七、剑桥STEP试卷样题

以下为2021年STEP 2真题：

[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, 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 .

八、剑桥STEP考试计分方式是怎样的？

STEP考试计分方式

考生作答6道题，每题均为20分，全卷满分120分。

尽管只需要做6道题，但不限制考生的答题数量。考生答题超过6道时，每道题都会判分，但只取得分最高的6道题计入总分。

2019年改革后STEP不再给出bonus mark，而是严格按照评分标准判分，也即每道题最高得分不超过20分。

STEP考试成绩等级

历年STEP考试成绩等级划分标准

需要注意的是，尽管STEP 2、3的满分、等级都一样，但每种考试每年各个等级对应的分数阈值都不一样。

历年STEP考试成绩等级划分标准及各等级人数占比等数据可扫下方动图二维码，或点击链接查看。

九、剑桥STEP考试范围是什么

为了适应近几年的A Level课程改革，STEP考试在2019年做了重大调整，最主要的变化是对STEP 2、3的考试范围和试卷结构进行了调整，但题型没有变化。主要变化简述如下：

1. 根据英国教育部对A Level数学和进阶数学的改革修订了STEP考试大纲；
2. STEP 2和3试卷的题量由13道减少为12道；
3. 出题风格不变，往年真题可用于备考。

2022年STEP考试沿用2021版大纲，该版大纲与2020版大纲几乎没有区别。