By considering the equation , show that there exist numbers and such that the three equations hold.
(iii) Let , , and be real numbers. Given that there are distinct numbers , and such that

show, using part (i), that is a root of the equation

and write down the other two roots.
Deduce that .
(iv) Find numbers , and such that

and verify that your solution satisfies the four equations .

[STEP 2, 2021Q3]

In this question, , and are real numbers.
Let denote the largest integer that satisfies and let denote the fractional part of , so that and . For example, if , then and and if , then and .
(i) Solve the simultaneous equations

(ii) Given that , and satisfy the simultaneous equations

show that and solve the equations.
(iii) Solve the simultaneous equations

[STEP 2, 2021Q4]

(i) Sketch the curve , giving the coordinates of any stationary points.
(ii) The function is defined by for , where is the minimum possible value such that has an inverse function. What is the value of ?
Let be the inverse of . Sketch the curve .
(iii) For each of the following equations, find a real root in terms of a value of the function , or demonstrate that the equation has no real root. If the equation has two real roots, determine whether the root you have found is greater than or less than the other root.

(iv) Given that the equation has a unique positive root, find this root in terms of a value of the function .

[STEP 2, 2021Q5]

(i) Use the substitution , where is a function of , to solve the differential equation

where is a constant.
(ii) The curve with equation has the property that, for all values of except , the tangent at the point passes through the point .
(a) Given that , find for .
Sketch for . You should find the co-ordinates of any stationary points and consider the gradient of as . You may assume that as .
(b) Given that , sketch for , giving the co-ordinates of any stationary points.

[STEP 2, 2021Q6]

A plane circular road is bounded by two concentric circles with centres at point . The inner circle has radius and the outer circle has radius . The points and lie on the outer circle, as shown in the diagram, with , and .
(i) Show that I cannot cycle from to in a straight line, while remaining on the road.
(ii) I take a path from to that is an arc of a circle. This circle is tangent to the inner edge of the road, and has radius (where ) and centre .
My path is represented by the dashed arc in the above diagram.
Let .
(a) Use the cosine rule to find in terms of , and .
(b) Find also an expression for in terms of , and .
You are now given that is much less than .
(iii) Show that and are also both much less than .
(iv) My friend cycles from to along the outer edge of the road.
Let my path be shorter than my friend’s path by distance . Show that

Hence show that is approximately a fraction

of the length of my friend’s path.

[STEP 2, 2021Q7]

(i) The matrix represents an anticlockwise rotation through angle () in two dimensions, and the matrix also represents a rotation in two dimensions. Determine the possible values of and deduce that .
(ii) Let be a real matrix with , but .
Show that . Given that

show that .
Hence prove that .
(iii) Let be a real matrix.
Show that if and represents a rotation, then also represents a rotation. What are the possible angles of the rotation represented by ?

[STEP 2, 2021Q8]

(i) Show that, for ,

(ii) The sequence is defined by

Show that, for ,

(iii) Evaluate and and deduce that, for , can be written in the form

where and are integers (which you should not attempt to evaluate).
(iv) Show that for . Given that is non-zero for all , deduce that tends to as tends to infinity.

[STEP 2, 2021Q9]

Two particles, of masses and where , are attached to the ends of a light, inextensible string. A particle of mass is fixed to a point on the string. The string passes over two small, smooth pulleys at and , where is horizontal, so that the particle of mass hangs vertically below and the particle of mass hangs vertically below . The particle of mass hangs between the two pulleys with the section of the string making an acute angle of with the upward vertical and the section of the string making an acute angle of with the upward vertical. is the point on vertically above . The system is in equilibrium.
(i) Using a triangle of forces, or otherwise, show that:
(a)

(b) divides in the ratio , where

(ii) You are now given that .
Show that and determine the ratio of to in terms of the masses only.

[STEP 2, 2021Q10]

A train moves westwards on a straight horizontal track with constant acceleration , where . Axes are chosen as follows: the origin is fixed in the train; the -axis is in the direction of the track with the positive -axis pointing to the East; and the positive -axis points vertically upwards.
A smooth wire is fixed in the train. It lies in the - plane and is bent in the shape given by , where is a positive constant. A small bead is threaded onto the wire. Initially, the bead is held at the origin. It is then released.
(i) Explain why the bead cannot remain stationary relative to the train at the origin.
(ii) Show that, in the subsequent motion, the coordinates of the bead satisfy

and deduce that is constant during the motion.
(iii) Find an expression for the maximum vertical displacement, , of the bead from its initial position in terms of , and .
(iv) Find the value of for which the speed of the bead relative to the train is greatest and give this maximum speed in terms of , and .

[STEP 2, 2021Q11]

A train has seats, where . For a particular journey, all seats have been sold, and each of the passengers has been allocated a seat.
The passengers arrive one at a time and are labelled according to the order in which they arrive: arrives first and arrives last. The seat allocated to is labelled .
Passenger ignores their allocation and decides to choose a seat at random (each of the seats being equally likely). However, for each , passenger sits in if it is available or, if is not available, chooses from the available seats at random.
(i) Let be the probability that, in a train with seats, sits in . Write down the value of and find the value of .
(ii) Explain why, for ,

and deduce that, for ,

(iii) Give the value of in its simplest form and prove your result by induction.
(iv) Let be the probability that, in a train with seats, sits in . Determine for .

[STEP 2, 2021Q12]

(i) A game for two players, and , can be won by player , with probability , won by player , with probability , where , or drawn. A match consists of a series of games and is won by the first player to win a game. Show that the probability that wins the match is

(ii) A second game for two players, and , can be won by player , with probability , or won by player , with probability . A match consists of a series of games and is won by the first player to have won two more games than the other. Show that the match is won after an even number of games, and that the probability that wins the match is

(iii) A third game, for only one player, consists of a series of rounds. The player starts the game with one token, wins the game if they have four tokens at the end of a round and loses the game if they have no tokens at the end of a round. There are two versions of the game. In the cautious version, in each round where the player has any tokens, the player wins one token with probability and loses one token with probability . In the bold version, in each round where the player has any tokens, the player’s tokens are doubled in number with probability and all lost with probability .
In each of the two versions of the game, find the probability that the player wins.
Hence show that the player is more likely to win in the cautious version if and more likely to win in the bold version if .

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。

2019年改革後STEP不再給出bonus mark，而是嚴(yan) 格按照評分標準判分，也即每道題最高得分不超過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)	約前5~15%

曆年STEP考試成績等級劃分標準

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。

曆年STEP考試成績等級劃分標準及各等級人數占比等數據可掃下方動圖二維碼，或點擊《備考指南｜劍橋STEP數學考試》鏈接查看。

全麵解讀｜劍橋STEP數學考試【2022】

劍橋STEP考試範圍是什麽(me)

考試	考試範圍
STEP 1 (已取消，但仍作為(wei) 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(nei) 容
STEP 2 (同樣作為(wei) 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(nei) 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(nei) 容