AST (Aptitude Scholastic Test)
Examination Programme
Mathematics
I Background
The AST (Aptitude Scholastic Test) is a screening test designed and administered by the
Ambright Education Group and taken by high-performing senior middle school graduates and
other candidates of an equivalent educational level. It is a combination of an Aptitude Test and a
Proficiency Test and was developed originally to pre-select candidates to be interviewed for
admission to undergraduate courses at the University of Cambridge. Although it continues to be
used for this purpose, it is increasingly being adopted by other world-leading universities to
admit students directly to undergraduate courses. The unique feature of the format and content
of the AST is that it combines the nature and characteristics of University matriculation
examinations in China (i.e. the Gaokao) with those of leading international examinations.
Participating universities continue to assess applicants holistically based on their standard
admissions criteria for international students but use the AST to inform and aid significantly
their decision-making admissions processes for international students.
The AST tests encompass the nature and characteristics of both Chinese college and
international entrance examinations and are set at a considered and appropriate level of
difficulty to enable outstanding students to demonstrate their problem-solving abilities and
academic literacy, primarily in the STEM subjects. Overall, the AST test papers are established
as a reliable, discriminatory and a valuable tool in the selection of talented students for entry to
world-leading universities.
II Content and Requirements
The content and requirements of the AST Mathematics examination are determined
primarily on the academic quality requirements for admission to domestic Chinese and
international
institutions
of
higher
learning,
informed
by
The
National Mathematics Curriculum for Senior High School (2017) 1 in China by and leading
international curricula, such as A-level, Advanced Placements (AP), the International
Baccalaureate (IB), Standard Assessment Tests (SAT) and the American College Tests (ACT).
The content of the Mathematics AST includes sets, common logic language, equality
1 The National Mathematics Curriculum for Senior High School (2017), Beijing: People’s Education Press, 2018
1
relation and inequality relation, function, trigonometric function, preliminary solid geometry,
plane analytic geometry, plane vector, space vector and solid geometry, sequence of number,
complex
number,
counting
principle,
probability,
statistics,
derivatives
of single
variable functions and its applications.
The detailed content is as follows:
1. Sets
(1) concept and presentation of a set
(2) basic relationships between sets
(3) basic operation of sets
2. Common logic language
(1) necessary condition, sufficient condition and necessary and sufficient condition
(2) universal quantifier and existential quantifier
(3) negation of universal quantifier proposition and existential quantifier proposition
3. Equality relation and inequality relation
(1) basic properties of inequality
(2) quadratic inequality with one unknown
(3) basic inequality and its simple application
4. Function
(1) concept of a function
(2) monotonicity, odevity, periodicity, maximum value, minimum value and
geometrical significance of a function
(3) simple piecewise function and its applications
(4) images and properties of exponential function
(5) images and properties of logarithmic function
(6) images and properties of power function
(7) function and equation
(8) function model and its applications
5. Trigonometric function
(1) concept of trigonometric functions and their properties
(2) induced formulae for trigonometric function
(3) basic relationships among trigonometric functions
(4) trigonometric formulae for the sum (difference) of two angles
(5) sine theorem, cosine theorem and their applications
6. Preliminary solid geometry
(1) basic solid figures
2
(2) location relation between basic figures
(3) location relation between two lines, a line and a plane, two planes
(4) calculation of angle and distance
7. Plane analytic geometry
(1) equation of a straight line
(2) location relation between two straight lines
(3) distance between two points, a point and a straight line
(4) standard equation and general equation of a circle
(5) location relation between two circles, a straight line and a circle
(6) standard equation of ellipse with its center on the origin of the coordinate system and
its geometric properties
(7) standard equation of hyperbola with its center on the origin of the coordinate system
and its geo metric properties
(8) standard equation of parabola with its vertex on the origin of the coordinate system
and its geometric properties
8. Plane vectors
(1) concept of a plane vector
(2) addition, subtraction of plane vectors and scalar multiplication of vectors
(3) plane vector coordinates representation
(4) scalar product of vectors
(5) parallelism and perpendicularity between two plane vectors
(6) application of plane vectors
9. Space vectors and solid geometry
(1) space rectangular coordinates system
(2) space vector and its operation
(3) fundamental theorem and coordinates representation of space vector
(4) application of space vectors
10. Sequence of number
(1) concept of a sequence
(2) arithmetic progression
(3) geometric progression
(4) comprehensive application of sequences
11. Complex numbers
(1) concept of a complex number
(2) operations of complex numbers
3
(3) trigonometric representation of a complex number
12. Counting principle
(1) addition and multiplication theorems
(2) arrangement and combination
(3) binomial theorem
13. Probability
(1) random events and probability
(2) independence of random events
(3) conditional probability of random events
(4) discrete random variables and their probability distributions
(5) normal distribution
14. Statistics
(1) basic access to data and related concepts
(2) sampling
(3) statistical graph
(4) correlation of paired data
(5) simple linear regression model
(6) contingency table of 2×2
15. Derivative and differential of single variable function
(1) concept of derivative
(2) calculation of derivative
(3) applications of derivative in the study of function
(4) concept and calculation of differential
16. Definite integral
(1) concept of definite integral
(2) fundamental theorem of calculus and its applications
(3) using definite integral to calculate the area of some closed figures and the volume of
the sphere, circular cone, circular truncated cone, triangular pyramid and the frustum of
a triangular pyramid
4
III Test Format and Structure
1. Test Format
Closed-book written examination
Time duration: 180 minutes
Total score: 300 marks
2. Ratio between levels of difficulty of questions
The test paper consists of questions of three different levels of difficulty: straight
forward questions (A), simple comprehensive essay questions (B)，comprehensive essay
questions (C), in a ratio of approximately 4:3:1.
3. Test Paper Structure
Question Type
Number
Score
Basic essay questions
Choose 8 from 10
120 marks
Choose 6 from 8
120 marks
Choose 2 from 4
60 marks
16
300 marks
Simple comprehensive
essay questions
Comprehensive essay
questions
Total
5
Ⅳ Sample Test
Aptitude Scholastic Test
Mathematics
TOTAL TIME ALLOWED：3 HOURS
TOTAL SCORE: 300
Section A Basic questions
Choose and complete 8 from the following 10 questions. Please answer in the designated
areas on the question paper and include explanations, proofs and all relevant steps to support
your mathematical calculations. (Full marks: 120; 15 marks for each question completed.)
1. Given two sets P＝{x : x2－16＜0} and Q＝{x : x＝2n, n ∈Z}, find P ∩ Q.
2. Consider the complex number m=(1＋i)(1－ni) (where m, n∈R , and i is the purely imaginary number of
modulus 1). Find the value of m－n.
3. A class of sixty students is numbered from 1 to 60 at random. If five students are selected by systematic
sampling, and their numbers are 4, m, 28, n and 52 respectively, find the value of m＋n.
6
4. The planar vectors a and b are such that a is perpendicular to b, a＝ (1,－1) and |b|＝2 2. Find b.
5. Find the domain of f(x) =
2－x
.
ln(x－1)
6. The equations of the lines l1 and l2 are x－2y－1=0 and ax－by＋1=0, respectively, where a, b∈{1,2,3,4}.
Find the probability that l1 is parallel to l2.
7. It is given that the line x＝
π
π
is an axis of symmetry of the function f(x)＝3sin(2x＋φ) (0＜φ＜ ). Find
6
2
the value of f(/2).
7
8. Shown in the figure is a cuboid ABCD -A1B1C1D1。Given that AB＝1, AA1＝2 and A1C＝3, find the volume
of the cuboid.
D1
C1
A1
B1
D
A
C
B
Figure 8
9. Suppose that
an  is an arithmetic sequence with common difference d ≠ 0. Suppose also that a1，a3，a7
form a geometric sequence. Find the value of
a1
.
d
8
10. It is given that tan(α+β)=
2
π 1
π
and tan(β－ )= . Find the value of tan(α＋ ).
5
4 4
4
9
Section B Intermediate questions
Choose and complete 6 of the following 8 questions. Please answer in the designated areas
on the question paper and include any explanatory notes, proofs and intermediate
mathematical steps. (Full marks: 120; 20 marks for each question completed.)
11.In the triangle ABC, the sides a, b and c of △ABC are opposite to the interior angles A, B and C,
respectively. It is given that a(sinA－sinB)＝(c－b)(sinB＋sinC).
(1) Find the angle at C.
(2) If 2a＝b＋ 2c,
find the value of sinA.
12. As shown in the figure, AB＝2 is the diameter of the semicircle and the point O is the centre. The point C
lies on the semicircle which is distinct from points A and B. Suppose P is a varying point on the line segment
→ → →
OC. Find the minimal value of ( PA ＋ PB )· PC .
C
P
A
O
Figure 12
10
B
13. Shown in the figure is a cube ABCD -A1B1C1D1. Points M, N, G are the midpoints
of AA1, D1C and AD, respectively.
(1) Prove that MN is parallel to the plane ABCD.
(2) Suppose α is a plane passing through MN. Prove that α is perpendicular to the
plane B1BG.
D1
A1
C1
B1
M
N
G
A
D
B
C
Figure 13
14. In the Yang Hui triangle, shown in the table, the jth number in row i is denoted by f(i，j) (i，j∈N＋，1≤j≤i).
For example, f(3, 2)＝2 and f(4, 1)＝1.
(1) Compute the four stated values:
f(3，3)，f(4，3)，f(5，4) and f(n，2).
1
1
(2) Find the value of f(2，1)+f(3，2)+f(4，3)+…+f(n，n-1).
1
1
1
.
11
2
3
4
1
1
3
6
4
……
Figure 14
1
1
15. Consider the function f (x)＝x2＋mx＋n, where m, n ∈R. If there exists a non-zero constant t such that
1
f (t)＋ f ( )＝- 2, find the minimum value of m2＋4n2.
t
16. Consider the circle ⊙C:(x＋2)2＋y2＝4. Two lines l1 and l2 both pass through the point A(m, 0) and are
perpendicular to each other.
(1) If both lines l1 and l2 are tangent to ⊙C, find the respective equations of the lines l1 and l2。
(2) Let m＝2 and consider a point M (1, a). If ⊙M (a circle centered at the point M) and ⊙C are
circumscribed and both lines l1 and l2 are tangent to ⊙M , find the equation of ⊙M.
17. Given the set A = {1, 2, 3, 4, 5} .
(1) Find the number of subsets of A. [alternative: Find the order of the power set of A.]
(2) Find the number of non-empty subsets of A which don’t contain the element 3.
(3) Assume three different numbers are selected at random from A. Find the probability that their sum is
even.
12
18. Consider the inequality mx2－2x－m＋1< 0 (m ∈R, x ∈R).
(1) Determine whether there exists m satisfying mx2－2x－m＋1<0 for each x ∈R . If m exists, find the
range of m; or if it does not exist, provide reasons.
(2) Suppose the inequality is satisfied for each m such that |m|≤2. Find the range of x.
13
Section C Comprehensive questions
Choose and finish 2 from the following 4 questions. Please answer in the designated areas on
the question paper and include any explanatory notes, proofs and intermediate
mathematical steps. (Full marks: 60; 30 marks for each question completed.)
19. As shown in the figure, an oval parterre centered at the point O. AB is the long axis of the ellipse and
the point C is the upper vertex of the ellipse. To lay the irrigation pipeline, the points E and F should
be determined in AB such that OE＝OF, where CE, CF and FA sum to give the overall length, denoted
u, of the pipeline. Suppose ∠CFO＝θ, OA＝20 m and OC＝10 m.
C
(1) Find the relationship between u and θ and give the range of cos θ.
(2) Find the maximal value of u.
B
E
O
A
F
Figure 19
20. Given the functions f ( x) = ln x , g ( x) = e .
x
(1) If  ( x) = f ( x) −
x +1
, find the interval of the real line in which  (x) is monotonic.
x −1
(2) Suppose A( x0 , f ( x0 )) is a point on the graph of y = f (x) and l is the line tangent to y = f (x) at the
point A( x0 , f ( x0 )) . Prove that there exists a unique point x0  1 such that l is tangent to y = g (x) .
14
21. It is given that the sequence {an} satisfies a2t－1＋a2s－1＝2at＋s－1＋2(t－s)2 for any positive integers t and
s. It is given that a1＝0 and a2 ＝2.
(1) Find the values of a3 and a5.
(2) Suppose bn＝a2n＋1－a2n－1 where n∈N*. Prove that {bn} is an arithmetic sequence.
(3) Suppose cn＝( an＋1－an) qn－1 where n∈N* and q≠0. Calculate Sn (the sum of the first n terms of {cn}).
22. Consider the ellipse
x2
y2
＋ 2 =1 (a > b >0). It is given that it has a focal length of 2 3. The symmetric
2
a
b
point of P(0,2) about the line y=－x lies on the ellipse.
(1) Find explicitly the equation of the ellipse.
(2) As shown in the figure, A and B are the upper and lower vertices of the ellipse. The line l passing
through P intersects the ellipse at distinct points C, D. (Note that point C lies on the segment PD).
→ →
① Find the range of OC ·OD.
② If AD intersects BC at point Q, decide whether the y-coordinate of the point Q is a fixed number.
If it is, find the y-coordinate, or if it isn’t, give reasons.
y
P
C
A
Q
O
D
B
Figure 22
15
x
Glossary
arithmetic sequence 等差数列
approximate equality relation 约等关系
arithmetic progression 等差级数，等差数列
binomial theorem [数] 二项式定理
calculation n. 计算
cuboid adj. 立方形的；立方体的 n. 长方体
cosine theorem 余弦定理
circumscribe vt. 在……上画圈；包围；（几何学）外接
complex number [数] 复数
contingency table [数] 列联表
domain n. 领域；域名
diameter n. 直径
equation n. 方程式，等式
even adj. 偶数的
ellipse n. [数] 椭圆形
explicitly adv. 明确地；明白地
existential quantifier [数] 存在量词
function n. [数] 函数
fixed number 定数
geometric sequence 等比序列
geometrical significance of a function 函数的几何意义
induced formulae 诱导公式
interval n. 间距
intersect vi. 相交，交叉
images and properties of exponential function 指数函数的图象和性质
line segment n. [数] 线段
long axis [数] 长轴
logarithmic function [数] 对数函数
modulus n. 系数
monotonic adj. 单调的；无变化的
monotonicity n. [数] 单调性
non-zero constant 非零常数
necessary condition [数] 必要条件，充分必要条件
necessary and sufficient condition [数] 充要条件；必要且充分的条件
normal distribution n. [数] 正态分布
oval parterre 椭圆形
odevity 奇偶性
purely imaginary number [数] 纯虚数；虚数
plane analytic geometry 平面解析几何
probability n. [数] 或然率
plane vector 平面向量
perpendicular adj. 垂直的，正交的 n. 垂直线，垂线；垂直的位置
16
positive integer [数] 正整数
power function [数] 幂函数；功率函数
piecewise function 分段函数
random events and probability 随机事件与概率
systematic sampling [数] 系统抽样
symmetry axis [数] 对称轴
segment n. 段，部分
subset n. [数] 子集
sufficient condition [数] 充分条件
symmetric point 对称点
sine theorem 正弦定理
standard equation [数] 标准方程
scalar product [数] 数积
solid geometry [数] 立体几何
space vector 空间矢量
simple linear regression model 简单线性回归模型
tangent n. 切线，切面；正切
trigonometric function n. 三角函数
trigonometric formulae 三角函数公式
universal quantifier [数] 全称量词
value n. 值
vertex n. 顶点
x-coordinate 横坐标
y-coordinate 纵坐标
17