Κίνα και μαθηματικά με εισαγωγικό τεστ

The Changing Educational Frameworkfor the

Teaching of Mathematics in China

by

Yanming Wang

Suzhou Railway Teachers� CollegeSuzhouChina

Changing Educational Framework in the Teaching of Mathematics in ChinaYanming Wang

1. Changes in schooling

First we look at some historical facts. Before 1949, most schools were run by either thestate or private sectors. The curriculum content and systems of schools were almostidentical. After 1949, the Chinese school system was similar to the USA scheme; that isthere were

* 6-grade elementary schools (age group 6-12, known as Years 1-6)

* 3-grade junior schools (age group 12-15, known as Years 7-9)

* 3-grade senior schools (age group 15-18, known as Years 10-12)

The junior schools and senior schools together comprised the middle schools. There werealso various technical, commercial, vocational and teacher training schools where studentsusually spent three years.

Students could enter the technical, commercial, vocational and teacher training schoolswhen they had graduated from junior school and passed the entrance examination. Studentscould enter college or university when they had graduated from senior school and passedthe entrance examination.

A large number of schools were government maintained. They belonged to the country orlocal provinces. The whole education system in China achieved excellent results during the1949 to 1966 period. But in 1966, when the cultural revolution began in China, theeducation system and teaching methods were changed and the students went to thefactories and countryside to work, so their basic scientific education was much reduced.

In 1978, the pre-1996 education system was reintroduced. Students had to pass anentrance examination before entering college or university. Education again became animportant characteristic in China. This period is called the Spring of Science.

In the 1990's, the education system was completely reformed so that there is now a 9-yearcompulsory education system with

* 6-grade elementary school

* 3-grade junior school

If students want to enter senior schools and receive higher education, they must pay forpart of the tuition.

Private elementary schools and middle schools have been established and there are alsosome private colleges and universities. During the 1980's some industrially-supportedschools were founded. So there are various types of schools in China, but most schools arestill government maintained.

The colleges and universities offer masters degrees and doctorates as well as undergraduatecourses, but most of the educational research programs and experiments are controlled bythe Education Ministry.

1


2. Changes in the curriculum of mathematics education

After 1949, the general schools had curricula issued by the Education Ministry. Theywere similar to the USSR curricula. The elementary schools and middle schools changedtheir curricula content about every four years. In addition there were some complementarycurricula issued for special mathematical classes.

After 1978, the curricula content and teaching methods were reformed and innovationswere brought in. A new curriculum was issued by the Education Minister which wasfollowed by new mathematics textbooks and other teaching and learning materials.Currently the new textbooks for the 9-year compulsory education period published by thePeople's Education Press are used by about 70% of elementary and middle schools whilstsome elementary and middle schools use their own resources.

3. The content of school mathematics

(a) Mathematics content for 6-year (compulsory) elementary school

Numbers and Operations

Measurements

Basic Algebra

Basic Geometry

Basic Application of Mathematics

(b) Mathematics content for 3-year (compulsory) junior school

Algebra: identities - laws of indices, laws of square root, logarithms;

Equations and Inequalities: first degree, quadratic, systems of equations (linear andquadratic), irrational, logarithmic;

Sequences and Series: arithmetic, geometric;

Geometry: congruency and similarity, notable lines and points of triangle, angles oftriangle, relation between the angles and sides, Pythagorean theorem, circle - Thalestheorem, circumferential and central angles, chord quadrangle, tangent quadrangle;

Probability and Statistics

(c) Mathematics content for senior school

Functions: Elementary functions and their features;Transformations.

Sets Theory: Notation of sets, property of sets, operation of sets.

Trigonometry: Definitions of trigonometric functions;Relations of trigonometric functions;Sine theorem, Cosine theorem;Sin(A+B), Cos(A+B), Tan(A+B), Cotan(A+B);Proof of Tangent theorem.

2


Vectors, Coordinate Geometry:Operation with vectors;Coordinates of a sharing point of a section;Equations of straight line, circle, parabola;Centroid of tetrahedron;Distance between point and line, between parallel lines;Equations of bisectrix;Equations of ellipse and hyperbola;General features of the conical sections;Standard equation of plane;Equations of sphere, cylinder, rotation surface.

Space Geometry:Concepts of points and lines and planes in space;Property of points and lines, lines and lines, lines and planes;Planes and planes;Property of Cube, Cuboid and cylinder and Sphere.

Complex Numbers:Concept of complex numbers, representation, conjugation;Operations with complex numbers;Trigonometric form of complex numbers;De Moivre's theorem;Extraction of roots of complex numbers;Roots of Unity

Analysis: Limit of Sequences, propositions of limit;Limit of functions in the finite and infinite;Continuity of functions;Rules of differentiation.

Series: Concept of series, convergence of series;Geometric series, sum of convergent geometric series.

Probability: Permutations and combinations;Elements of classical probability;Probability fields;Elements of statistics including mean and standard deviation

3


4. The training system for mathematics teachers in China

There are elementary teacher training schools and middle teacher training colleges oruniversities in China.

Students can enter the elementary teacher training schools after they finish study injunior schools, but they must pass the entrance examination. Students must study threeyears in the elementary teacher training school. They learn Chinese, mathematics, history,art and music and some ICT. Before graduating, they go to the elementary schools forteaching practice for about six months. Emphasis is placed on the experience of teaching.In the elementary schools, students first observe lessons and then teach the pupils andcheck homework. Some students have contact with the parents of the pupils and help thepupils to make more progress.

Students can enter the middle teacher training college or normal university after theyfinish their study in senior schools, but they must pass the entrance examination like thosewho want to enter other colleges and universities. At teacher training colleges, mathsteachers are trained at university level in mathematical subjects. But the mathematicalteaching methodology, its training system and practice is stronger at these colleges than inuniversities. The students usually have 6 to 8 weeks teaching practice in middle schools.They first observe and then teach pupils, and must complete a report about their practice.

5. Teaching approaches to mathematics

Chinese mathematics teachers in general agree that the teaching and learning ofmathematics must:

* extend pupils' ways of thinking;

* develop pupils' abilities in problem handling and solving;

* provide applicable mathematical knowledge, expertise and skillsfor pupils' future needs.

The good practice of teaching involves the pupils being active in the process and makesthem respect and, hopefully, enjoy mathematics, recognizing its usefulness andappreciating the subject in its own right.

We would like to give all our pupils a depth of mathematical understanding because wethink it is a worthwhile and necessary part of their culture. But alongside the traditionalteaching approaches, teachers are now innovating with whole class interactive teachingwith the focus on the teacher.

Our interactive teaching means that the teachers try to instruct pupils in discoveringmathematical facts by themselves. We call it the method of discovery. We like this methodand use it in class teaching. Many teachers now also do their best to introduce newteaching methods, such as "flexible teaching" and "different degree teaching" in which thecontent is differentiated according to the ability of the pupils.

4


So they try to help make pupils more interested in learning mathematics. For example, by:

* using visual objects;

* using computers.

They try to get pupils motivated, wanting to learn mathematics by themselves. They wantto develop the mathematical skills and abilities of pupils and encourage their creativity inmathematics.

Pupils gain excellent results for their hard work. The mathematical abilities of Chinesepupils are comparatively at a high level. They have, for example, good results in theMathematical Olympiad.

6. Assessment system for mathematics in China

In general, pupils have to pass a mathematics test, set by their teacher, at the end of everyterm. As an example, Appendix 1 gives a sample of the end of elementary school test inmathematics (for pupils age 12+).

There are also two important examinations in China. The first examination is a test forpupils entering senior school after they have finished their studies at junior school (atage 15+) . This is a local (not national) examination and a sample paper is given inAppendix 2.

The second and the most important examination is the national entrance examination forcollege or university. If pupils want to enter higher education institutions, they must takethe examination of mathematics. It is difficult for pupils but parents want their children togo to good universities and pupils need to get high scores in mathematics for this.Appendix 3 gives a sample test for ALL pupils at the end of Senior school (age 18+) andAppendix 4 gives a sample national test paper in mathematics for University entrance.

7. Current issues and problems in mathematics education in China

There are a number of issues and concerns in mathematics education presently; theseinclude:

(1) some teachers feel that the textbooks are too difficult for the pupils and needrewriting in a more motivational style;

(2) some people hope for more reform and innovation in the teaching of mathematics;

(3) there is tremendous pressure from parents and society for pupils to achieve highlevels of attainment in both primary and middle schools;

and

(4) the biggest pressure on both pupils and parents is that of entry to higher education,where there are not nearly enough places for the demand and only the highestattainers get places.

Overall though, we are proud of our maths teaching and the levels of attainment achievedby most pupils at school in China.

5

CHINA, Examination for Entry to Junior School (Age 12+) 1999

6

PART A

1. Express 413

hours in hours and minutes. hrs mins

2. 3 tons 50 kilograms = tons

3. : 8 = 0.75 = 15

= % =

16

4. A = 20, B = 50. A is % of B.

B is more than A by %.

5. Which one of the following,

π , 3 14 318

3 15. , , . %

is (a) the largest ?

(b) the smallest ?

6. 1 ÷ =0 2.

7. The number of lines of symmetry of a square is .

CHINA July 1999 Examination for Entry to Junior School(Age 12+)

APPENDIX 1


7

8. The sum of all the edges of a cube is 48 cm.

The surface area of the cube = cm 2

The volume of the cube = cm 3

9. The length of a rectangular playground is 80 m and its width is 60 m.

The area of the playground = m 2

10. Two numbers A and B are such that A is 20% greater than B.

The ratio A : B = :


8

PART B

For each of the following statements, write in the box whether it is TRUE or FALSE.

1. The only fraction between 25

and 45

is 35

.

2. For cylinders with the same base areas, the volumesare in the ratio of their heights.

3. Two triangles with the same base and height can beput together to form a parallelogram.

4. For a cone and a cylinder, both with the same height and base

area, the volume of the cone is 13

the volume of the cylinder.

PART C

Calculate:

1. 101 65× =

2. 7 335

− =

3. 5 4 115

. + =

4. 118

− =

5. 634

3÷ =

6. 4 2518

× =

7. 4 0 5 0 5 4×( ) ÷ ×( ). . =

8. 1 6 1 6 1 6 1 6 25. . . .+ + +( ) × =


9

PART D

Solve for x:

1. x : 5 = 0.6 : 123

x =

2. 134

− × =x 0 75 4 32. x =

3. Four times x is 2.5 more than 10. x =

PART E

Calculate:

1. 101 21 4 21 4× − =. .

2. 25 1 25 4 8× × × =.

3. 1235

7 25 17

20− − =.

4. 7 6 534

2− ÷ + × =

5. 13 125

6 5 213

123

× + −

÷

=.


10

PART F

1. The radius of a circle is 3 cm.

Calculate its circumference and area. Circumference = cm

Area = cm 2

2. Train A leaves station C and travels to station D at a speed of 50 km/h.

At the same time, train B leaves station D and travels to station C at aspeed of 45 km/h.

The trains cross when they have been travelling for 4 hours.

What is the distance between stations C and D? Distance =

3. A factory must produce a total of 1200 tons of goods in five days.It produces 80 tons of goods per day for the first 3 days.

How many tons of goods must be produced per day in the final two days? tons

4. The number of cars made by a company last year was expectedto be 7200. In fact, 15% more cars than this were made.

What was the total number of cars made? cars

5. A completes a task in 10 days. It takes B 15 days to completethe same task.

If A and B work together, how many days will it take them to

complete 13

of the task?

6. A man has an order from a shop for some toys. If he makes250 toys per day he can complete the order in 12 days.

How many days will it take him to complete the order if hemakes 300 toys per day?

7. A cylindrical bucket has a volume of 24 cu dm. The area

of the base of the bucket is 7.5 m 2 . Water is poured into

the bucket until it is 34

full. What height is the top of the

bucket above the surface of the water?

km

days

days

per day

dms

CHINA, Mathematics Test Paper for Entry to Senior School (Age 15+) 1999

11

1. 4 = ?

(a) 2 (b) – 2 (c) ± 2 (d) 16

(4 marks)

2. An angle is 36°. What is the size of its complementary angle?

(a) 64° (b) 54° (c) 144° (d) 36°(4 marks)

3. What are the coordinates of point P', if P' and P (1, – 2) have rotational symmetry?

(a) (– 1, 2) (b) (– 1, – 2) (c) (– 2, – 1) (d) (1, 2)

(4 marks)

4. In function yx

=−

12

, x ∈ ?

(a) x ≥ 2 (b) x > 2 (c) x > −2 (d) x ≠ 2(4 marks)

5. Which shape has line symmetry but not rotational symmetry?

(a) rhombus (b) rectangle

(c) equilateral triangle (d) circle

(4 marks)

6. What is 19 990 in scientific notation?

(a) 19 99 10 3. × (b) 199 9 10 2. × (c) 1 999 10 4. × (d) 1 999 10 4. × −

(4 marks)

7. Which statement is correct?.

(a) a a a2 3 6× = (b) − −( ) = −5 5 (c)1 1 2a b a b

+ =+

(d) 319

2− =

(4 marks)

8. If the mean value of 1, 3 and x is 3, what is the value of x ?

(a) 5 (b) 3 (c) 2 (d) – 1(4 marks)

CHINA July 1999 Examination for Entry to Senior School(Age 15+)

APPENDIX 2


12

9. If the radii of two circles are 3 cm and 5 cm, and the distance between thecentres of the two circles is 2 cm, what can you say about the two circles?

(a) they are separate (b) they touch externally

(c) they intersect (d) they touch internally(4 marks)

10. In the diagram, if ∠ = ° ABC 60 , then ∠ = EDC ?

(a) 120 ° (b) 60 °

(c) 40 ° (d) 30 °

(4 marks)

11. The graph of the function yk

x= passes through the point (– 4, – 5).

What is the function?

(a) yx

= − 20(b) y

x=20

(c) yx

= 20(d) y

x= −20

(4 marks)

12. If the sum of the interior angles of a polygon is equal to the sumof its exterior angles, how many sides does it have?

(a) 3 (b) 4 (c) 5 (d) 6(4 marks)

13. If the graph of function y k x b= + passes through the first, thirdand fourth quadrants, which statement is correct?

(a) k b> >0 0, (b) k b> <0 0,

(c) k b< >0 0, (d) k b< <0 0,(4 marks)

14. If the diameter of the base face of a cylinder is 4, and the height is 2,what is the area of the curved surface of the cylinder?

(a) 8π (b) 4π (c) 16π (d) 8(4 marks)

15. In the diagram, PA is a tangent to the circle, centre O.

PB = BC.

If PA = 3 2 , what is the length of BC ?

(a) 3 2 (b) 3

(c) 3 (d) 2 3

(4 marks)

A

D

E

CB

A

PBCO


13

16. In triangle ABC, if ∠ = ° =c A9035

and sin , cot B = ?

(a)35

(b)54

(c)34

(d)43

(4 marks)

17. In the diagram shown, what is a b a b− + + ?

(a) 2 a (b) −2 a (c) 0 (d) 2b(4 marks)

18. If x x1 2, are the roots of the equation x z m x m2 1 0− − − = , what

can you say about x x1 2, ?

(a) x x x xl ≠ ∈ℜ2 1 2, (b) x x x xl = ∈ℜ2 1 2,

(c) x xl 2 ∉ℜ (d) uncertain(4 marks)

19. In the graph of y a x b x c= + +2 , as shown, which of thefollowing is true?

(a) b c a+ − = 0 (b) b c a+ − > 0

(c) b c a+ − < 0 (d) uncertain

(4 marks)

20.130

52

2 1sin °+ −( ) ° +

+=π ?

(4 marks)

21. In the rectangle ABCD shown opposite, E is themid-point of CD.

Prove that ∠ = ∠EAB EBA .

(4 marks)

22. Solve

3 15 2 2 5 1 02 2x x x x+ − + + + =(6 marks)

a o b

y

xO

D E C

A B


14

23. The distance between two places, A and B, is 15 km.

At 0600 hours, James leaves A to walk to B.

20 minutes later, Rachel leaves B to cycle to A.

Rachel stays at A for 40 minutes and then cycles back to B, arriving there at the same time asJames.

If Rachel's speed is 10 km per hour faster than James' speed, at what time do they arrive at B ?

(6 marks)

24. In triangle ABC, ∠ = °ACB 90 . CD is the perpendicular from C to AB.

AD has length m and BD has length n. AC BC2 : :2 2 1= .

The roots of the equation 14

2 1 122 2x n x m+ −( ) + = and x x1 2 and and x x1 22 192−( ) < .

If m, n are integers, what is the value of y m x n= + ?

(8 marks)

25. If the graphs of function y x a x b= + − +2 2 2 1

y x a x b= − + −( ) + −2 23 2 1

pass through the points M and N, where M and N are on the x-axis?

What are the true values of a and b ?(7 marks)

26. In the diagram, the length of the chord AB is 4.

P is a moving point on the circumference of the

circle, centre O, and cos∠ =APB13

.

Explain why there exists a triangle APB withmaximum area.

What is the maximum area of triangle APB ?

(9 marks)

A

P

B

O

CHINA, Mathematics Test Paper for Graduation from Senior School (Age 18+) 1999

15

1. tan 405° = ?

(a) 1 (b) – 1 (c)12

(d) − 12

(3 marks)

2. log log2 251

10+ = ?

(a) 1 (b) – 1 (c) 2 (d) – 2(3 marks)

3. If α π= 137

,

then

(a) sin cosα α> >0 0 and (b) sin cosα α> <0 0 and

(c) sin cosα α< <0 0 and (d) sin cosα α< >0 0 and ?(3 marks)

4. Which of the functions,

f x x1 12

( ) = log , f x x2 2( ) = , f x x2 3 1( ) = + , f x x4

2 3( ) = − ,

is a decreasing function on (0, + ∞ ) ?

(a) f x1 ( )

(b) f x2 ( )

(c) f x3 ( )

(d) f x4 ( )(3 marks)

5. Which of the following lines passes through the point (0, 1) and is perpendicularto y x= − +2 3 ?

(a) 2 1 0x y− − = (b) x y− + =2 2 0

(c) 2 1 0x y− + = (d) x y− − =2 2 0(3 marks)

6. In the diagram, l r= 2 . What is the size of angle α ?

(a) 30° (b) 45°

(c) 60° (d) 150°

(3 marks)

r

l

α

CHINA July 1999 Examination for Graduation from Senior School(Age 18+) (Three hours allowed)

APPENDIX 3


16

y

x

F1

F2

7. In triangle ABC, A B = 2, BC = 3 and AC = 7 .

What is the size of angle B ?

(a) 60° (b) 120° (c) 30° (d) 150°(3 marks)

8. Which of the following is the solution of x + <1 5 ?

(a) − ∞( ), 4 (b) − ∞( ), 24

(c) [ , )−1 4 (d) [ , )−1 24(3 marks)

9. Using Cartesian coordinates, the graph of the function y x= −1 x x∈ℜ ≠( )0 , has

(a) symmetry about the x-axis (b) no rotational symmetry

(c) symmetric about the y-axis (d) symmetric about y x=(3 marks)

10. The smallest period of function y x= −2 12sin is:

(a) 4π (b) 2π (c) π (d)π2

(3 marks)

11. Which of the following is the equation of the ellipse whose graphsatisfies the following conditions:

F and F1 2 are focus points; F F1 2 = 6 ; eccentricity e = 35

.

(a)x y2 2

4 51+ = (b)

x y2 2

16 251+ =

(c)x y2 2

5 41+ = (d)

x y2 2

5 161+ =

(3 marks)

12. Limn

n n

n→∞− +

+

≈1 24

2

2 ?

(a) 2 (b) 1 (c) 1 (d) − 14

(3 marks)

13. If sin , cosα β π α β π= = −35

45

and < , < ,

then cos ?α β−( ) =

(a)π25

(b) − 725

(c) 1 (d) 0

(3 marks)


17

14. If complex numbers z i z i1 21 2 1 3= + −,

then the complex conjugate of z z1 2+ is

(a) 2 − i (b) 2 + i (c) − −2 i (d) 2 + i ?(3 marks)

15. If a, b, c > 0, and 1, a, b, c, 16 is a geometric series,then b = ?

(a) 2 (b) 4 (c) 8 (d)172

(3 marks)

16. If function f x x a x f f( ) = + ( ) = ( )2 10 2, a constant, satisfies , which of thestatements below is correct?

(a) f f f12

1 3

< ( ) < ( ) (b) f f f

12

3 1

< ( ) < ( )

(c) f f f112

3( ) <

< ( ) (d) f f f1 3

12

( ) < ( ) <

(3 marks)

17. Which of the following statements is correct?

(a) For two straight lines, a and b, where a is perpendicular to plane α ,and b is perpendicular to plane β , then a is parallel to b.

(b) If the straight line a is parallel to the plane α , then all straight lines, l,lying in the plane α are parallel to a.

(c) If α β and are planes and α is perpendicular to β , then there exists

a straight line, l, in α , such that l is perpendicular to β .

(d) If α β and are planes and a is a straight line such that a is perpendicular

to α and β , then α is parallel to β .(3 marks)

18. In an arithmetic series a n{ }, if a a4 5 12+ =then S 8 = ?

(a) 12 (b) 24 (c) 36 (d) 48(3 marks)

19. If x > −1 and y xx

= + ++

21

1,

then y min = ?

(a) 0 (b) 1 (c) 2 (d) 3(3 marks)


18

20. The height, h, of the cylinder shown is 4 and the areaof the curved surface is 8π .

What is the length of the line A C ?

(a) 17 (b) 5

(c) 2 5 (d) 20(3 marks)

21. In the binomial expansion of x −( )3 10 , the coefficient of x 6 is:

(a) −27 10 C 6 (b) 27 40 C 4 (c) −9 10 C 6 (d) 9 10 C 4

(3 marks)

22. In a batch of 100 lights, 2 are defective. You choose three lights at random from thebatch of 100. In how many ways can you make your choice to include exactly onedefective light?

(a) 100

3P (b) 100

3C (c) 2

1

982P P (d) 2

1

9821

C C

(3 marks)

23. In the diagram, ABCD is a rectangle and EC is a parabola.EF = 5 m and BC = 2 m .

A car of height 4 m and width 3 m, is to pass through theABCDE shape.

What is the least value of AB for which this can happen?Give your answer to one decimal place.

(a) 5.0 m (b) 5.2 m

(c) 5.4 m (d) 5.6 m(3 marks)

24. The diagram shows a right prism, ABCDA1B

1 C

1 D

1.

E is the midpoint of D1 D.

(a) Prove that D1 B is perpendicular to AC.

(b) If A B = 3 2 and A1 A = 6 , what is the volume

of the triangle-based pyramid EACD ?

(10 marks)

A

C

h = 4

E

5 mD C

AF

B

2 m

A1

D1 C1

B1

A B

CD

E


19

25. f x( ) is the inverse function of function y x=+

−210 1

1 x ∈ℜ( ) and the graph

of g x( ) and the graph of yx

x= −

−4 2

1 are symmetric about y x= − 1.

Write F x f x g x( ) = ( ) + ( ).(a) What is F x( ) and its domain?

(b) A and B are two points on the graph of F x( ).The straight line A B and the y-axis are vertical. Why?

(11 marks)

26. A hyperbola, C, passes through point M (1, 2), with they-axis as its right directrix and its right focus, F, lies on

x y−( ) + −( ) =1 2 42 2 x >( )0 .

(a) What is the eccentricity of hyperbola C ?

(b) What is the equation of the hyperbola whenMF is parallel to the x-axis?

(c) What is the equation of N, the intersection pointof the line MF and the right part of the hyperbola, C ?

(10 marks)

y

x

M

O

CHINA, National Mathematics Test Paper (College or University Entrance) 1999

20

1. I is a universal set and M, P and S are subsets of set I.The shaded part of the diagram is:

(a) M P S∩( ) ∩

(b) M P S∩( ) ∪

(c) M P S∩( ) ∩

(d) M P S∩( ) ∪(4 marks)

2. If map f a f a a: A B A B A→ ∀ ∈ ( ) = ∈ = − − −{ , , , , , }3 2 1 1 2 3 4and set B is the image of A under f, how many members are there in set B?

(a) 4

(b) 5

(c) 6

(d) 7(4 marks)

3. If the function y g x= ( ) is the inverse function of y f x= ( ) and f a b ab( ) = ≠, 0 ,

then g b( ) =

(a) a

(b) a −1

(c) b

(d) b −1 ?(4 marks)

4. Function f x m w x q( ) = +( )sin , w >( )0 is an increasing function on a b,[ ] and

f a m( ) = − f b m( ) = .

Does the function g x m w x q( ) = +( )cos on a b,[ ](a) increase

(b) decrease

(c) take the value m

(d) take the value –m ?(4 marks)

5. If f x( ) is an odd function and has period π , then f x( ) could be(a) sin x

(b) cos x

(c) sin2x

(d) cos2x ?(4 marks)

M P

S

I

CHINA July 1999 National Mathematics Test Paper(for College or University Entrance)

APPENDIX 4


21

6. In polar coordinates, the curve r = −

4

3sin θ π

has

(a) the straight line θ π=3

as a line of symmetry,

(b) the straight line θ π= 56

as a line of symmetry,

(c) symmetry about the point 23

,π

,

(d) has rotational symmetry about the pole.(4 marks)

7. Water in a circular cylinder is such that r h= =2 cm and 6 cm , as shown in the diagram.

The same volume of water is put into a circular cone, as shown, with AB AO BO= = .What is the value of H ?

(a) 6 3 cm

(b) 6 cm

(c) 2 18

(d) 3 123 cm

(4 marks)

8. If 2 3 40 1 2

23

34

4x a a x a x a x a x+( ) = + + + + , then

a a a a a0 2 42

1 32+ +( ) − +( ) = ?

(a) 1

(b) – 1

(c) 0

(d) 2 ?

(4 marks)

9. Line 3 2 3 0x y+ − = and circle x y2 2 4+ = intersect.The area of the sector of the minor arc is equal to

(a)π6

(b)π4

(c)π3

(d)π2

?

(4 marks)

rh

A B

O

H


22

A B

C

FE

D

10. In the polyhedron ABCDEF shown below, ABCD is a right square and AB = 3.

EF is parallel to AB, EF = 32

. The distance between EF and plane AC is 2.

The volume of the polyhedron ABCDEF is

(a)92

(b) 5

(c) 6

(d)152

?

(4 marks)

11. If sin tan cotα α α> > − < <

π α π2 2

, then α ∈

(a) −

π π2 2

,

(b) −

π4

0,

(c) 04

, −

π

(d)π π4 2

,

?

(5 marks)

12. The diagram shows the frustum of a right circular cone with radius = 5.S S1 2 and are the curved surfaces of two small frustums. S S1 2 1 2: := .What is the value of R ?

(a) 10

(b) 15

(c) 20

(d) 25(5 marks)

5

R

S1

S2


23

A

B

at least 6

13. Points M and N are given by M N154

454

, , ,

− −

and four curves have equations

1. 4 2 1 0x y+ − = 2. x y2 2 3+ = 3.x

y2

2

21+ = 4.

xy

22

21− =

For which curves does there exist point p on the curve, such that MP NP= ?

(a) Equations 1. and 3.

(b) Equations 2. and 4.

(c) Equations 1. 2. and 3.

(d) Equations 2. 3. and 4.(5 marks)

14. A man wants to buy at least 3 videos and a least 2 boxes of floppy disks. Videos cost60 yuan each and a box of disks costs 70 yuan. He plans to spend no more than 500 yuan.What is the maximum number of videos and boxes of floppy disks that he can buy?

(a) 5

(b) 6

(c) 7

(d) 8(5 marks)

15. The ellipse x

a

y

ba b

2

2

2

2 1 0+ = > >( ) has right focus F1,

right directrix is l 1, as shown in the diagram.

A B F C= 1 ,

Determine the eccentricity of the ellipse.

(4 marks)

16. A field is divided into ten strips as shown in the diagram.You want to plant two crops, A and B in separate strips,so that there are at least 6 strips separating them.In how many different ways can you do this?

(4 marks)

y

xO

A

B

F1 C

l 1


24

α β

17. If a b ab a b> > = + +0 0 3, and , then ab ∈ ?(4 marks)

18. α β and are different planes and m n and are straight lines.

m ∉α , m ∉β , n ∉α , n ∉β .

We now have four results:1. m n⊥ 2. α β⊥ 3. n ⊥ β 4. m ⊥ α

Which of these results is correct, so that from three of them you can imply the rest?(4 marks)

19. Solve

3 2 2 1 0 1log log ,a ax x a a− < − > ≠( )(10 marks)

20. If complex number z i= +3 2cos sinθ θ , and

y z= − < <

θ θ π

arg 02

,

then y max = ? and θ = ? (when y y= max).(12 marks)

21. In the diagram, right prism ABCD – A B C D1 1 1 1

point E is on D D1 , plane EAC is parallel to D B1 .

The angle between planes E A C and A B C D is 45 °.

A B = a .

(a) What is the surface area of EAC ?

(b) What is the distance between the straight lines A B1 1

and A C ?

(c) What is the volume of the pyramid B1 EAC ?(12 marks)

22. (a) In the diagram, α = input steel thickness

β = output steel thickness.

If each pair of rollers reducesthe thicknessby a factor r , how many pairs of rollers

are needed?

[ i.e. r = −α βα

o o

o

α o input,= β o output= ]

A1

D1 C1

B1

A B

CD

E

β 0α 0


25

(b) A reduction mill consists of 4 pairs of rollers, each making a percentage reduction inthickness of 20%. The circumference of each of the rollers is 1600 mm.

There is a fault on the nth roller, causing a black mark on the steel at regular intervals.Let L n be the distance between each mark on the steel.

Complete this table:

(14 marks)

23. The graph image of y f x= ( ) is a step-graph line.

The gradient of the line is b n when

n y n n≤ ≤ + =( )1 0 1 2, , , ... .

If x n{ } is defined by f x n nn( ) = =( )1 2, , ... ,

(a) what are x x x n1 2, and ?

(b) f x( ) = ? What is the domain of f x( )

(c) Prove that, if x y,( ) is the intersecting point between

y f x= ( ) and y x= , then x ≤ 1.(14 marks)

24. In the diagram, A is the point (a, O) and the straight line lis given by x = −1.

Point B moves on the straight line l.

OC is a bisector of angle BOA.

AB and OC intersect at C.

What is the equation of C ?

Discuss the type of equation and its relationship with a.(14 marks)

1

y

x

y f x= ( )

y

xO

Bl

A

Fault on Roll, n 1 2 3 4

L n (mm) 1600