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HKDSE 數學必修部分卷二模擬試卷（多項選擇題）

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試卷結構

甲部30 題 · 30 分
乙部15 題 · 15 分

呢份試卷涵蓋嘅課題

Quadratic Equations in One Unknown
Functions and Graphs
Inequalities and Linear Programming
Exponential and Logarithmic Functions
Arithmetic and Geometric Sequences and Their Summations
Variations
Equations of Straight Lines
Basic Properties of Circles
More about Trigonometry (2D and 3D Problems)
Permutations and Combinations
More about Probability
Measures of Dispersion
More about Graphs of Functions
More about Polynomials
More about Equations
Equations of Circles
Uses and Abuses of Statistics
Further Applications
Loci

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香港中學文憑考試模擬試卷

數學 (必修部分)

Paper 2: Multiple Choice

作答時間： 1 hr 15 min總分： 45

考生須知：

本試卷分為多個部分。
本試卷所有題目均須作答。
請在指定位置作答。
除另有說明外，須清楚列出所有計算步驟。
除另有說明外，數值答案須為準確值或準確至三位有效數字。

由 studify.life 生成 • 16/10/2026

A

題目 1

Solve

x^2-5x+6=0

.(1 分)

x=1

x=6

x=2

x=3

x=-2

x=-3

x=2

only

題目 2

The function

f

is defined by

f(x)=2x-3

. Find

f(4)

.(1 分)

-5

題目 3

Solve

3x+7<16

.(1 分)

x<3

x>3

\displaystyle x<\frac{23}{3}

x\leq 3

頁次：2

題目 4

Simplify

\displaystyle \dfrac{2^3\times 2^4}{2^2}

.(1 分)

2^5

2^9

2^3

2^2

題目 5

The first three terms of an arithmetic sequence are

7, 11, 15

. Find its 10th term.(1 分)

題目 6

A quantity

y

varies directly as

x

. When

x=4

y=18

. Find

y

when

x=10

.(1 分)

7.2

頁次：3

題目 7

The points

A(1,2)

and

B(5,10)

lie on a straight line. Find its gradient.(1 分)

\displaystyle \frac{1}{2}

-2

題目 8

A circle has centre

O

and radius 5 cm. A chord is 8 cm long. What is the perpendicular distance from

O

to the chord?(1 分)

2 cm

3 cm

4 cm

\sqrt{41}

題目 9

In a right-angled triangle, the acute angle

\theta

has opposite side 6 and hypotenuse 10. Find

\sin\theta

.(1 分)

\displaystyle \frac{3}{5}

\displaystyle \frac{4}{5}

\displaystyle \frac{5}{3}

\displaystyle \frac{3}{4}

頁次：4

題目 10

How many different arrangements can be made using all the letters of the word MATH?(1 分)

題目 11

A fair six-sided die is rolled once. What is the probability of obtaining an even number?(1 分)

\displaystyle \frac{1}{6}

\displaystyle \frac{1}{3}

\displaystyle \frac{1}{2}

\displaystyle \frac{2}{3}

題目 12

The values

2,4,4,6

have what population standard deviation?(1 分)

\sqrt{2}

\sqrt{8}

頁次：5

題目 13

Which is the equation of a line with gradient

-3

and

y

-intercept 2?(1 分)

y=3x+2

y=-3x+2

y=-3x-2

y=2x-3

題目 14

f(x)=x^2+1

, which expression is

f(x-2)

?(1 分)

x^2-3

(x-2)^2+1

(x+2)^2+1

x^2-2+1

題目 15

A fair coin is tossed twice. What is the probability of getting exactly one head?(1 分)

\displaystyle \frac{1}{4}

\displaystyle \frac{1}{2}

\displaystyle \frac{3}{4}

頁次：6

題目 16

The data are

3,5,7,9

. What is the range?(1 分)

題目 17

Solve

\log_{10}x=2

.(1 分)

x=20

x=100

x=10

x=0.01

題目 18

Which is a factorisation of

x^2-9

?(1 分)

(x-3)^2

(x+3)^2

(x-3)(x+3)

(x-9)(x+1)

頁次：7

題目 19

The line

y=2x+1

is translated 3 units upwards. What is the equation of its image?(1 分)

y=2x+4

y=2x-2

y=5x+1

y=2(x+3)+1

題目 20

The two roots of

x^2-7x+10=0

are

\alpha

and

\beta

. Find

\alpha+\beta

.(1 分)

-10

-7

題目 21

y

varies inversely as

x

, and

y=6

when

x=2

, find

y

when

x=3

.(1 分)

頁次：8

題目 22

A sector has radius 6 cm and central angle

60^\circ

. Find its arc length.(1 分)

2\pi

6\pi

\pi

12\pi

題目 23

Find the 5th term of the geometric sequence

3,6,12,\ldots

.(1 分)

題目 24

Solve

2(x-1)=3x+4

.(1 分)

x=-6

x=2

x=-2

x=6

頁次：9

題目 25

The circle has equation

(x-2)^2+(y+1)^2=16

. What are its centre and radius?(1 分)

Centre

(2,-1)

, radius 4

Centre

(-2,1)

, radius 4

Centre

(2,-1)

, radius 16

Centre

(2,1)

, radius 16

題目 26

In the diagram,

AB

is a diameter of the circle and

C

lies on the circumference. Find

\angle ACB

.(1 分)

45^\circ

90^\circ

180^\circ

60^\circ

頁次：10

題目 27

Which value is the median of

1, 3, 4, 8, 10

?(1 分)

5.2

題目 28

\tan\theta=1

and

0^\circ<\theta<90^\circ

, find

\theta

.(1 分)

30^\circ

45^\circ

60^\circ

90^\circ

題目 29

Which point lies on the graph

y=x^2

?(1 分)

(2,4)

(2,2)

(4,2)

(-2,-4)

頁次：11

題目 30

A linear programming feasible region is bounded by

x\geq0

y\geq0

x+y\leq6

and

x\leq4

. What is the maximum value of

P=2x+y

?(1 分)

頁次：12

B

題目 31

The roots of

x^2-4x+8=0

are

\alpha

and

\beta

. Find

\alpha^2+\beta^2

.(1 分)

-8

題目 32

Solve

\log_2(x-1)+\log_2(x-3)=3

.(1 分)

x=5

x=-1

x=4

x=3

題目 33

A polynomial

p(x)

leaves remainder 5 when divided by

x-2

, and remainder

-1

when divided by

x+1

. What is the remainder when

p(x)

is divided by

(x-2)(x+1)

?(1 分)

2x+1

-2x+1

2x-1

x+3

頁次：13

題目 34

Solve

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

.(1 分)

x=1\pm\sqrt{2}

x=\pm\sqrt{2}

x=2

only

x=0

題目 35

The sum of the first 20 terms of an arithmetic sequence is 610. Its first term is 2. Find its common difference.(1 分)

\displaystyle \frac{61}{10}

\displaystyle \frac{59}{19}

題目 36

A positive quantity

y

varies directly as

x^2

and inversely as

\sqrt z

. When

x=2

and

z=9

y=8

. Find

y

when

x=3

and

z=4

.(1 分)

\displaystyle \frac{27}{2}

頁次：14

題目 37

A circular sector has radius 10 cm and arc length 6 cm. Find its area.(1 分)

30 cm

^2

60 cm

^2

16 cm

^2

3 cm

^2

題目 38

A vertical tower

AB

stands on horizontal ground. From point

C

, 20 m from its foot

A

, the angle of elevation of its top

B

35^\circ

. Find the height of the tower, to the nearest metre.(1 分)

11 m

14 m

29 m

16 m

頁次：15

題目 39

From 7 distinct people, a committee of 3 is selected. Two particular people, P and Q, must not both be on the committee. How many committees are possible?(1 分)

題目 40

A bag contains 4 red and 3 blue counters. Two counters are drawn at random without replacement. Find the probability that they are of different colours.(1 分)

\displaystyle \frac{4}{7}

\displaystyle \frac{3}{7}

\displaystyle \frac{1}{2}

\displaystyle \frac{2}{7}

題目 41

The scores of a group have mean 50 and population standard deviation 8. Each score

x

is transformed to

y=3x-10

. What are the mean and population standard deviation of the transformed scores?(1 分)

Mean 140, standard deviation 24

Mean 140, standard deviation 14

Mean 40, standard deviation 24

Mean 150, standard deviation 8

頁次：16

題目 42

For

x\ne0

, the graph of

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

has which vertical and horizontal asymptotes?(1 分)

x=1

and

y=3

x=-1

and

y=3

x=1

and

y=2

x=3

and

y=1

題目 43

A circle has centre

(1,2)

and passes through

(4,6)

. Which is its equation?(1 分)

(x-1)^2+(y-2)^2=25

(x+1)^2+(y+2)^2=25

(x-1)^2+(y-2)^2=5

(x-4)^2+(y-6)^2=25

頁次：17

題目 44

The locus of a point

P

satisfying

PA=PB

, where

A(-2,0)

and

B(4,0)

, is which line?(1 分)

x=1

y=0

x=2

y=1

題目 45

A rectangular garden has perimeter 40 m. Its length is

x

m and width is

(20-x)

m. Which value of

x

gives the maximum area?(1 分)

頁次：18

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

The factorisation is

x^2-5x+6=(x-2)(x-3)

. Hence

x=2

x=3

(A1). Option B is correct. A and C do not satisfy the equation; D omits one root.

第 2 題1 分

正確答案： A

Substitute

x=4

f(4)=2(4)-3=5

(A1). Thus A. B omits the subtraction, C adds 3, and D has an incorrect sign.

第 3 題1 分

正確答案： A

Subtract 7 to obtain

3x<9

; dividing by positive 3 gives

x<3

(A1). Thus A. B reverses the direction, C fails to subtract 7 correctly, and D incorrectly includes equality.

第 4 題1 分

正確答案： A

Using index laws, add exponents when multiplying and subtract when dividing:

2^{3+4-2}=2^5

(A1). A is correct. B adds rather than subtracting the denominator exponent; C and D use incorrect exponent arithmetic.

第 5 題1 分

正確答案： B

The common difference is 4. The 10th term is

7+9(4)=43

(A1). B is correct. A uses only eight increments, C uses ten increments, and D uses seven increments.

第 6 題1 分

正確答案： B

Since

y=kx

k=18/4=4.5

. At

x=10

y=4.5(10)=45

(A1). B is correct. A applies inverse scaling, while C and D use incorrect scale factors.

第 7 題1 分

正確答案： A

The gradient is

(10-2)/(5-1)=8/4=2

(A1). Therefore A. B inverts the ratio, C uses only the vertical change, and D gives the wrong sign.

第 8 題1 分

正確答案： B

The perpendicular from the centre bisects the chord, giving a right triangle with hypotenuse 5 and one leg 4. Thus the distance is

\sqrt{5^2-4^2}=3

cm (A1). B is correct. A confuses a radius difference, C uses the half-chord, and D adds rather than subtracts squares.

第 9 題1 分

正確答案： A

By definition,

\sin\theta=\text{opposite}/\text{hypotenuse}=6/10=3/5

(A1). A is correct. B is the adjacent-to-hypotenuse ratio, C is greater than 1, and D uses the wrong denominator.

第 10 題1 分

正確答案： C

The four letters are distinct, so the number of arrangements is

4!=24

(A1). C is correct. A omits a factor, B squares 4, and D counts only the choice of a first letter.

第 11 題1 分

正確答案： C

The even outcomes are 2, 4 and 6, so the probability is

3/6=1/2

(A1). C is correct. A counts only one outcome; B and D use an incorrect number of favourable outcomes.

第 12 題1 分

正確答案： A

The mean is 4. The population variance is

[(2-4)^2+0^2+0^2+(6-4)^2]/4=8/4=2

, so the standard deviation is

\sqrt{2}

(A1). A is correct. B reports the range, C omits division by the population size, and D is not the square root of the variance.

第 13 題1 分

正確答案： B

In slope-intercept form

y=mx+c

m=-3

and

c=2

, hence

y=-3x+2

(A1). B is correct. A has the wrong gradient, C the wrong intercept, and D interchanges the two constants.

第 14 題1 分

正確答案： B

Replace every input

x

x-2

f(x-2)=(x-2)^2+1

(A1). B is correct. A and D incorrectly expand or substitute only part of the expression; C shifts in the opposite direction.

第 15 題1 分

正確答案： B

The four equally likely outcomes are HH, HT, TH and TT. Exactly one head occurs in HT and TH, so the probability is

2/4=1/2

(A1). B is correct. The other options miscount favourable outcomes or assert certainty.

第 16 題1 分

正確答案： B

Range = maximum minus minimum

=9-3=6

(A1). B is correct. A is half the range, C is the maximum, and D is the sum of the extremes.

第 17 題1 分

正確答案： B

By the definition of logarithm,

x=10^2=100

(A1). B is correct. A multiplies instead of exponentiating, C uses exponent 1, and D uses exponent

-2

第 18 題1 分

正確答案： C

Use the difference of two squares:

x^2-9=x^2-3^2=(x-3)(x+3)

(A1). C is correct. A and B expand to trinomials with middle terms; D expands to a different polynomial.

第 19 題1 分

正確答案： A

A vertical translation upwards by 3 adds 3 to every

y

-value:

y=2x+1+3=2x+4

(A1). A is correct. B shifts down, C changes the gradient, and D represents a horizontal shift.

第 20 題1 分

正確答案： C

For

ax^2+bx+c=0

, the sum of roots is

-b/a

. Here it is

-(-7)/1=7

(A1). C is correct. A and D confuse product and sum; B uses the coefficient with the wrong sign.

第 21 題1 分

正確答案： A

For inverse variation,

xy=k

. From the given values

k=12

, so when

x=3

y=12/3=4

(A1). A is correct. B applies direct variation, while C and D do not preserve the product.

第 22 題1 分

正確答案： A

Arc length

=(60/360)(2\pi\times6)=2\pi

cm (A1). A is correct. B is a semicircle length, C is half the correct arc length, and D is the full circumference.

第 23 題1 分

正確答案： C

The common ratio is 2. The terms are

3,6,12,24,48

, so the 5th term is 48 (A1). C is correct. A is the 4th term, B does not follow the ratio, and D applies an extra doubling.

第 24 題1 分

正確答案： A

Expand and collect:

2x-2=3x+4

, hence

-6=x

(A1). A is correct. The other options result from sign or transposition errors.

第 25 題1 分

正確答案： A

Compare with

(x-h)^2+(y-k)^2=r^2

. The centre is

(2,-1)

and

r=\sqrt{16}=4

(A1). A is correct. B reverses both centre signs, C mistakes

r^2

for

r

, and D has the wrong

y

-coordinate and radius.

第 26 題1 分

正確答案： B

The angle subtended by a diameter at the circumference is a right angle, so

\angle ACB=90^\circ

(A1). B is correct. The other choices contradict the angle-in-a-semicircle theorem.

第 27 題1 分

正確答案： B

The ordered data set has five values; its middle (third) value is 4 (A1). B is correct. A and D are adjacent order statistics, while C is the mean rather than the median.

第 28 題1 分

正確答案： B

In the acute range,

\tan45^\circ=1

, hence

\theta=45^\circ

(A1). B is correct. The tangent values at 30° and 60° differ from 1, and tangent is undefined at 90°.

第 29 題1 分

正確答案： A

For

x=2

y=x^2=4

, so

(2,4)

lies on the graph (A1). A is correct. B and C do not satisfy

y=x^2

, and the square of

-2

is positive, not

-4

第 30 題1 分

正確答案： B

The vertices are

(0,0),(4,0),(4,2),(0,6)

. The corresponding

P

values are

0,8,10,6

. Thus the maximum is 10 at

(4,2)

(A1). B is correct; the other options are values at other vertices or an incorrect evaluation.

第 31 題1 分

正確答案： D

By Vieta's relations,

\alpha+\beta=4

and

\alpha\beta=8

. Hence

\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=16-16=0

(A1). D is correct. A gives the product, B omits the product correction, and C incorrectly negates the final result.

第 32 題1 分

正確答案： A

The domain requires

x>3

. Combine logs:

(x-1)(x-3)=2^3=8

, so

x^2-4x-5=0

, giving

x=5

-1

. Only

x=5

satisfies the domain (A1). A is correct. B is extraneous; C and D do not satisfy the original equation.

第 33 題1 分

正確答案： A

The remainder is linear, say

ax+b

. It satisfies

2a+b=5

and

-a+b=-1

. Subtracting gives

3a=6

, so

a=2

b=1

. The remainder is

2x+1

(A1). A is correct; the other options fail at least one of the two required values.

第 34 題1 分

正確答案： A

For

x\ne\pm1

, combine fractions:

2x/(x^2-1)=1

, giving

x^2-2x-1=0

. Thus

x=1\pm\sqrt2

, neither excluded (A1). A is correct. B solves a different quadratic, and C and D do not satisfy the equation.

第 35 題1 分

正確答案： B

Use

\displaystyle S_{20}=\frac{20}{2}[2(2)+19d]

. Thus

610=10(4+19d)

, so

19d=57

and

d=3

(A1). B is correct. A confuses first term with difference; C and D result from incorrect division or term counts.

第 36 題1 分

正確答案： B

Write

y=kx^2/\sqrt z

. From the initial values,

8=k(4)/3

, so

k=6

. At

(x,z)=(3,4)

y=6(9)/2=27

(A1). B is correct. A omits the square scaling, C reverses or misapplies the square-root factor, and D uses an incorrect ratio.

第 37 題1 分

正確答案： A

For a sector,

A=\frac12 r\ell

when radius is

r

and arc length is

\ell

. Hence

A=\frac12(10)(6)=30

^2

(A1). A is correct. B omits the factor one-half; C and D use unrelated combinations of the given values.

第 38 題1 分

正確答案： B

Use

\tan35^\circ=h/20

, so

h=20\tan35^\circ\approx14.0

m (A1). B is correct. A uses sine, C uses the reciprocal tangent ratio, and D is an inaccurate rounding or estimate.

第 39 題1 分

正確答案： B

There are

\binom73=35

committees in total. Those containing both P and Q are formed by choosing one of the other five, giving 5. Therefore

35-5=30

(A1). B is correct. A subtracts the wrong number; C ignores the restriction, and D undercounts valid committees.

第 40 題1 分

正確答案： A

The probability is

P(RB)+P(BR)=(4/7)(3/6)+(3/7)(4/6)=24/42=4/7

(A1). A is correct. B counts only one order; C assumes an incorrect equally likely split; D fails to account for both orders.

第 41 題1 分

正確答案： A

The mean transforms as

3(50)-10=140

. Subtracting a constant does not change dispersion, and multiplication by 3 multiplies the standard deviation by 3, giving 24 (A1). A is correct. B incorrectly subtracts 10 from the standard deviation; C transforms the mean wrongly; D ignores the scaling.

第 42 題1 分

正確答案： A

The denominator is zero at

x=1

, giving the vertical asymptote; as

x

becomes large, the fraction tends to 0, giving horizontal asymptote

y=3

(A1). A is correct. The other options confuse numerator, denominator shift, or vertical translation.

第 43 題1 分

正確答案： A

The squared radius is

(4-1)^2+(6-2)^2=9+16=25

. Hence

(x-1)^2+(y-2)^2=25

(A1). A is correct. B has the wrong centre, C uses the radius rather than its square, and D puts the centre at the point on the circumference.

第 44 題1 分

正確答案： A

The locus of points equidistant from A and B is the perpendicular bisector of

AB

. The segment has midpoint

(({-2+4})/2,0)=(1,0)

and is horizontal, so its perpendicular bisector is

x=1

(A1). A is correct; the other lines are not the perpendicular bisector.

第 45 題1 分

正確答案： B

The area is

A=x(20-x)=-x^2+20x=-(x-10)^2+100

, which is maximized when

x=10

(A1). B is correct. The remaining options do not give the vertex;

x=20

gives zero area and the choices 5 and 15 give smaller equal areas.