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一元二次方程
函數及其圖像
不等式與線性規畫
指數函數與對數函數
等差數列與等比數列及其求和法
變分
直線方程
圓的基本性質
續三角學
排列與組合
續概率
離差的度量
續函數圖像
續多項式
續方程
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DSEMATH CPPAPER 2

STUDIFY MOCK EXAMINATION

HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION · PRACTICE PAPER

MATHEMATICS Compulsory PartPAPER 2

11.30 am – 12.45 pm (1¼ hours)

INSTRUCTIONS

(1)Read carefully the instructions on the Answer Sheet. After the announcement of the start of the examination, you should first stick a barcode label and insert the information required in the spaces provided. No extra time will be given for sticking on the barcode label after the 'Time is up' announcement.
(2)When told to open this book, you should check that all the questions are there. Look for the words 'END OF PAPER' after the last question.
(3)All questions carry equal marks.
(4)ANSWER ALL QUESTIONS. You are advised to use an HB pencil to mark all the answers on the Answer Sheet, so that wrong marks can be completely erased with a clean rubber. You must mark the answers clearly; otherwise you will lose marks if the answers cannot be captured.
(5)You should mark only ONE answer for each question. If you mark more than one answer, you will receive NO MARKS for that question.
(6)No marks will be deducted for wrong answers.

Not to be taken away before the end of the examination session

Studify practice paper. Not an official HKEAA examination paper.

There are 30 questions in Section A and 15 questions in Section B.
The diagrams in this paper are not necessarily drawn to scale.
Choose the best answer for each question.

Section A

\displaystyle \dfrac{(8^{n})^{2}}{2^{3n-1}}=

2^{3n-1}

2^{3n+1}

4^{3n+1}

2^{9n-1}

3(p-q)=p(q+1)

, then

q=

\displaystyle \dfrac{2p}{p+3}

\displaystyle \dfrac{2p}{p-3}

\displaystyle \dfrac{4p}{p+3}

\displaystyle -\dfrac{2p}{p+3}

x^{2}-y^{2}-4x+4y=

(x+y)(x-y-4)

(x+y)(x-y+4)

(x-y)(x+y-4)

(x-y)(x+y+4)

\displaystyle \dfrac{3}{2x-1}-\dfrac{2}{x+1}=

\displaystyle \dfrac{5-x}{(2x-1)(x+1)}

\displaystyle \dfrac{1-x}{(2x-1)(x+1)}

\displaystyle \dfrac{x+5}{(2x-1)(x+1)}

\displaystyle \dfrac{x-5}{(2x-1)(x+1)}

y=0.0358

(correct to 3 significant figures), find the range of values of

y

0.0357\leq y<0.0359

0.03575<y\leq 0.03585

0.03575\leq y<0.03585

0.0358\leq y<0.0359

a

and

b

are constants such that

(x-2)(x+a)\equiv x^{2}+bx-10

, then

b=

3 .

5 .

7 .

10 .

Let

f(x)=2x^{2}+kx-6

, where

k

is a constant. If

f(-2)=f(3)

, then

f(1)=

-6

-4

-2

2 .

Let

p(x)=x^{3}+ax^{2}-5x+2

, where

a

is a constant. When

p(x)

is divided by

x-2

, the remainder is 4. Find the remainder when

p(x)

is divided by

x+1

-1

A sum of money is deposited in a bank at an interest rate of 4% per annum for 2 years, compounded quarterly. The interest received is about

8.00% of the principal.

8.16% of the principal.

8.24% of the principal.

8.29% of the principal.

10.

The solution of

5-3x>2(x-5)

and

4x+7\geq -1

x<3

x\geq -2

-2\leq x<3

-2<x\leq 3

11.

3a=4b

and

5b=6c

, then

a:c=

5:8

2:3

3:2

8:5

12.

It is given that

z

varies directly as

x^{2}

and inversely as

y

. If

x

is increased by 20% and

y

is decreased by 10%, then

z

increased by 30% .

increased by 33% .

increased by 44% .

increased by 60% .

13.

Consider the sequence

a_{1}, a_{2}, a_{3}, \ldots

with

a_{1}=2

a_{2}=5

and

a_{n+2}=a_{n+1}+2a_{n}

for any positive integer

n

. Find

a_{6}

151

14.

Let

h

and

k

be positive constants. Which of the following must be true about the graph of

y=k-(x+h)^{2}

? I. The graph opens downwards. II. The axis of symmetry of the graph is

x=-h

. III. The

y

-intercept of the graph is positive.

I and II only

I and III only

II and III only

I, II and III

15.

A solid right circular cylinder of base radius 3 cm and height 8 cm is melted and recast into a solid sphere. Find the radius of the sphere.

2.62 cm

3.00 cm

3.30 cm

3.78 cm

16.

The volumes of two similar solid right circular cones are 54 cm

^{3}

and 128 cm

^{3}

. If the base area of the smaller cone is 18 cm

^{2}

, then the base area of the larger cone is

13.5 cm

^{2}

24 cm

^{2}

32 cm

^{2}

\displaystyle 42\frac{2}{3}

^{2}

17.

The area of the sector

OAB

6\pi

^{2}

, where

O

is the centre of the sector. If

\angle AOB=135^\circ

, which of the following are true? I. The radius of the sector is 4 cm. II. The arc length

AB

3\pi

cm. III. The area of

\triangle OAB

is 8 cm

^{2}

I and II only

I and III only

II and III only

I, II and III

18.

In the figure,

ABCD

is a parallelogram.

E

is a point lying on

CD

such that

AE

bisects

\angle DAB

. If

\angle ABC=64^\circ

, then

\angle AEC=

116^\circ

118^\circ

122^\circ

128^\circ

19.

In the figure,

AC

is a diameter of the circle

ABCD

. If

\angle BAC=34^\circ

, then

\angle ADB=

34^\circ

56^\circ

62^\circ

68^\circ

20.

The coordinates of the point

P

are

(-3,\,8)

P

is rotated anticlockwise about the origin through

90^\circ

to the point

Q

Q

is then reflected with respect to the

x

-axis to the point

R

. Find the

y

-coordinate of

R

-8

-3

21.

If each interior angle of a regular

n

-sided polygon is

156^\circ

, then

n=

12 .

13 .

15 .

18 .

22.

\displaystyle \dfrac{\sin(90^\circ-\theta)}{\cos(180^\circ+\theta)}+\tan\theta\tan(90^\circ-\theta)=

-2

0 .

1 .

2 .

23.

The straight line

3x-4y+12=0

cuts the

x

-axis and the

y

-axis at

A

and

B

respectively. If

O

is the origin, then the area of

\triangle OAB

6 .

7.5 .

12 .

24 .

24.

The coordinates of the points

A

and

B

are

(1,\,4)

and

(5,\,-2)

respectively. Let

P

be a moving point in the rectangular coordinate plane such that

AP=BP

. The equation of the locus of

P

2x-3y-3=0

2x-3y+3=0

3x+2y-11=0

3x-2y-7=0

25.

Which of the following are true about the circle

x^{2}+y^{2}-6x+8y-11=0

? I. The coordinates of the centre are

(3,\,-4)

. II. The radius is 6. III. The origin lies inside the circle.

I and II only

I and III only

II and III only

I, II and III

26.

If the straight line

y=x+k

is a tangent to the circle

x^{2}+y^{2}=8

, then

k=

\pm 2\sqrt{2}

\pm 4

\pm 4\sqrt{2}

\pm 8

27.

The stem-and-leaf diagram below shows the distribution of the ages of a group of hikers. Find the inter-quartile range of the distribution.

Stem (tens)	Leaf (units)
2	3 5 8
3	0 4 4 7
4	1 2 6 9
5	5

28.

The mean of 40 numbers is 25. If the mean of 15 of these numbers is 31, then the mean of the remaining 25 numbers is

19.6 .

20.8 .

21.4 .

28.0 .

29.

Two fair dice are thrown. Find the probability that the sum of the two numbers thrown is 8 or both numbers thrown are odd.

\displaystyle \dfrac{7}{36}

\displaystyle \dfrac{1}{4}

\displaystyle \dfrac{5}{18}

\displaystyle \dfrac{1}{3}

30.

Each number of a group of numbers is multiplied by 2 and then 3 is subtracted from it. Let

m

be the mean of the original group. Which of the following must be true? I. The mean of the new group is

2m-3

. II. The range of the new group is twice that of the original group. III. The variance of the new group is twice that of the original group.

I only

I and II only

II and III only

I, II and III

Section B

31.

The L.C.M. of

6a^{2}b^{3}c

9ab^{4}

and

4a^{3}b

36a^{3}b^{4}c

18a^{3}b^{4}c

ab

36a^{6}b^{8}c

32.

The graph in the figure shows the linear relation between

x

and

\log_{2}y

. Which of the following must be true?

\displaystyle y=8\cdot 2^{-\frac{x}{2}}

\displaystyle y=3\cdot 2^{-\frac{x}{2}}

y=8\cdot 2^{-2x}

y=2^{3-2x}

33.

3^{2x+1}-10(3^{x})+3=0

, then

x=

-1

or 1 .

\displaystyle \dfrac{1}{3}

or 3 .

0 or 1 .

-1

only .

34.

Let

a

be a real number and

\displaystyle z=\dfrac{a+3i}{1-i}

. If the real part of

z

is 0, which of the following are true? I.

a=3

II. The imaginary part of

z

is 3. III.

z^{2}

is a real number.

I and II only

I and III only

II and III only

I, II and III

35.

a

b

c

is an arithmetic sequence with a non-zero common difference, which of the following must be true? I.

a+c=2b

II.

2^{a}

2^{b}

2^{c}

is a geometric sequence. III.

a^{2}

b^{2}

c^{2}

is an arithmetic sequence.

I only

I and II only

I and III only

II and III only

36.

Find the range of values of

k

such that

x^{2}+(k-2)x+4>0

for any real number

x

-6<k<2

-2<k<6

k<-2

k>6

k<-6

k>2

37.

(x,\,y)

is a point satisfying

x\geq 0

y\geq 0

x+y\leq 6

and

x+2y\leq 8

, then the greatest value of

3x+4y

18 .

20 .

22 .

24 .

38.

The base of the right pyramid

VABCD

is a square of side 10 cm. The height of the pyramid is 12 cm. Find the angle between the plane

VAB

and the base.

50.2^\circ

59.5^\circ

67.4^\circ

73.3^\circ

39.

Let

O

be the origin. The coordinates of the points

A

and

B

are

(8,\,0)

and

(0,\,6)

respectively. Which of the following are true? I. The circumcentre of

\triangle OAB

(4,\,3)

. II. The orthocentre of

\triangle OAB

(4,\,3)

. III. The in-centre of

\triangle OAB

lies on the straight line

y=x

I and II only

I and III only

II and III only

I, II and III

40.

In the figure,

PT

is the tangent to the circle

ABC

A

. If

AB=AC

and

\angle PAB=52^\circ

, then

\angle BAC=

52^\circ

64^\circ

76^\circ

104^\circ

41.

A committee of 5 members is to be formed from 8 boys and 6 girls. If the committee must consist of at least 2 girls, how many different committees can be formed?

1 176

1 526

1 582

1 946

42.

A bag contains 4 red balls and 3 white balls. Balls are drawn randomly one by one from the bag without replacement until a white ball is drawn. Find the probability that at most two draws are needed.

\displaystyle \dfrac{2}{7}

\displaystyle \dfrac{3}{7}

\displaystyle \dfrac{4}{7}

\displaystyle \dfrac{5}{7}

43.

In a test, the standard score of Ann is 1.25 and her score is 70 marks, while the standard score of Ben is −0.5 and his score is 56 marks. The mean of the test scores is

56 marks .

58 marks .

60 marks .

62 marks .

44.

The mean and the standard deviation of a group of 10 numbers are 20 and 4 respectively. Two numbers 20 and 20 are added to the group. Which of the following must be true? I. The mean is unchanged. II. The standard deviation decreases. III. The range is unchanged.

I and II only

I and III only

II and III only

I, II and III

45.

Let

T(n)

be the

n

th term of an arithmetic sequence with a positive common difference

d

. Let

x_{1}

and

y_{1}

be the mean and the standard deviation of

\{T(1), T(2), \ldots, T(20)\}

, while

x_{2}

and

y_{2}

are the mean and the standard deviation of

\{T(21), T(22), \ldots, T(40)\}

. Which of the following must be true? I.

x_{2}-x_{1}=20d

II.

y_{1}=y_{2}

III.

y_{2}=2y_{1}

I only

I and II only

I and III only

II and III only

END OF PAPER

參考答案

揀一條題目就會顯示答案同評分指引。建議先自己試做。

第 1 題1 分

正確答案： B

1: B.

(8^{n})^{2}=2^{6n}

, so the expression is

2^{6n-(3n-1)}=2^{3n+1}

第 2 題1 分

正確答案： A

2: A.

3p-3q=pq+p

, so

2p=q(p+3)

and

\displaystyle q=\dfrac{2p}{p+3}

第 3 題1 分

正確答案： C

3: C.

(x-y)(x+y)-4(x-y)=(x-y)(x+y-4)

第 4 題1 分

正確答案： A

4: A.

\displaystyle \dfrac{3(x+1)-2(2x-1)}{(2x-1)(x+1)}=\dfrac{5-x}{(2x-1)(x+1)}

. B forgets to change the sign of

-1

第 5 題1 分

正確答案： C

5: C. The value is rounded to the nearest 0.0001, so the lower limit 0.03575 is included and the upper limit 0.03585 is not.

第 6 題1 分

正確答案： A

6: A.

x^{2}+(a-2)x-2a\equiv x^{2}+bx-10

, so

a=5

and

b=a-2=3

第 7 題1 分

正確答案： A

7: A.

8-2k-6=18+3k-6

gives

k=-2

, so

f(1)=2-2-6=-6

第 8 題1 分

正確答案： D

8: D.

p(2)=8+4a-10+2=4

, so

a=1

. The remainder is

p(-1)=-1+1+5+2=7

第 9 題1 分

正確答案： D

9: D. Amount

=P(1+1\%)^{8}\approx 1.0829P

. B compounds yearly and C half-yearly.

第 10 題1 分

正確答案： C

10: C. The first inequality gives

x<3

and the second gives

x\geq -2

. 'And' takes the common part.

第 11 題1 分

正確答案： D

11: D.

a:b=4:3=8:6

and

b:c=6:5

, so

a:c=8:5

第 12 題1 分

正確答案： D

12: D. New

\displaystyle z=\dfrac{(1.2)^{2}}{0.9}z=1.6z

, an increase of 60%.

第 13 題1 分

正確答案： C

13: C.

a_{3}=9

a_{4}=19

a_{5}=37

and

a_{6}=37+2(19)=75

第 14 題1 分

正確答案： A

14: A. The coefficient of

x^{2}

-1

(I) and the vertex is

(-h,\,k)

(II). The

y

-intercept is

k-h^{2}

, which may be negative, so III need not be true.

第 15 題1 分

正確答案： D

15: D.

\displaystyle \dfrac{4}{3}\pi r^{3}=\pi(3)^{2}(8)=72\pi

, so

r^{3}=54

and

r\approx 3.78

cm.

第 16 題1 分

正確答案： C

16: C. Ratio of lengths

\displaystyle =\sqrt[3]{\dfrac{54}{128}}=\dfrac{3}{4}

, so the ratio of areas is

9:16

and the larger base area is

\displaystyle 18\times\dfrac{16}{9}=32

^{2}

第 17 題1 分

正確答案： A

17: A.

\displaystyle \dfrac{135}{360}\pi r^{2}=6\pi

gives

r=4

(I). Arc

\displaystyle =\dfrac{135}{360}(2\pi)(4)=3\pi

(II). Area of

\displaystyle \triangle OAB=\dfrac{1}{2}(4)(4)\sin 135^\circ\approx 5.66

^{2}

, so III is false.

第 18 題1 分

正確答案： C

18: C.

\angle DAB=180^\circ-64^\circ=116^\circ

, so

\angle EAB=58^\circ

\angle AED=\angle EAB=58^\circ

(alt.

\angle

AB/\!/DC

), so

\angle AEC=180^\circ-58^\circ=122^\circ

第 19 題1 分

正確答案： B

19: B.

\angle ABC=90^\circ

(

\angle

in semi-circle), so

\angle ACB=56^\circ

\angle ADB=\angle ACB=56^\circ

(

\angle

s in the same segment).

第 20 題1 分

正確答案： C

20: C.

Q=(-8,\,-3)

and

R=(-8,\,3)

第 21 題1 分

正確答案： C

21: C. Each exterior angle is

24^\circ

, so

\displaystyle n=\dfrac{360^\circ}{24^\circ}=15

第 22 題1 分

正確答案： B

22: B.

\displaystyle \dfrac{\cos\theta}{-\cos\theta}+\tan\theta\cdot\dfrac{1}{\tan\theta}=-1+1=0

第 23 題1 分

正確答案： A

23: A.

A=(-4,\,0)

and

B=(0,\,3)

, so the area is

\displaystyle \dfrac{1}{2}(4)(3)=6

第 24 題1 分

正確答案： A

24: A. The locus is the perpendicular bisector of

AB

: through

(3,\,1)

with slope

\displaystyle \dfrac{2}{3}

, i.e.

2x-3y-3=0

第 25 題1 分

正確答案： D

25: D. Centre

(3,\,-4)

; radius

=\sqrt{9+16+11}=6

; substituting

(0,\,0)

gives

-11<0

, so the origin lies inside.

第 26 題1 分

正確答案： B

26: B. Substituting gives

2x^{2}+2kx+k^{2}-8=0

\Delta=4k^{2}-8(k^{2}-8)=0

, so

k^{2}=16

and

k=\pm 4

第 27 題1 分

正確答案： B

27: B. The 12 ages are 23, 25, 28, 30, 34, 34, 37, 41, 42, 46, 49, 55. Lower quartile = (28 + 30) ÷ 2 = 29; upper quartile = (42 + 46) ÷ 2 = 44; inter-quartile range = 15.

第 28 題1 分

正確答案： C

28: C.

\displaystyle \dfrac{40(25)-15(31)}{25}=\dfrac{535}{25}=21.4

第 29 題1 分

正確答案： D

29: D.

\displaystyle P(\text{sum }8)=\dfrac{5}{36}

\displaystyle P(\text{both odd})=\dfrac{9}{36}

and both happen for (3, 5) and (5, 3). Required probability

\displaystyle =\dfrac{5+9-2}{36}=\dfrac{1}{3}

第 30 題1 分

正確答案： B

30: B. The mean becomes 2m − 3 and the range and standard deviation are doubled, so the variance is multiplied by 4. III is false.

第 31 題1 分

正確答案： A

31: A. L.C.M. of 6, 9 and 4 is 36; take the highest power of each variable:

a^{3}b^{4}c

. C is the H.C.F.

第 32 題1 分

正確答案： A

32: A.

\displaystyle \log_{2}y=3-\dfrac{x}{2}

, so

\displaystyle y=2^{3-\frac{x}{2}}=8\cdot 2^{-\frac{x}{2}}

第 33 題1 分

正確答案： A

33: A. Let

u=3^{x}

3u^{2}-10u+3=0

, so

u=3

\displaystyle \dfrac{1}{3}

and

x=1

-1

. B gives

u

, not

x

第 34 題1 分

正確答案： D

34: D.

\displaystyle z=\dfrac{(a+3i)(1+i)}{2}=\dfrac{(a-3)+(a+3)i}{2}

. Real part 0 gives

a=3

, so

z=3i

and

z^{2}=-9

, a real number.

第 35 題1 分

正確答案： B

35: B. I holds for any arithmetic sequence.

\displaystyle \dfrac{2^{b}}{2^{a}}=\dfrac{2^{c}}{2^{b}}=2^{d}

, so II holds.

b^{2}-a^{2}=d(a+b)

but

c^{2}-b^{2}=d(b+c)

, which are not equal, so III is false.

第 36 題1 分

正確答案： B

36: B. Need

\Delta<0

(k-2)^{2}-16<0

, so

-4<k-2<4

and

-2<k<6

第 37 題1 分

正確答案： B

37: B. The vertices are

(0,\,0)

(6,\,0)

(4,\,2)

and

(0,\,4)

, giving 0, 18, 20 and 16. The greatest value is 20.

第 38 題1 分

正確答案： C

38: C. Let

M

be the mid-point of

AB

and

N

the centre of the base.

\displaystyle \tan\angle VMN=\dfrac{12}{5}

, so the angle is

67.4^\circ

. B is the angle between

VA

and the base.

第 39 題1 分

正確答案： B

39: B.

\angle AOB=90^\circ

, so the circumcentre is the mid-point of

AB

(4,\,3)

, and the orthocentre is

O

. The in-radius is

\displaystyle \dfrac{6+8-10}{2}=2

, so the in-centre is

(2,\,2)

, which lies on

y=x

第 40 題1 分

正確答案： C

40: C.

\angle ACB=\angle PAB=52^\circ

(

\angle

in alt. segment).

\angle ABC=\angle ACB=52^\circ

(base

\angle

s, isos.

\triangle

), so

\angle BAC=180^\circ-104^\circ=76^\circ

第 41 題1 分

正確答案： B

41: B.

C^{14}_{5}-C^{8}_{5}-C^{6}_{1}C^{8}_{4}=2002-56-420=1526

第 42 題1 分

正確答案： D

42: D.

\displaystyle \dfrac{3}{7}+\dfrac{4}{7}\times\dfrac{3}{6}=\dfrac{5}{7}

第 43 題1 分

正確答案： C

43: C.

70=\mu+1.25\sigma

and

56=\mu-0.5\sigma

, so

\sigma=8

and

\mu=60

第 44 題1 分

正確答案： D

44: D. Adding numbers equal to the mean keeps the mean, adds nothing to the sum of squared deviations and increases the count, so the standard deviation falls. Since the mean is 20 and the standard deviation is not 0, the group has numbers both below and above 20, so the range is unchanged.

第 45 題1 分

正確答案： B

45: B. Each term of the second group is

20d

more than the corresponding term of the first group, so the mean increases by

20d

and the standard deviation is unchanged. III contradicts II.