Post on 19-Mar-2018
HKCEE Mathematics Multiple Choices
CONTENTS
1. Percentages P.1-5
2. Estimation and Error P.6
3. Algebraic Expression
3.1 LCM and HCF P.7-8
3.2 Factorization P.9-12
3.3 Algebraic Simplification P.13-18
4. Polynomials
4.1 Function and Graph P.19-21
4.2 Remainder and Factor Theorem P.22-24
4.3 Identities P.25-26
5. Equations
5.1 Simultaneous Equations P.27-30
5.2 Quadratic Equations and Graphs P.31-40
6. Indices, Logarithms and Surds
6.1 Indices P.41-43
6.2 Logarithms P.44-45
6.3 Surds P.46-47
7. Ratio, Ratio and Variation P.48-53
8. Inequality
8.1 Linear Inequality P.54-55
8.2 Linear Programming P.56-61
9. Decimal, Binary and Hexadecimal Number P.62
10. Mensurations P.63-75
11. Plane Geometry
11.1 Circles P.76-103
11.2 Similar and Congruent Triangles P.104-105
11.3 Polygons P.106-122
11.4 Centers in a triangle P.123
11.5 Rotational and Reflectional Symmetries P.124-125
12. Trigonometry
12.1 Trigonometric Equations P.126-133
12.2 Trigonometric Graph P.134-137
12.3 2D P.138-164
12.4 3D P.165-173
13. Coordination Geometry
13.1 Straight lines P.174-183
13.2 Circles P.184-190
13.3 Polar Coordinates P.191
14. Sequence P.192-197
15. Probability P.198-203
16. Statistics P.204-213
Multiple-Choices Answers (By Topics) P.214-218
Multiple-Choices Answers (By Years) P.219
1
1. 1990/II/14
Find the amount (correct to the nearest dollar) of $10 000 at 12% p.a., compounded monthly, for 2 years.
A. 10 201 B. 12 400 C. 12 544 D. 12 697 E. 151 786
2. 1990/II/15
If a flat is sold for $720 000, the gain is 20%. Find the percentage loss if the flat is sold for $540 000.
A. 5% B. 64
1% C. 10% D. 11
9
1% E. 33
3
1%
3. 1990/II/42
If A is 30% greater than B and B is 30% less than C, then
A. A is 9% less than C B. C is 9% less than A C. A = C
D. A is 9% greater than C E. C is 9% greater than A
4. 1991/II/11
A blanket loses 10% of its length and 8% of its width after washing. The percentage loss in area is
A. 18.8% B. 18% C. 17.2% D. 9% E. 8%
5. 1991/II/15
A man borrows $10 000 from a bank at 12% per annum compounded monthly. He repays the bank $2000 at the
end of each month. How much does he still owe the bank just after the second repayment?
A. $6 181 B. $6 200 C. $6 201 D. $8 304 E. $8 400
6. 1991/II/43
P sold an article to Q at a profit of 25%. Q sold it to R also at a profit of 25%. If Q gained $500, how much
did P gain(in $)?
A. 250 B. 320 C. 333 D. 400 E. 500
7. 1992/II/12
A sum of $10000 is deposited at 4% p.a., compounded yearly. Find the interest earned in the second year.
A. $16 B. $400 C. $416 D. $800 E. $816
8. 1992/II/44
By selling an article at 10% discount off the marked price, a shop still makes 20% profit. If the cost price
of the article is $19800, then the marked price is
A. $21600 B. $26136 C. $26400 D. $27225 E. $27500
HKCEE MATHEMATICS | 1 Percentages | P.1
2
9. 1993/II/18
A merchant marks his goods 25% above the cost. He allows 10% discount on the marked price for a cash sale.
Find the percentage profit the merchant makes for a cash sale.
A. 12.5% B. 15% C. 22.5% D. 35% E. 37.5%
10. 1993/II/43
Which of the following gives the compound interest on $10000 at 6% p.a. for one year, compounded monthly?
A. 1212
06.010000$ ×× B. ( )106.110000$ 12
− C.
12
12
06.0110000$
+
D.
−
+ 1
12
06.0110000$
12
E.
−
+ 1
12
6.0110000$
12
11. 1993/II/44
Originally3
2of the students in a class failed in an examination. After taking a re-examination, 40% of the failed
students passed. Find the total pass percentage of the class.
A. %3
226 B. %
3
133 C. 40% D. 60% E. %
3
173
12. 1994/II/9
Mr. Chan bought a car for $143 900. If the value of the car goes down by 10% each year, find its value at the
end of the third year. (Give your answer correct to the nearest hundred dollars.)
A. $94 400 B. $100 700 C. $104 900 D. $115 100 E. $116 600
13. 1994/II/10
A wholesaler sells an article to a retailer at a profit of 20%. The retailer sells it to a customer for $3 600 at a
profit of $720. Find the original cost of the article to the wholesaler.
A. $2 304 B. $2 400 C. $2 880 D. $3 000 E. $3 456
14. 1995/II/13
Find the interest on P$ at %r p.a. for n years, compounded half-yearly.
A. ( ) PrPn
$%21$ −+ B. ( ) PrPn
$%1$ −+ C. ( ) PrPn
$%1$2 −+
D. Pr
P
n
$%2
1$ −
+ E. Pr
P
n
$%2
1$
2
−
+
HKCEE MATHEMATICS | 1 Percentages | P.2
3
15. 1995/II/44
The marked price of a toy is $120 and the percentage profit is 60%. If the toy is sold at a discount of 20%, the
profit is
A. $14.40 B. $21.00 C. $24.00 D. $33.60 E. $48.00
16. 1996/II/14
Shop A offers a 10% discount on a book marked at $P. Shop B offers a 15% discount on the same book marked
at $Q. If the selling price of the book is the same in both shops, express Q in terms of P.
A. Q = P + 5 B. Q = 17
18P C. Q =
20
21P D. Q =
21
20P E. Q =
18
17P
17. 1996/II/43
The length of a rectangle is decreased by 20%. If the area remains unchanged, find the percentage increase of its
width.
A. 1
1 %4
B. 1
12 %2
C. 2
16 %3
D. 20 % E. 25 %
18. 1997/II/10
There are 1 200 students in a school, of which 640 are boys and 560 are girls. If 55% of the boys and 40% of the
girls wear glasses, what percentage of students in the school wear glasses?
A. 47% B. 47.5% C. 48% D. 52% E. 53%
19. 1997/II/38
Find the interest on $10 000 at 16% per annum for 2 years, compounded half-yearly. Give the answer correct to
the nearest dollar.
A. $1 664 B. $3 456 C. $3 605 D. $7 424 E. $8 106
20. 1998/II/14
A man bought a box of 200 apples for $ 500. 10 of the apples were rotten and the rest were sold at $ 4 each. Find
his percentage profit correct to 2 significant figures.
A. 34% B. 38% C. 52% D. 57% E. 60%
21. 1999/II/11
In a class, students study either History or Geography, but not both. If the number of students studying Geography
is 50% more than those studying History, what is the percentage of students studying History?
A. 25% B. 331
3% C. 40% D. 60% E. 66
2
3%
HKCEE MATHEMATICS | 1 Percentages | P.3
4
22. 2000/II/14
A man bought two books at $30 and $70 respectively. He sold the first one at a profit of 20% and the second
one at a loss of 10%. On the whole, he
A. lost 1% B. lost 10% C. gained 1% D. gained 10% E. gained 13%
23. 2001/II/16
A bank offers loans at an interest rate of 18% per annum, compounded monthly. A man took a loan of $20 000
and repays the bank in monthly installments of $ 4 000. Find the outstanding balance after his first installment.
A. $16 000 B. $16 240 C. $16 300 D. $18 880 E. $19 600
24. 2001/II/27
40% of the students in a class failed in a test. They had to sit for another test in which 70% of them failed
again. Find the percentage of students who failed in both tests.
A. 10% B. 12% C. 18% D. 28% E. 30%
25. 2002/II/12
The simple interest on a sum of money at r% p.a. for 4 years is equal to the compound interest on the same
amount at 4% p.a. for 4 years compounded half-yearly. The value of r, correct to 2 significant figures, is
A. 2.1 B. 4.2 C. 4.3 D. 9.2
26. 2002/II/14
The cost price of a toy is $100 and the marked price is $140. If the toy is sold at 10% discount of the marked
price, the profit is
A. $26 B. $30 C. $36 D. $50
27. 2003/II/11
John’s daily working hours have increased from 8 hours to 10 hours but his hourly pay has decreased by 25%.
Find the percentage change in John’s daily income.
A. A decrease of 6.67% B. A decrease of 6.25% C. 0% D. An increase of 6.67%
28. 2003/II/12
A sum of $8 000 is deposited at 1% p.a., compounded yearly. Find the interest earned after 4 years. Give the
answer correct to the nearest dollar.
A. $ 303 B. $ 320 C. $ 324 D. $ 325
29. 2004/II/12
The marked price of a book is 20% above the cost. If the book is sold at a 10% discount on the marked price, then
the percentage profit is
A. 2% B. 8% C. 10% D. 18%
HKCEE MATHEMATICS | 1 Percentages | P.4
5
30. 2005/II/12
Peter sold two flats for $ 999 999 each. He lost 10% on one and gained 10% on the other. After the two
transactions, Peter
A. gained $ 10 101 B. gained $ 20 202 C. lost $ 10 101 D. lost $ 20 202
31. 2006/II/10
The marked price of a car is 50% higher than the cost. If the car is sold at a 20% discount on the marked price,
then the percentage profit is
A. 10% B. 20% C. 30% D. 40%
32. 2006/II/11
A sum of $ 14 000 is deposited at 4% per annum for 5 years, compounded yearly. Find the interest correct to
the nearest dollar.
A. $ 2 378 B. $ 2 800 C. $ 3 033 D. $ 3 034
33. 2007/II/10
If the bus fare is increased from $ 4 to $ 5, then the percentage increase of the fare is
A. 20% B. 25% C. 80% D. 125%
34. 2007/II/11
A sum of $ 30 000 is deposited at an interest rate of 12% per annum for 4 years, compounded monthly.
Find the amount correct to the nearest dollar.
A. $ 44 400 B. $ 47 206 C. $ 48 141 D. $ 48 367
35. 2008/II/12
The marked price of a bag is $ 900. If the bag is sold at the marked price, then the percentage profit is 50%.
If the bag is sold at a 20% discount on the marked price, then the profit is
A. $ 120 B. $ 180 C. $ 210 D. $ 270
HKCEE MATHEMATICS | 1 Percentages | P.5
6
1. 1995/II/1
Round off the number 0.044449 to 3 significant figures.
A. 0.04 B. 0.044 C. 0.045 D. 0.0444 E. 0.0445
2. 1996/II/1
Evaluate 1.15 ÷ 15 correct to 3 significant figures.
A. 0.076 B. 0.077 C. 0.0766 D. 0.0767 E. 0.076
3. 1997/II/1
Express π2 as a decimal correct to 3 significant figures.
A. 9.86 B. 9.87 C. 9.88 D. 9.860 E. 9.870
4. 2003/II/9
If 8452.08448.0 << a , which of the following must be true?
A. 9.0=a (correct to 1 sig.t figure) B. 85.0=a (correct to 2 sig. figures)
C. 845.0=a (correct to 3 sig. figures) D. 8450.0=a (correct to 4 sig. figures)
5. 2007/II/12
Express 2007 as a decimal correct to 5 significant figures.
A. 44.790 B. 44.799 C. 44.79955 D. 44.800
6. 2008/II/17
0.0498765 =
A. 0.050(correct to 2 decimal places). B. 0.050(correct to 3 significant figures).
C. 0.0499(correct to 4 decimal places). D. 0.0499(correct to 5 significant figures).
HKCEE MATHEMATICS | 2 Estimation and Error | P.1
7
1. 1990/II/37
The H.C.F. and L.C.M. of three expressions are xyz2 and x
3y
5z
4 respectively. If two of the expressions are
x2y
3z
3 and x
3yz
2, find the third expression.
A. x2y
3z
3 B. x
2y
5z
3 C. xy
3z
3 D. xy
5z
4 E. xy
3z
4
2. 1991/II/6
The L.C.M. of x, 2x2, 3x
3, 4x
4, 5x
5 is
A. x B. 5x5 C. 60x
5 . D. 120x
5 E. 120x
15
3. 1992/II/40
The L.C.M. of P and Q is 2312 cab . The L.C.M. of X , Y and Z is cba3230 . What is the L.C.M. of P , Q , X , Y
and Z?
A. 363360 cba B. 23260 cba C. 2360 cab D. cba326 E. cab
36
4. 1993/II/11
Find the H.C.F. and L.C.M. of cab2 and 3
abc .
H.C.F. L.C.M.
A. a 432cba
B. abc 32cab
C. abc 432cba
D. 32cab abc
E. 432cba abc
5. 1994/II/3
The L.C.M. of (x – 1)2, x
2 – 1 and x
3 – 1 is
A. x – 1 B. (x – 1)4(x + 1)(x
2 + x + 1) C. (x – 1)
2(x + 1) (x
2 + x + 1)
D. (x – 1)2(x + 1) (x
2 – x + 1) E. (x – 1)(x + 1) (x
2 + x + 1)
HKCEE MATHEMATICS | 3.1 Algebraic Expression - LCM and HCF | P.1
8
6. 1995/II/6
The L.C.M. of xx −3 and 14
−x is
A. 1−x B. ( )( )11 +− xx C. ( )( )( )111 2 ++− xxxx
D. ( )( )( )( )1111 22 ++++− xxxxx E. ( ) ( ) ( )111 222 ++− xxxx
7. 1996/II/3
Find the L.C.M. of 4x2yz and 6xy
3.
A. 2xy B. 12x2y
3 C. 12x
2y
3z D. 24x
2y
3z E. 24x
3y
4z
8. 1996/II/37
m and n are multiples of 3 and 4 respectively. Which of the following must be true?
I. mn is a multiple of 12.
II. The H.C.F. of m and n is even.
III. The L.C.M. of m and n is even.
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
9. 2003/II/38
The L.C.M. of 2210xy and yzx 230 is
A. xy30 B. xyz70 C. zyx 22210 D. zyx 33630
10. 2004/II/38
The L.C.M. of b−2 , 24 b− and 38 b− is
A. )44)(2)(2( 2bbbb +−+− B. )44)(2)(2( 2bbbb +++−
C. )24)(2)(2( 2bbbb +−+− D. )24)(2)(2( 2bbbb +++−
11. 2005/II/38
The H.C.F. of x2(x + 1)(x + 2) and x(x + 1)
3 is
A. x(x + 1) B. x(x + 1)(x + 2) C. x2(x + 1)
3 D. x
2(x + 1)
3(x + 2)
HKCEE MATHEMATICS | 3.1 Algebraic Expression - LCM and HCF | P.2
9
1. 1990/II/7
a3 + 8a
−3 =
A. (a − a
2)(a
2 + 2 +
2
4
a) B. (a −
a2
1)(a
2 + 1 +
24
1
a) C. (a +
a2
1)(a
2 −
2
1 +
24
1
a)
D. (a + a
2)(a
2 − 4 +
2
4
a) E. (a +
a
2)(a
2 − 2 +
2
4
a)
2. 1993/II/3
Simplify ( )( )1313 22 +++− xxxx
A. 14 +x B. 124 +− xx C. 124 ++ xx
D. 1323 24−−− xxx E. 13323 234 ++−+ xxxx
3. 1994/II/35
Factorize bababa +−+− 22 2 .
A. ( )( )1−−− baba B. ( )( )1+−− baba C. ( )( )1−+− baba
D. ( )( )1+−+ baba E. ( )21−− ba
4. 1995/II/36
Factorize 11 3072 −+−−
nnnaaa .
A. ( )( )526 +− aa n B. ( )( )526 −+ aaa n C. ( )( )526 +− aaa n
D. ( )( )5261 −+− aaa n E. ( )( )5261 +−− aaa n
5. 1997/II/4
9 – a2 – b
2 + 2ab =
A. (3 – a – b)(3 – a + b) B. (3 – a – b)(3 + a – b) C. (3 – a – b)(3 + a + b)
D. (3 – a + b)(3 + a – b) E. (3 – a + b)(3 + a + b)
6. 1998/II/8
Factorize x2 – y
2 + 2x + 1.
A. (x + y + 1)(x + y – 1) B. (x + y + 1)(x – y + 1) C. (x + y – 1)(x – y + 1)
D. (x + y – 1)(x – y – 1) E. (x – y + 1)(x – y – 1)
7. 1999/II/2
x2 – y
2 – x + y =
A. (x – y)(x – y – 1) B. (x – y)(x + y – 1) C. (x – y)(x + y + 1)
D. (x + y)(x – y – 1) E. (x + y)(x – y + 1)
HKCEE MATHEMATICS | 3.2 Algebraic Expression - Factorization | P.1
10
8. 1999/II/40
2 2
2 1
1 2 3
x
x x x
−− =
− − −
A. 2 2 5
( 1)( 1)( 3)
x x
x x x
− + +
− + + B.
2 2 7
( 1)( 1)( 3)
x x
x x x
− + +
− + + C.
2 5
( 3)( 1)( 1)
x
x x x
− −
− − +
D. 2 5
( 3)( 1)( 1)
x
x x x
−
− − + E.
2 4 7
( 3)( 1)( 1)
x x
x x x
− + −
− − +
9. 2000/II/2
Factorize yxyxx +−−2 .
A. )1)(( −− xyx B. )1)(( +− xyx C. )1)(( −+ xyx D. ))(1( yxx +− E. ))(1( xyx −+
10. 2000/II/37
Simplify 22
2
ba
ab
ab
b
ba
a
−+
−+
+.
A. ba
ba
−
+ B.
ba
ba
+
−− C.
22
22 4
ba
abba
−
++− D.
22
22
ba
ba
−
+ E. 1
11. 2001/II/2
=−+− )32)(132( 2 xxx
A. 2356 23 +−− xxx B. 29136 23 −−− xxx C. 29136 23 +−+− xxx
D. 2356 23 +−−− xxx E. 2956 23 +−−− xxx
12. 2001/II/47
=+
−+
−+
−
1
1
54
12 x
x
xx
x
A. )5)(1(
632
++
−+
xx
xx B.
)5)(1(
452
++
−+
xx
xx C.
)5)(1(
)1)(4(
++
−+
xx
xx D.
)5)(1(
)4)(1(
−+
−−
xx
xx E.
)5)(1(
)6)(1(
−+
−−
xx
xx
13. 2003/II/37
=−
−−+ 2
2
6
102 xxx
A. 3
2
+x B.
3
2
+
−
x C.
)2)(3(
213
−+
−
xx
x D.
)2)(3(
216
−+
−
xx
x
HKCEE MATHEMATICS | 3.2 Algebraic Expression - Factorization | P.2
11
14. 2003/II/39
=−3
3 27
xx
A. )9
6)(3
(2
2
xx
xx +−+ B. )
93)(
3(
2
2
xx
xx +−+ C. )
96)(
3(
2
2
xx
xx ++− D. )
93)(
3(
2
2
xx
xx ++−
15. 2004/II/37
=
−
−
x
y
y
x
yx
94
23
A. yx 32
1
−
B. yx 32
1
+ C.
yx 32
1
−
− D.
yx 32
1
+
−
16. 2006/II/3
1 1
1 1x x− =
+ −
A. 2
2
1 x−
B. 2
2
1x −
C. 2
2
1
x
x−
D. 2
2
1
x
x −
17. 2006/II/4
pr + qr – ps – qs =
A. (p + q)(r – s) B. (p + q)(s – r) C. (p – q)(r – s) D. (p – q)(s – r)
18. 2007/II/2
1 1
3 3n n−
+ −=
A. 2
6
9 n− B.
2
6
9n − C.
2
2
9
n
n− D.
2
2
9
n
n −
19. 2007/II/3
(x + x)(y + y + y) =
A. 6xy B. 2x + 3y C. x2y
3 D. 6x
2y
3
20. 2008/II/3
1
1 1
k
k k
−− =
− −
A. 1 B. 1
1
k
k
+
− C.
1
1
k
k
+
− D.
2
2
1
1
k
k
+
−
HKCEE MATHEMATICS | 3.2 Algebraic Expression - Factorization | P.3
12
21. 2008/II/4
(2x2 – 3x + 1) – 2(x
2 + 2x – 1) =
A. x – 1 B. –7x + 3 C. 4x2 + x – 1 D. 4x
2 – 7x + 3
22. 2008/II/5
Which of the following must have x + y as a factor?
I. x2 – y
2
II. x2 + y
2
III. x(x + y) – x – y
A. I only B. II only C. I and III only D. II and III only
HKCEE MATHEMATICS | 3.2 Algebraic Expression - Factorization | P.4
13
1. 1990/II/2
=
−
+−
+
−−
yx
yx
yx
yx
1
1
A. yx
xy
+
− B.
yx
yx
+
− C.
y
x D. x + y E. x − y
2. 1990/II/3
If x =ba
ab
−
+1, then b =
A. xa
ax
+
−1 B.
xa
ax
−
−1 C.
xa
ax
+
−1 D.
xa
ax
+
−1 E.
xa
ax
−
+1
3. 1991/II/2
21
1
x− −
2)1(
1
x+ =
A. )1)(1(
222 xx +−
B. )1)(1(
222
2
xx
x
+− C.
22
2
)1)(1(
2
xx
x
+− D.
2)1)(1(
2
xx +− E.
2)1)(1(
2
xx
x
+−
4. 1991/II/5
=+
+
yx
yx
11
1122
A. 2
1
x+
2
1
y B.
2
1
x+
xy
1+
2
1
y C.
2
1
x+
xy
2+
2
1
y D.
2
1
x−
xy
2+
2
1
y E.
2
1
x−
xy
1+
2
1
y
5. 1992/II/1
=+ba
11
A. ab
ba + B.
ba
ab
+ C.
ab
1 D.
ba +
2 E.
ba +
1
6. 1992/II/2
If b
a−
−=1
11 , then =b
A. a−
−1
11 B.
a+−
1
11 C.
a−+
1
11 D.
a++
1
11 E.
a−+−
1
11
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.1
14
7. 1992/II/7
If kd
c
b
a== and dcba , , , are positive, then which of the following must be true?
A. kdb
ca=
+
+ B. kcdab == C. kbdac == D. kca == E. k
bd
ac=
8. 1992/II/43
If the price of an orange rises by $1, then 5 fewer oranges could be bought for $100. Which of the following
equations gives the original price x$ of an orange?
A. 51
100=
+x B. 5
100
1
100=−
+ xx C. 5
1
100100=
+−
xx D. 5
100
1
100=−
− xx E. 5
1
100100=
−−
xx
9. 1993/II/2
If ( )[ ]dnan
s 122
−+= , then d =
A. ( )( )1
2
−
−
nn
ans B.
( )1
2
−
−
n
ans C. ( )1−nn
s D. ( )1−
−
na
nas E.
( )( )1
4
−
−
nn
ans
10. 1994/II/2
If 2
12
+
−=
x
xy , then =x
A. 2
31 y+ B.
y
y
+
+
2
21 C.
y
y
−
+
2
21 D.
y
y
+
−
2
21 E.
y
y
−
−
2
21
11. 1994/II/36
=−
−
y
x
x
y
yx
4
12
A. xy −2 B. xy +2 C. xy −2
1 D.
xy +2
1 E.
xy −4
1
12. 1995/II/2
If 1=+
xy
yx, then y =
A. x
x−1 B.
x
x 1− C.
x
x
−1 D.
1−x
x E.
x
x
+
−
1
1
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.2
15
13. 1995/II/37
Simplify
x
y
y
x
y
x
x
y
−
−
− 11
.
A. yx
yx
+
− B.
yx
yx
+
−− C.
yx
yx
−
+ D.
yx
yx
−
+− E. −1
14. 1996/II/4
If A = 2πr2 + 2πrh, then h =
A. A – r B. A
r C.
2
Ar
rπ− D.
2
Ar
rπ− E. 22
2
Ar
rπ
π−
15. 1996/II/36
Simplify 2
1 1 3 1
1 1 1
x
x x x
−+ +
− + −.
A. 1
1 x− B.
1
1 x+ C.
1
1 x−
+ D.
2
3 1
1
x
x
+
− E.
2
1 5
1
x
x
−
−
16. 1997/II/3
If a x c
b x d
+=
+ (c ≠ d), then x =
A. c a
d b− B.
a b
c d
−
−
C. b a
c d
−
−
D. ad bc
c d
−
−
E. bc ad
c d
−
−
17. 1997/II/28
Simplify 2 2
4 3
4 2x x x−
− − −
A. 1
( 1)( 2)x x+ + B.
1
( 1)( 2)x x+ − C.
1
( 1)( 2)x x− −
D. 10
( 1)( 2)( 2)
x
x x x
+
+ − + E.
10
( 1)( 2)( 2)
x
x x x
−
− − +
18. 1998/II/1
If ( 3)
3
y zx
z
−= , then z =
A. 3
3x y−
B. 3
3x y
−
−
C. 3
3
y
x y−
D. 3
3
y
x y
−
−
E. 3
3
x y
y
−
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.3
16
19. 1998/II/16
If 0a c
b d= ≠ , which of the following must be true?
I. a b
c d= II.
a b c d
b d
+ += III.
a b c d
b d
− −=
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
20. 1998/II/37
Let a and b be two consecutive positive integers. Which of the following must be true?
I. a + b is odd II. ab is odd III. a2 + b2 is odd
A. III only B. I and II only C. I and III only D. II and III only E. I, II and III
21. 1998/II/39
2 2
2 3
1 2x x x− =
− − −
A. 1
( 1)( 2)x x
−
− −
B. 1
( 1)( 2)x x
−
+ − C.
1
( 1)( 2)x x
−
+ +
D. 1
( 1)( 1)( 2)x x x
−
− + − E.
7
( 1)( 1)( 2)
x
x x x
− −
− + −
22. 1999/II/3
If 1
1
ba
b
+=
−, then b =
A. 1
2
a − B.
1
2
a
a
− C.
1
1
a
a
+
− D.
1
1
a
a
−
+ E.
1
1
a
a
−
+
23. 1999/II/14
Let m be a positive integer. Which of the following must be true?
I. m2 is even.
II. m(m + 1) is even.
III. m(m + 2) is even.
A. I only B. II only C. III only D. I and III only E. II and III only
24. 2000/II/1
If )(2
bah
A += , then =b
A. ahA −2 B. )(2
aAh
− C. h
aA −2 D.
h
Aa
2− E. a
h
A−
2
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.4
17
25. 2001/II/1
If b
a+
−=1
12 , then =b
A. 2
1
−
−
a
a B.
2
1
−
−
a
a C.
2
1
−
+
a
a D.
2
3
−
−−
a
a E.
a
a−1
26. 2001/II/13
A piece of wire of length 36 cm is cut into two parts. One part, x cm long, is bent into a square and the other
part is bent into a circle. If the length of a side of the square is equal to the radius of the circle, which of the
following equations can be used to find x?
A. 36 4
2
xx
π
−= B.
36
2
xx
π
−= C.
36 4
4 2
x x
π
−= D.
36
4
x x
π
−= E.
36
4 2
x x
π
−=
27. 2001/II/28
If 3
2
2
3=
−
+
yx
yx, then =
+
−
yx
yx
A. 6
5− B.
5
3− C.
5
3 D.
4
3 E.
6
5
28. 2002/II/1
If 1 1
x a
x a=
+ −, then x =
A. a B. 2
1
a
a−
C. 1 2
a
a+ D.
1 2
a
a−
29. 2002/II/37
=
−
−
xx
x
1
21
A. 1
3
−
−
x
x B.
1
32
2
−
−
x
x C.
1
12
2
−
+
x
x D.
1
12
2
−
+−
x
x
30. 2003/II/3
If 2
1
−
−=
b
ba , then =b
A. 1
12
−
−
a
a B.
1
12
+
−
a
a C.
1
1
−a D.
1
1
+a
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.5
18
31. 2004/II/2
If y
xyx
2
2−= , then =y
A. x
x
21
2
− B.
12
2
−x
x C.
x
x
2
21− D.
x
x
2
12 −
32. 2005/II/2
If a = 1 – 2b, then b =
A. 1
2
a − B.
1
2
a + C.
1
2
a− − D.
1
2
a−
33. 2006/II/2
If 2x – 5y = 7, then y =
A. 5
2 7x − B.
5
2 7x + C.
2 7
5
x − D.
2 7
5
x +
34. 2007/II/4
Let x be the smaller one of two consecutive integers. If the sum of the squares of the two integers is less than
three times the product of the two integers by 1, then
A. x2 + (x + 1)2 = 3x(x + 1) – 1 B. x
2 + (x + 1)2 = 3x(x + 1) + 1
C. ( )2 23 ( 1) ( 1) 1x x x x+ + = + − D. ( )2 23 ( 1) ( 1) 1x x x x+ + = + +
35. 2008/II/2
If m = 7 – 3n , then n =
A. 7
3
m− B.
7
3
m+ C.
3
7 m− D.
3
7 m+
36. 2008/II/41
If the sum and the product of two numbers are 34 and 120 respectively, then the difference between the two
numbers is
A. 24 B. 26 C. 28 D. 30
HKCEE MATHEMATICS | 3.3 Algebraic Expression - Algebraic Simplification | P.6
19
1. 1990/II/4
If f(n) =2
1n(n − 1), then f(n + 1) − f(n) =
A. f(1) B. f(n) C. 2
n D. 1 E. n
2. 1991/II/35
If f(x) = x −x
1, then f(x) − f
x
1 =
A. 0 B. 2x C. x
2− D. 2
−
xx
1 E. 2
− x
x
1
3. 1993/II/1
If f ( ) 210 xx = , then f ( )4y =
A. y410 B. y4210 + C. y810 D. y40 E. y240
4. 1994/II/1
If f ( ) 2 2x x x= + , then f ( )1x − =
A. 2x B. 12
−x C. 122 −+ xx D. 322 −+ xx E. 142 −+ xx
5. 1995/II/35
If f ( )1
xx
x=
−, then f
1
x
f(–x) =
A. 2
1− B. −1 C.
x
x
+
−−
1
1 D.
21 x
x
− E.
12 −x
x
6. 1997/II/27
If f(x) = 3x2 + bx + 1 and f(x) = f(–x), then f(–3)
A. –26 B. 0 C. 3 D. 25 E. 28
7. 1998/II/2
If f(x) = x2
– 3x – 1, then f(a) + f(–a) =
A. 2a2
B. 2a2 – 2 C. 6a D. –6a E. –2
8. 1999/II/1
If f(x) = x2 – 1, then f(a – 1) =
A. a2
– 2a B. a2 – 3a C. a
2 – 3a – 2 D. a
2 – 1 E. a
2 – 2
HKCEE MATHEMATICS | 4.1 Polynomials - Function and Graph | P.1
20
9. 2000/II/4
Let f(x) = 3x2 + ax – 7. If f(–1) = 0, find f(–2).
A. –27 B. –11 C. –3 D. 1 E. 13
10. 2002/II/2
Let f(x) = x2 – x – 3 . If f(k) = k , then k =
A. 1 B. –1 or 3 C. –3 or 1 D. – 3 or 3
11. 2003/II/1
If f ( ) 22 1x x kx= + − and 1
f ( 2) f2
− = ,then =k
A. 3
17− B. −5 C. 3 D.
5
31
12. 2004/II/3
If 2f ( ) 1x x x= − + , then f ( 1) f ( )x x+ − =
A. 0 B. 2 C. 2x D. 4x
13. 2005/II/3
If 2f ( ) 2 3 4x x x= − + , then f (1) f ( 1)− − =
A. −6 B. −2 C. 2 D. 6
14. 2006/II/5
If f(x) =1
x
x+, then f(3)f(
1
3) =
A. 3
16 B.
1
2 C.
3
4 D. 1
15. 2006/II/37
The figure shows the graph of y = 4x. The coordinates of P are
A. (1, 0) B. (0, 1) C. (4, 0) D. (0, 4)
HKCEE MATHEMATICS | 4.1 Polynomials - Function and Graph | P.2
21
16. 2007/II/8
Let f(x) = x2 − ax + 2a, where a is a constant. If f(−3) = 29, then a =
A. −38 B. −20 C. −4 D. 4
17. 2007/II/38
Which of the following may represent the graph of y = f(x) and the graph of y = f(x + 1) on the same rectangular
coordinate system?
18. 2008/II/6
Let f(x) = x2 + kx + 7 , where k is a constant. If f(4) – f(3) = 21 , then k =
A. 0 B. 4 C. 14 D. 28
19. 2008/II/37
Which of the following may represent the graph of y = f(x) and the graph of y = f(x) + 2 on the same rectangular
coordinate system?
20. 2008/II/38
The figure shows the graph of y = ax , the graph of y = b
x and the graph of y = c
x on the same rectangular
coordinate system, where a , b and c are positive constants. Which of the following must be true?
I. a > b II. b > c III. a > 1 IV. c > 1
A. I and III only B. I and IV only C. II and III only D. II and IV only
HKCEE MATHEMATICS | 4.1 Polynomials - Function and Graph | P.2
22
1. 1990/II/34
Let f(x) = 3x3 − 4x + k. If f(x) is divisible by x − k, find the remainder when f(x) is divided by x + k.
A. 2k B. k C. 0 D. – k E. −k − 1
2. 1991/II/3
Which one of the following is a factor of x3 – 4x
2 + x + 6?
A. (x + 1)(x – 2) B. (x + 1)(x + 2) C. (x – 1)(x + 2) D. (x – 1)(x – 3) E. (x – 1)(x + 3)
3. 1992/II/6
Which of the following is a factor of 4(a + b)2 – 9(a – b)
2?
A. 5b – a B. 5a + b C. –a – b D. 13b – 5a E. 13a – 5b
4. 1992/II/41
If a polynomial ( )xf is divisible by 1−x , then ( )1−xf is divisible by
A. 2−x B. 2+x C. 1−x D. 1+x E. x
5. 1993/II/9
The expression kxx +− 22 is divisible by ( )1+x . Find the remainder when it is divided by ( )3+x .
A. 1 B. 4 C. 12 D. 16 E. 18
6. 1993/II/39
In factorizing the expression 4224bbaa ++ , we find that
A. ( )22 ba − is a factor. B. ( )22 ba + is a factor. C. ( )22 baba −− is a factor.
D. ( )22 baba +− is a factor. E. it cannot be factorized.
7. 1994/II/37
P(x) is a polynomial. When P(x) is divided by ( )25 −x , the remainder is R . If P(x) is divided by (2−5x), then
the remainder is
A. R B. −R C. R5
2 D.
5
2 E. −
5
2
8. 1995/II/3
If ( ) kxxxf ++= 9999 is divisible by x + 1 , then k =
A. −100 B. −98 C. 98 D. 100 E. 198
9. 1996/II/5
Find the remainder when x3 - x
2 + 1 is divided by 2x + 1.
A. -11 B. 5
8 C.
7
8 D.
9
8 E. 5
HKCEE MATHEMATICS | 4.2 Polynomials - Remainder and Factor Theorem | P.1
23
10. 1996/II/6
Which of the following expressions has/have b - c as a factor?
I. ab – ac II. a(b – c) – b + c III. a(b – c) – b – c
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
11. 1997/II/6
If 2x2 + x + m is divisible by x - 2, then it is also divisible by
A. x + 3 B. 2x – 3 C. 2x +3 D. 2x – 5 E. 2x + 5
12. 1998/II/6
Let f(x) = 2x3 - x
2 - 7x + 6. It is known that f(-2) = 0 and f(1) = 0. f(x) can be factorized as
A. (x + 1)(x + 2)(2x - 3) B. (x + 1)(x - 2)(2x + 3) C. (x - 1)(x + 2)(2x + 3)
D. (x - 1)(x + 2)(2x - 3) E. (x - 1)(x - 2)(2x + 3)
13. 1999/II/38
It is given that F(x) = x3 – 4x
2 + ax + b. F(x) is divisible by x – 1. When it is divided by x + 1, the remainder is 12.
Find a and b.
A. a = 5, b = 10 B. a =1, b = 2 C. a = -3, b = 6 D. a = -4, b = 7 E. a = -7, b = 10
14. 2000/II/9
Let 652)( 23 +−−= xxxxf . It is known that 0)1( =f . )(xf can be factorized as
A. )6()1( 2 +− xx B. )6)(1)(1( ++− xxx C. )3)(2)(1( +−− xxx
D. )3)(2)(1( −+− xxx E. )3)(2)(1( −−+ xxx
15. 2001/II/3
Let 12)1)(12()( +++−= xxxxf . Find the remainder when )(xf is divided by 12 +x .
A. −1 B. −1
2 C. 0 D. 1 E. 2
16. 2001/II/22
Which of the following is a factor of 222)(2 baba +−− ?
A. ba 3− B. ba 2− C. ba + D. ba 3+ E. ba −3
17. 2001/II/48
Let baxxxxf +++= 23 2)( . If )(xf is divisible by 1+x and 2−x , )(xf can be factorized as
A. )2)(1)(1( −+− xxx B. )2()1( 2 −+ xx C. )2)(1)(3( −+− xxx
D. )2)(1)(3( −++ xxx E. )2)(1( −+ xxx
HKCEE MATHEMATICS | 4.2 Polynomials - Remainder and Factor Theorem | P.2
24
18. 2002/II/38
The remainder when baxx ++2 is divided by 2+x is −4. The remainder when 12 ++ bxax is divided by
2−x is 9. The value of a is
A. −3 B. −1 C. 1 D. 3
19. 2003/II/2
Let ,2)( 23 kxxxf ++= where k is a constant. If ,0)1( =−f find the reminder when )(xf is divided by 1−x .
A. −1 B. 0 C. 2 D. 6
20. 2004/II/40
If 67)( 3 +−= xxxf is divisible by kxx +− 32 , then =k
A. −2 B. 2 C. −3 D. 3
21. 2005/II/40
Let k be a positive integer. When 2 1kx kx k+ + + is divided by x+1, the remainder is
A. −1 B. 1 C. 2k −1 D. 2k +1
22. 2006/II/40
Let k be a non-zero constant. When x3 + kx
2 + 2kx + 3k is divided by x + k, the remainder is k. Find k.
A. -1 B. 1 C. -2 D. 2
23. 2007/II/40
Let f(x) be a polynomial. If f(x) is divisible by x - 1, which of the following must be a factor of f(2x + 1)?
A. x B. x – 3 C. 2x – 1 D. 2x + 1
HKCEE MATHEMATICS | 4.2 Polynomials - Remainder and Factor Theorem | P.3
25
1. 1991/II/36
If p(x2 − x) + q(x
2 + x) = 4x
2 + 8x, find p and q.
A. p = 4, q = 8 B. p = −8, q = 4 C. p = −2, q = 6 D. p = 2, q = 6 E. p = 6, q = −2
2. 1993/II/5
If ( )( ) 32153 2 −−−≡−+ xbxaxx , then
A. 3 ,5 −=−= ba B. 3 ,5 =−= ba C. 5 ,3 −=−= ba D. 3 ,5 −== ba E. 5 ,3 == ba
3. 1994/II/7
Which of the following is/are an identity/identities?
I. (x + 2)(x − 2) = x2 − 4
II. (x + 2)(x − 2) = 0
III. (x + 2)3 = x
3 + 8
A. I only B. II only C. III only D. I and II only E. II and III only
4. 1995/II/10
If ( ) cbxxx ++≡++ 22 3163 , then c =
A. − 8 B. − 2 C. 0 D. 3
1 E. 1
5. 1996/II/8
If 2
2
1 1 1
a b
x x x≡ +
− + −, find a and b.
A. a = 2, b = 1 B. a = 1, b = 2 C. a = 1, b = 1 D. a = 1, b = −1 E. a = −1, b = 1
6. 1997/II/7
Which of the following is/are an identity/identities?
I. x2 = 4 II. (2x + 3)
2 = 4x
2 + 12x + 9 III. (x + 1)
2 = x
2 + 1
A. I only B. II only C. III only D. I and II only E. II and III only
7. 1998/II/5
If (x + 3)2 − (x + 1)(x – 3) ≡ P(x + 1) + Q, find P and Q.
A. P = 2, Q = 4 B. P = 2, Q = 10 C. P = 4, Q = 2 D. P = 4, Q = 8 E. P = 8, Q = 4
8. 1999/II/6
If (3x – 1)(x – a) ≡ 3x2 + bx – 2 , then
A. a = 2, b = –1 B. a = 2, b = –7 C. a = –2, b = 5 D. a = –2, b = –5 E. a = –2, b = –7
HKCEE MATHEMATICS | 4.3 Polynomials - Identities | P.1
26
9. 2000/II/10
If bxaxx +−≡++ 22 )2(373 , then
A. 12−=a , 5−=b B. 12−=a , 7=b C. 4−=a , 3=b
D. 0=a , 5−=b E. 0=a , 19=b
10. 2001/II/11
Which of the following is an identity/ are identities?
I. x2 + 2x + 1 = 0 II. x
2 + 2x + 1 = (x + 1)
2 III. x
2 + 1 > 0
A. I only B. II only C. III only D. I and III only E. II and III only
11. 2002/II/6
If (x +1)2 + P(x +1) ≡ x
2 + Q, then
A. P = –2, Q = –1 B. P = –2, Q = 1 C. P = 2, Q = –1 D. P = 2, Q = 1
12. 2003/II/6
If )1(2))(32( 2 ++≡−+ xbxaxx , then
A. 3−=a and b 9= B. 3
1−=a and
3
11=b C.
3
1=a and
3
7=b D. 3=a and 9−=b
13. 2004/II/10
If xxxxbxxa 45)2()2( 222 +−≡−+− , then =a
A. −1 B. 1 C. −2 D. 2
14. 2005/II/4
(2x − 3)(x2 + 3x − 2)=
A. 2x3 + 3x
2 + 5x − 6 B. 2x
3 + 3x
2 + 5x + 6 C. 2x
3 + 3x
2 − 13x − 6 D. 2x
3 + 3x
2 − 13x + 6
15. 2005/II/10
If x2
+ 2 ax + 8 ≡ (x + a)2 + b, then b =
A. 8 B. a2 + 8 C. a
2 − 8 D. 8 − a
2
16. 2006/II/6
Which of the following is an identity/ are identities?
I. x2 – 4 = 0 II. x
2 – 4 = (x – 2)
2 III. x
2 – 4 = (x + 2)(x – 2)
A. II only B. III only C. I and II only D. I and III only
HKCEE MATHEMATICS | 4.3 Polynomials - Identities | P.2
27
1. 1990/II/35
Let m be a constant. Find the value of x such that
=−=++m
x
mxx26
1
12
A. 1 B. 2 C. 3 D. 4 E. 5
2. 1991/II/8
Solve the following equations :x − 1 = y + 2 = x + y − 5
A. x = 1, y = −2 B. x = 1, y = 4 C. x = 4, y = 1 D. x = 7, y = −2 E. x = 7, y = 4
3. 1993/II/13
If the simultaneous equations
=−=
xy
kxy2
have only one solution, find k .
A. –1 B. 4
1− C. – 4 D.
4
1 E. 1
4. 1994/II/8
If βα ≠ and =−−
=−−03
032
2
bh
bh
ββαα
, then =+ βα
A. −3
b B.
3
b C. h D.
3
h− E.
3
h
5. 1994/II/39
If x = 3 , y = 2 satisfy the simultaneous equations
=−=+
3
2
aybx
byax, find the values of a and b.
A. 0=a , 1=b B. a = 0 , b = −1 C. a =6
5, b = −
4
1
D. a = −13
1, b =
39
37 E. a = −
13
12, b =
13
5
HKCEE MATHEMATICS | 5.1 Equations - Simultaneous Equations | P.1
28
6. 1995/II/7
Solve the simultaneous equations:
−=+
=−
16
2
63
4
yx
yx
A. 12 ,2
1−=−= yx B. 12 ,
2
1=−= yx C. 12 ,
2
1−== yx
D. 12 ,2
1== yx E.
2
7 ,
24
5−== yx
7. 1996/II/10
Solve 2 2 13
1
x y
x y
+ = + =
A. 2
3
x
y
= − = B. 6
7
x
y
= − = C. 2
1
x
y
= = − or 3
4
x
y
= − =
D. 2
3
x
y
= − = or 3
2
x
y
= = − E. 6
7
x
y
= − = or 7
6
x
y
= = −
8. 1997/II/8
Solve
31
12 1
2
yx
yx
− = − =
A. x =5
4, y =
7
4 B. x =
11
4, y =
1
11 C. x =
11
4, y =
13
22 D. x =
11
6, y =
7
11 E. x =
6
11, y =
7
11
9. 1998/II/4
Solve
32 1
17
xy
xy
+ = − − =
A. (0, –3) B. (1, –1) C. (4, –1
3) D. (4, –3) E. (22, –
1
15)
HKCEE MATHEMATICS | 5.1 Equations - Simultaneous Equations | P.2
29
10. 1999/II/8
If 2 3 2
3
y x x
y x
= + − = − +, then
A. x = –1 B. x = –1 or 5 C. x = –2 or 1 D. x = –5 or 1 E. x = –5 or 8
11. 2000/II/5
If 2 1
2 2
y x
y x
= − = −, then y =
A. –4 B. 0 C. 1 D. 0 or 8 E. –4 or 4
12. 2001/II/12
If 2 4 44
2 4
y x x
y x
= − − = − +, then y =
A. –32 or 52 B. –12 or 16 C. –12 or 96 D. –8 or 20 E. 12 or 24
13. 2001/II/39
If a, b are distinct real numbers and 2
2
4 1 0
4 1 0
a a
b b
+ + = + + =, find a2 + b2.
A. 1 B. 9 C. 14 D. 16 E. 18
14. 2002/II/8
If )1,2(),( −=yx is a solution of the simultaneous equations
=++=+−
01
08
aybx
byax,then =a
A. −3 B. 2 C. 4
9 D. 3
15. 2003/II/7
If 2 4
3 4
y x
y x
= + = − +, then y =
A. 0 B. 13 C. 0 or −3 D. 4 or 13
HKCEE MATHEMATICS | 5.1 Equations - Simultaneous Equations | P.3
30
16. 2004/II/8
If 2 10
4 14
pq q
p q
+ = + = , then q =
A. 2 B. 3 C. 3
2
− or 3 D. 2 or 20
17. 2004/II/42
If α β≠ and
2
2
4 3
4 3
α αβ β
= + = +, then ( 1)( 1)α β+ + =
A. –6 B. 0 C. 2 D. 8
18. 2005/II/7
If 2 3
4 3
β αβ α
= − = −, then β =
A. 4 B. 13 C. 0 or 4 D. –3 or 13
19. 2007/II/7
The price of 6 oranges and 3 apples is $ 42 while the price of 8 oranges and 5 apples is $ 60. Find the price of
an apple.
A. $ 3 B. $ 4 C. $ 5 D. $ 6
20. 2008/II/8
If m + 2 = n – 1 = 3m + n – 46 , then n =
A. 15 B. 16 C. 17 D. 18
HKCEE MATHEMATICS | 5.1 Equations - Simultaneous Equations | P.4
31
1. 1990/II/8
If p and q are the roots of the equation x2 − x + 3 = 0, then (2
p − 2)(2
q − 2) =
A. 32
1 B.
8
1 C.
2
1 D. 8 E. 32
2. 1990/II/31
The graph of y = ax2 + bx + c is given as shown. Which of the following is/are true?
I. a < 0
II. b < 0
III. c < 0
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III only
3. 1991/II/39
If (x − 2)(x − 3) = (a − 2)(a − 3), solve for x.
A. x = 0 or 5 B. x = 2 or 3 C. x = a or 2 D. x = a or 3 E. x = a or 5 − a
4. 1992/II/3
For what value(s) of x does the equality ( )( )
12
21+=
−
−+x
x
xx hold?
A. −1 only B. 2 only C. Any value D. Any value except –1 E. Any value except 2
5. 1992/II/36
Which of the following intervals must contain a root of 2x3 – x
2 – x – 3 = 0?
I. –1 < x < 1 II. 0 < x < 2 III. 1 < x < 3
A. I only B. II only C. III only D. I and II only E. II and III only
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.1
32
6. 1992/II/38
From the figure, if βα ≤≤ x , then
A. ( ) ( ) 02 ≤−+−+ kcxmbax B. ( ) ( ) 02 <−+−+ kcxmbax
C. ( ) ( ) 02 =−+−+ kcxmbax D. ( ) ( ) 02 >−+−+ kcxmbax
E. ( ) ( ) 02 ≥−+−+ kcxmbax
7. 1993/II/7
The diagram shows the graphs of bxaxy += 2 and dcxy += . The solutions of the equation
dcxbxax +=+2 are
A. –1, 1 B. –1, 2 C. 0, 1 D. 0, 3 E. 1, 3
8. 1993/II/12
If α and β are the roots of the quadratic equation 0132 =−− xx , find the value of βα11+ .
. A. –3 B. –1 C. 3
1− D.
3
2 E. 3
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.2
33
9. 1994/II/38
In the figure, the line kmxy += cuts the curve cbxxy ++= 2 at α=x and β=x . Find the value of αβ .
A. −b B. c C. m − b D. k − c E. c − k
10. 1995/II/39
If βα , are the roots of the equation 0342 =−− xx , then =++ 22 βαβα
A. −13 B. 5 C. 13 D. 16 E. 19
11. 1995/II/40
Find the range of values of k such that the equation ( ) 0122 =+−+ xkx has real roots.
A. 4=k B. 40 << k C. 40 ≤≤ k D. 4or 0 >< kk E. 4or 0 ≥≤ kk
12. 1995/II/41
Which of the following may represent the graph of 1032 ++−= xxy ?
13. 1996/II/11
If α and β are the roots of the equation 2x2 + 4x − 3 = 0, find
α ββ α+ .
A. 22
3− B.
16
3− C.
14
3− D.
8
3− E.
2
3
14. 1996/II/40
If 3 is a root of the equation x2 – x + c = 0 , solve x
2 – x + c > 0.
A. x < –2 or x > 3 B. x < 2 or x > 3 C. x > –6 D. –2 < x < 3 E. 2 < x < 3
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.3
34
15. 1996/II/41
The curve in the figure is the graph of y = −x2 + bx + c. Find the area of the rectangle OPQR.
A. bc B. b2 C. c
2 D. b
2 − 4c E. b
2 + 4c
16. 1997/II/30
The difference of the roots of the equation 2x2 − 5x + k = 0 is
7
2. Find k.
A. −6 B. −3 C. 3
2− D. 3 E.
51
16
17. 1997/II/31
In the figure, find the coordinates of the mid-point of AB.
A. 7 35
( , )2 2
− B. 5 25
( , )2 4
− C. 5 37
( , )2 2
− D. 5 13
( , )2 2
E. 7 35
( , )2 2
18. 1997/II/34
The figure shows the graph of a quadratic function f(x). If the vertex of the graph is (1,3), then f(x) =
A. −3(x – 1)2 + 3 B. −3(x + 1)
2 + 3 C. −(x – 1)
2 + 3
D. −(x + 1)2 + 3 E. 3(x – 1)
2 − 3
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.4
35
19. 1998/II/9
The figure shows the graph of y = ax2 + bx + c. Which of the following is true?
A. a > 0, c > 0 and b2 − 4ac > 0 B. a > 0, c > 0 and b
2 − 4ac < 0 C. a > 0, c < 0 and b
2 − 4ac < 0
D. a < 0, c > 0 and b2 − 4ac > 0 E. a < 0, c < 0 and b
2 − 4ac > 0
20. 1998/II/10
Solve (x – 1)(x – 3) = x – 3.
A. x = 1 B. x = 2 C. x = 0 or 3 D. x = 1 or 3 E. x = 2 or 3
21. 1998/II/11
In the figure, ABCD is a square of side 10 cm. If AE = AF and the area of ∆CEF is 20 cm2, which of the
following equations can be used to find AF?
A. x2 + 10(10 – x) + 20 = 100 B. x
2 + 20(10 – x) + 20 = 100 C.
1
2x
2 + 10x + 20 = 100
D. 1
2x
2 + 10(10 – x) + 20 = 100 E.
1
2x
2 +
10(10 )
2
x− + 20 = 100
22. 1998/II/12
The figure shows the graph of y = ax2 + bx + c. Which of the following is true?
A. a > 0 , c > 0 and b2 – 4ac > 0 B. a > 0 , c > 0 and b
2 – 4ac < 0 C. a > 0 , c < 0 and b
2 – 4ac < 0
D. a < 0 , c > 0 and b2 – 4ac > 0 E. a < 0 , c < 0 and b
2 – 4ac > 0
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.5
36
23. 1999/II/5
In the figure, the graph of y = x2 − 6x + k touches the axis. Find k.
A. k ≥ 0 B. k ≥ 9 C. k = −9 D. k = 0 E. k = 9
24. 1999/II/9
Which of the following may represent the graph of y = x2 − 3x − 18?
25. 1999/II/42
John goes to school and returns home at speed x km/h and (x + 1) km/h respectively. The school is 2 km from
John’s home and the total time for the two journeys is 54 minutes. Which of the following equations can be
used to find x?
A. 1 54
2 2 60
x x ++ = B.
2 2 54
1 60x x+ =+
C. [ ]1
( 1)542
4 60
x x+ +=
D.
[ ]4 54
1 60( 1)
2x x
=+ +
E. 54
2 2( 1)60
x x+ + =
26. 2000/II/34
09 22=− ba and 0<ab , then =
+
−
ba
ba
A. 2− B. 2
1− C. 0 D.
2
1 E. 2
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.6
37
27. 2000/II/39
Which of the following may represent the graph of y = –x2 + 2x – 3?
28. 2001/II/23
In the figure, the graph of y = 2x2 − 9x + 4 cuts the x-axis at A and B, and the y-axis at C. Find the area of ∆ABC .
A. 4 B. 6 C. 7 D. 8 E. 14
29. 2001/II/40
Suppose the graph of y = x2 – 2x – 3 is given. In order to solve the quadratic equation 2x
2 – 6x – 3 = 0, which
of the following straight lines should be added to the given graph?
A. y = 4x B. y = x –3
2 C. y = –x +
3
2 D. y = 2x – 3 E. y = –2x + 3
30. 2002/II/5
The figure shows the graph of y = x2 + bx + c . Find b.
A. 11
2
− B. –5 C. 5 D.
11
2
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.7
38
31. 2002/II/7
Which of the following equations has/have equal roots?
I. x2 = x II. x
2 + 2x + 1 = 0 III. (x + 3)
2 = 1
A. II only B. III only C. I and II only D. I and III only
32. 2003/II/5
If the equation 142 =+− kxx has no real roots, then the range of values of k is
A. 4>k B. 4≥k C. 5>k D. 5≥k
33. 2003/II/41
Let k be a constant. If α and β are the roots of the equation x2 – 3x + k = 0, then 2α +3β =
A. 3 – k B. 3 + k C. 9 – k D. 9 + k
34. 2003/II/42
The figure shows the graph of y = –x2 + ax + b . Which of the following is true?
A. a < 0 and b < 0 B. a < 0 and b > 0 C. a > 0 and b < 0 D. a > 0 and b > 0
35. 2004/II/5
In the figure, the graph of y = 2x2 – 4x + c passes through the point (1, k). Find the value of k.
A. –5 B. –4 C. –3 D. –2
36. 2004/II/6
If the equation 4x2 + kx + 9 = 0 has equal positive roots, then k =
A. –6 B. 6 C. –12 D. 12
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.8
39
37. 2004/II/7
Solve x(x – 6) = x.
A. x = 6 B. x = 7 C. x = 0 or x = 6 D. x = 0 or x = 7
38. 2005/II/5
In the figure, the area of trapezium is 12 cm2. Which of the following equations can be used to find x?
A. x(x + 2) = 12 B. x(x + 2) = 24 C. x2 – x(x – 2) = 12 D. x
2 – x(x – 2) = 24
39. 2005/II/6
The figure shows the graph of y = ax2 + x + b. Which of the following is true?
A. a > 0 and b < 0 B. a > 0 and b > 0 C. a < 0 and b < 0 D. a < 0 and b > 0
40. 2005/II/8
If the quadratic equation kx2 + 6x + (6 – k) = 0 has equal roots, then k =
A. –6 B. –3 C. 3 D. 6
41. 2006/II/7
The figure shows the graph of y = f(x). If f(x) is a quadratic function, then f(x) =
A. 1
2(x + 1)(x − 4) B. 2(x + 1)(x − 4) C.
1
2(x − 1)(x + 4) D. 2(x − 1)(x + 4)
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.9
40
42. 2006/II/8
Solve 3x2 = 21x
A. x = 3 B. x = 7 C. x = 0 or x = 3 D. x = 0 or x = 7
43. 2006/II/9
Find the range of values of k such that the quadratic equation x2 + 2x – k = 2 has two distinct real roots.
A. k > –3 B. 3k ≥ − C. k > –1 D. 1k ≥ −
44. 2007/II/5
Which of the following statements about the graph of y = (x + 1)2 – 4 is true?
A. The coordinates of the vertex of the graph are (–1, 4).
B. The equation of the axis of symmetry of the graph is x = 1.
C. The x-intercepts of the graph are –1 and 3.
D. The y-intercept of the graph is –3.
45. 2007/II/42
If p = q2 – 12q + 6 = 2q – 7 , then p =
A. 1 or 13 B. –1 or –13 C. –5 or 19 D. –9 or –33
46. 2008/II/9
The figure shows the graph of y = –(x + h)2 + k . Which of the following must be true?
A. h > 0 and k > 0 B. h > 0 and k < 0 C. h < 0 and k > 0 D. h < 0 and k < 0
47. 2008/II/10
The figure shows the graph of y = f(x) , where f(x) is a quadratic function. The solution of f(x) < 3 is
A. a < x < d B. b < x < c C. x < a or x > d D. x < b or x > c
HKCEE MATHEMATICS | 5.2 Equations - Quadratic Equations and Graphs | P.10
41
1. 1990/II/1
(a2n
)3 =
A. a6n
B. a8n
C. 32n
a D. 36n
a E. 38n
a
2. 1991/II/1
(a2a
)(3a4a
) =
A. 3a6a
B. (3a)6a
C. 3a8a
D. 4a6a
E. (34a
)(a6a
)
3. 1992/II/8
Simplify �������
�����
terms
times
...
...
n
n
nnn
nnn
+++
×××.
A. 2−nn B. 2
n
n C. 2−n D. 2
n E. 1
4. 1992/II/9
If a and b are greater than 1, which of the following statements is/are true?
I. a b a b+ = +
II. (a-1
+ b-1
)-1
= a + b
III. a2b
3 = (ab)
6
A. I only B. II only C. III only D. I and II only E. None of them
5. 1993/II/34
If 369 2=
+x , then =x3
A. 3
2 B.
3
4 C. 2 D. 6 E. 9
6. 1994/II/33
(3 x ) 2 =
A. 3 )( 2x B. 3 2+x C. 3 x2 D. 6 x E. 9 x2
7. 1995/II/4
Simplify 3
2
12
6−
b
a
A. 4
8
a
b B.
9
18
a
b C.
8
4
b
a D.
18
9
b
a E.
124
1
ba
HKCEE MATHEMATICS | 6.1 Indices | P.1
42
8. 1995/II/38
If cba 1025 == and a, b, c are non-zero, then =+b
c
a
c
A. 10
7 B. 1 C. 7 D. 7log E.
5log
1
2log
1+
9. 1996/II/2
27
3
x
y=
A. 9x
y B. 9
x
y C. 9x y− D.
3
3
x
y E. 33 x y−
10. 1997/II/2
If 2x. 8
x = 64 , then x =
A. 3
2 B.
3
4 C.
6
5 D. 2 E. 4
11. 1998/II/7
2(2 )
8
m
m=
A. 2
3 B. 2
-m C. 2
m D.
2 32m m− E. 22 32 m m−
12. 1999/II/4
If 4x = a , then 16
x =
A. 4a B. a2 C. a
4 D. 2
a E. 4
a
13. 2000/II/3
Simplify 421
213
)(
)(
ba
ba−
−−
.
A. 3
1
ab B.
32
1
ba C.
62
1
ba D.
92
1
ba E.
6
4
b
a
14. 2001/II/10
=+−
−−
2
12
n
nn
a
aa
A. 1−na B. )1(2 aa n +− C. 11 −+ n
a D. a
11+ E. a+1
HKCEE MATHEMATICS | 6.1 Indices | P.2
43
15. 2002/II/3
=⋅yx 82
A. yx 32 + B. xy32 C. yx+16 D. xy16
16. 2003/II/4
=⋅yx 93
A. yx 23 + B. yx 33 + C. yx+27 D. xy27
17. 2004/II/1
=⋅
n
nn
3
922
A. n26 B. n36 C. n12 D. n212
18. 2005/II/1
a.a( a + a ) =
A. a4
B. 2a3
C. a3 + a D. 3a
2 + a
19. 2006/II/1
(2x)3 . x
3 =
A. 6x6 B. 8x
6 C. 6x
9 D. 8x
9
20. 2007/II/1
If n is a positive integer, then 32n
. 4n =
A. 62n
B. 63n
C. 122n
D. 123n
21. 2007/II/39
Which of the following is the greatest?
A. 5003000
B. 20002500
C. 25002000
D. 3000500
22. 2008/II/1
888
8871( 2)
2
− =
A. –2 B. –0.5 C. 0 D. 0.5
HKCEE MATHEMATICS | 6.1 Indices | P.3
44
1. 1990/II/5
If 2 = 10p, 3 = 10
q, express log
6
1 in terms of p and q.
A. −p − q B. pq
1 C.
qp +
1 D. pq E. p + q
2. 1991/II/34
If log x : log y = m : n, then x =
A. n
my B. (m − n)y C. m − n + y D. n
m
y E. n
ym log
3. 1992/II/5
If ab 1010 log2
11log += , then =b
A. a10 B. a+10 C. a5 D. 2
a E.
21
a+
4. 1993/II/8
If ( ) qpqp logloglog +=+ , then
A. 1== qp B. 1−
=q
qp C.
1+=
q
qp D.
q
qp
1+= E.
q
qp
1−=
5. 1994/II/34
If log 2 = a and log 9 = b, then log 12=
A. 2a + 3
b B. 2a +
2
b C.
3
2a +
3
2b D. a 2 + b 2
1
E. a 2 b 2
1
6. 1995/II/43
If the geometric mean of two positive numbers a and b is 10, then log a + log b =
A. 2
1 B. 1 C. 2 D. 10 E. 100
7. 1996/II/38
Let x > y > 0 . If log(x + y) = a and log(x – y) = b , then log 2 2x y− =
A. 2
a b+ B.
2
ab C. a b+ D. ab E. a b+
8. 1997/II/5
If log(x + a) = 2 , then x =
A. 2 – a B. 100 – a C. 100
a D. 2 – log a E. 100 – log a
HKCEE MATHEMATICS | 6.2 Logarithms | P.1
45
9. 1998/II/40
Suppose log10 2 = a and log10 3 = b . Express log10 15 in terms of a and b.
A. –a + b + 1 B. –a + 10b C. a + 2b D. (a + b)b E. 10b
a
10. 1999/II/39
If 1
2log y = 1 + log x , then
A. 10y x= B. y = 100 + x2 C. y = (10 + x)
2 D. y = 10x
2 E. y = 100x
2
11. 2000/II/38
If 3)log( =− ax , then =x
A. a+310 B. 3a C. a1000 D. a+1000 E. a+30
12. 2001/II/37
If 22 )(loglog xx = , then =x
A. 1 B. 10 C. 100 D. 1 or 10 E. 1 or 100
13. 2002/II/40
If 13loglog 2 += xx , then =x
A. 2 B. 5 C. 30 D. 0 or 30
14. 2003/II/40
If cba=
+10 , then =b
A. ac −log B. ca log− C. ac−
10 D. a
c 10−
15. 2004/II/39
If a105 = and b107 = , then =50
7log
A. 1−− ab B. 1+− ab C. a
b D.
1+a
b
16. 2005/II/39
If a and b are positive integers, then log( )b aa b
A. log( )ab ab B. (log )(log )ab a b C. ( ) log( )a b a b+ + D. log logb a a b+
17. 2006/II/38
Let a and b be positive numbers. If log10
a= 2 log b , then a =
A. 10b2 B. 20b C. b
2 + 10 D. 2b + 10
HKCEE MATHEMATICS | 6.2 Logarithms | P.2
46
1. 1990/II/33
21
1
+ +
32
1
+ +
43
1
+ +
54
1
+ =
A. 51
1
−
B. 15
1
−
C. 1 + 5 D. 1 − 5 E. −1 + 5
2. 1991/II/4
If y =mx
mx
−
+
1
1, then x =
A. 1
)1(
+
−
y
ym B.
)1(
1
+
−
ym
y C.
)1(
)1(2
2
ym
y
+
− D.
)1(
)1(2
2
+
−
y
ym E.
)1(
)1(2
2
+
−
ym
y
3. 1991/II/33
If ( 3 + 1) x = 2, then x =
A. 2 − 3 B. 3 − 1 C. 1 D. 2(2 − 3 ) E. 4 − 3
4. 1992/II/4
=+
−−
−
+
15
15
15
15
A. 0 B. 2
1 C. 3 D. 5 E. 5
2
1+
5. 1993/II/4
Simplify ba
a
ba
b
++
−.
A. ba −
1 B.
ba
baba
−
−+ 2 C.
a
ab
2
+ D.
ba
aabb
−
−+ 2 E.
ba
ba
−
+
6. 1994/II/4
If 23 +=a , then =−a
a1
A. 0 B. 22 C. 32 D. 3 − 2 E. 2
2
3
32+
7. 1995/II/5
=−
−+ 62
1
62
1
A. 6− B. 2
6− C. 0 D.
2
6 E. 6
HKCEE MATHEMATICS | 6.3 Surds | P.1
47
8. 1996/II/39
If 3 1
13 2
x − =
, then x =
A. 3 1
3 2− + B.
3 1
3 2+ C. 4 3 6− − D. 4 3 6− E. 4 3 6+
9. 1997/II/29
1 1
2 1 3 2− =
− −
A. 1 3− + B. 1 3− C. 1 2 2 3− + − D. 1 2 2 3− + E. 1 2 2 3+ −
10. 2000/II/40
If 212
5 =
+ x , then =x
A. 2102 − B. 24102 − C. 24102 + D. 2
110 − E.
3
24102 −
11. 2002/II/39
If ,4)13)(1( =−+x then x =
A. 332 − B. 132 + C. 232 + D. 2
134 −
12. 2004/II/4
=− aa 425
A. a3 B. a7 C. a21 D. a21
13. 2005/II/37
If n is a positive integer, then 1 1
1 2 1 2n n−
+ −=
A. 4
1 4
n
n− B.
4
1 4
n
n
−
+ C.
4
4 1
n
n + D.
4
4 1
n
n −
14. 2007/II/37
If a > 0 , then 3
2 4
a a
a− =
A. 1 B. 2
a C. a D. 2 a
15. 2008/II/39
If a > 0 , then 49 25a a− =
A. 2 a B. 12 a C. 24a D. 74a
HKCEE MATHEMATICS | 6.3 Surds | P.2
48
1. 1990/II/9
If a : b = 3 : 4 and b : c = 2 : 5, then a2 : c2 =
A. 3 : 10 B. 9 : 25 C. 9 : 100 D. 36 : 25 E. 36 : 100
2. 1990/II/10
If 1 U.S. dollar is equivalent to 7.8 H.K. dollars and 1 000 Japanese yen are equivalent to 53.3 H.K. dollars,
how many Japanese yen are equivalent to 50 U.S. dollars?
A. 1 463 B. 3 417 C. 7 317 D. 8 315 E. 20 787
3. 1990/II/43
Which of the following graphs shows that y is partly constant and partly varies inversely as x?
A. B. C.
O
y
x
O
y
x
O
y
x
D. E.
O
y
x
O
y
x
4. 1991/II/9
Let y vary partly as x
1 and partly as x. When x = 1, y = 5 and when x = 4, y =
2
25. Find y when x = 2 .
A. 2
5 B. 4 C.
4
25 D. 7 E.
2
17
5. 1991/II/10
If a
1:b
1 = 2 : 3 and a : c = 4 : 1, then a : b : c =
A. 12 : 8 : 3 B. 8 : 3 : 2 . C. 4 : 6 : 1 D. 2 : 3 : 8 E. 2 : 3 : 4
6. 1991/II/42
3 kg of a solution contains 40% of alcohol by weight. How much alcohol should be added to obtain a solution
containing 50% of alcohol by weight?
A. 0.3 kg B. 0.6 kg C. 0.75 kg D. 1.5 kg E. 3.75 kg
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.1
49
7. 1992/II/10
If a : b = 2 : 3, a : c = 3 : 4 and b : d = 5 : 2, find c : d.
A. 1 : 5 B. 16 : 45 C. 10 : 3 D. 20 : 9 E. 5 : 1
8. 1992/II/11
Suppose x varies directly as y2 and inversely as z. Find the percentage increase of x when y is increased by
20% and z is decreased by 20%.
A. 15.2% B. 20% C. 50% D. 72.8% E. 80%
9. 1992/II/45
Coffee A and coffee B are mixed in the ratio x:y by weight. A costs $50/kg and B costs $40/kg. If the cost
of A is increased by 10% while that of B is decreased by 15%, the cost of the mixture per kg remains unchanged.
Find x : y .
A. 2 : 3 B. 5 : 6 C. 6 : 5 D. 3 : 2 E. 55 : 34
10. 1993/II/35
If 3:2: =ba and 3:5: =cb , then =+−
++
cba
cba
A. –2 B. 2
5 C. 4 D.
2
17 E. 31
11. 1994/II/42
If 3:2: =ba , 4:3: =ca and 5:4: =da , then dcb :: =
A. 2:3:4 B. 3:4:5 C. 3:6:10 D. 18:16:15 E. 40:45:48
12. 1994/II/43
Let x vary inversely as y . If y is increased by 69%, then x will be
A. increased by 23.1%(3 sig. fig.) B. increased by 30% C. decreased by 23.1%(3 sig. fig.)
D. decreased by 30% E. decreased by 76.9%(3 sig. Fig)
13. 1995/II/11
x and y are two variables. The table below shows some values of x and their corresponding values of y.
X 2 3 6 12
Y 36 16 4 1
Which of the following may be a relation between x and y?
A. yx ∝ B. yx ∝ C. y
x1
∝ D. y
x1
∝ E. 2
1
yx ∝
14. 1995/II/12
If 125x = 25y and x, y are non-zero, find yx : .
A. 1 : 25 B. 1 : 5 C. 2 : 3 D. 3 : 2 E. 5 : 1
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.2
50
15. 1996/II/44
The following table shows the compositions of Tea A and Tea B which are mixtures of Chinese tea and Indian
tea:
If 4 kg of tea A and 10 kg of tea B are mixed, find the ratio of Chinese tea and Indian tea in the mixture.
A. 2 : 5 B. 16 : 17 C. 1 : 1 D. 5 : 4 E. 23 : 17
16. 1997/II/11
In a map of scale 1 : 500 , the length and breath of a rectangular field are 2 cm and 3 cm respectively. Find
the actual area of this field.
A. 30 m2 B. 150 m2 C. 1 500 m2 D. 3 000 m2 E. 15 000 m2
17. 1997/II/39
Suppose x varies directly as y and inversely as z . When y = 2 and z = 3 , x = 7. When y = 6 and z = 7 , x =
A. 1 B. 49
9 C. 9 D.
49
4 E. 49
18. 1998/II/15
If 2
53 4
x y
x y
+=
−, then x : y =
A. 3 : 7 B. 7 : 3 C. 7 : 11 D. 9 : 7 E. 11 : 7
19. 1998/II/17
If x varies inversely as y and directly as z2 , then
A. 2
x
yz is a constant B.
2
xy
z is a constant C.
2xz
y is a constant
D. 2
z
y is a constant E. 21
zy+ is a constant
20. 1999/II/12
If x : y = 3 : 4 and 2x + 5y = 598 , find x.
A. 23 B. 26 C. 69 D. 78 E. 104
21. 1999/II/13
If 1 Australian dollar is equivalent to 4.69 H.K. dollars and 100 Japanese yen equivalent to 5.35 H.K. dollars,
how many Japanese yen are equivalent to 1 Australian dollar? Give your answer correct to the nearest Japanese
yen.
A. 4 B. 25 C. 88 D. 114 E. 2509
Ratio of Chinese tea and Indian tea by weight
Tea A 3 : 1
Tea B 2 : 3
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.3
51
22. 1999/II/45
It is given that y varies inversely as x3 . If x is increased by 100% , then y is
A. increased by 800% B. increased by 700% C. decreased by 300%
D. decreased by 87.5% E. decreased by 12.5%
23. 2000/II/35
y varies directly as 2x and inversely as z . If 1=y when 2=x and 9=z , find y when 1=x and 4=z .
A. 3
2 B.
3
8 C.
6
1 D.
8
3 E.
26
9
24. 2000/II/36
Tea A and tea B are mixed in the ratio x : y by weight. A costs $80/kg and B costs $100/kg. If the cost of A is
increased by 10% and that of B is decreased by 12%, the cost of the mixture per kg remains unchanged. Find x : y.
A. 1 : 1 B. 2 : 3 C. 3 : 2 D. 5 : 6 E. 6 : 5
25. 2001/II/29
Suppose y is partly constant and partly varies inversely as x. When 1=x , 7=y and when 3=x , 3=y .
Find y when 2=x .
A. 2.5 B. 3.5 C. 4 D. 5 E. 6.5
26. 2002/II/10
If 1 Euro is equivalent to 6.94 H.K. dollars and 1 U.S. dollar is equivalent to 7.78 H.K. dollars, how many
Euros are equivalent to 100 U.S. dollars? Give your answer correct to the nearest Euro.
A. 89 B. 112 C. 129 D. 144
27. 2002/II/13
If ,432 zyx == then =+−
−+
zyx
zyx
A. 5
1 B.
3
1 C.
3
5 D.
5
7
28. 2002/II/15
It is given that y varies inversely as x. If x is increased by 50%, then y is decreased by
A. %3
133 B. 50% C. %
3
266 D. 100%
29. 2003/II/13
If yx 22781 = and x, y are non-zero integers, then =yx :
A. 2 : 3 B. 3 : 4 C. 4 : 3 D. 3 : 2
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.4
52
30. 2003/II/14
Suppose z varies directly as 2x and inversely y. When 4=x and 3=y , 2=z . When 2=x and 3=z , =y
A. 2
1 B. 1 C. 2 D. 18
31. 2003/II/15
The scale of a map is 1 : 4 000. If the actual area of a sports field is 8 000 m2, find its area on the map.
A. 02.0 cm2 B. 05.0 cm2 C. 2 cm2 D. 5 cm2
32. 2004/II/13
If 3:2)2(:)( =−− abba , then =ba :
A. 3 : 5 B. 5 : 3 C. 5 : 7 D. 7 : 5
33. 2004/II/14
A box contains two kinds of coins: $ 5 and $ 2. The ratio of the number of $ 5 coins to the number of $ 2 coins is
4 : 5. If the total value of the coins is $ 90, then the total number of coins in the box is
A. 9 B. 18 C. 27 D. 36
34. 2004/II/15
The scale of a map is 1 : 20 000. If two buildings are 3.8 cm apart on the map, then the actual distance between
the two buildings is
A. 0.076 km B. 0.76 km C. 7.6 km D. 76 km
35. 2004/II/16
It is known that y varies partly as x and partly as x . When 1=x , 4=y and when 4=x , 10=y . Find y
when 16=x .
A. 28 B. 52 C. 80 D. 256
36. 2005/II/13
Let x and y be non-zero numbers. If 2x − 3y = 0, then (x + 3y) : (x +2y) =
A. 3 : 2 B. 4 : 3 C. 9 : 7 D. 11 : 8
37. 2005/II/14
If z varies directly as y2 and inversely as x, which of the following must be constant?
A. xy2z B. 2y z
x C.
2
xz
y D.
2
z
xy
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.5
53
38. 2006/II/13
Let x , y and z be non-zero numbers. If x : y = 1 : 2 and y : z = 3 : 1 , then (x + y) : (y + z) =
A. 3 : 4 B. 4 : 3 C. 8 : 9 D. 9 : 8
39. 2006/II/14
It is given that x varies directly as y and inversely as z2 . If y is decreased by 10% and z is increased by 20%,
then x is decreased by
A. 10% B. 23.6% C. 25% D. 37.5%
40. 2006/II/15
The scale of a map is 1 : 8 000 . If the area of a park on the map is 2 cm2 , then the actual area of the park is
A. 4 000 m2 B. 6 400 m2 C. 12 800 m2 D. 16 000 m2
41. 2007/II/13
Let a and b be non-zero numbers. If 7a + 5b = 3a + 8b , then a : b =
A. 3 : 4 B. 4 : 3 C. 10 : 13 D. 13 : 10
42. 2007/II/14
It is given that y is partly constant and partly varies directly as x . When x = 2 , y = 17 and when x = 4 , y = 11 .
Find the value of x when y = 5 .
A. –3 B. 6 C. 8 D. 112
43. 2008/II/13
The costs of rice of brand A and rice of brand B are $8 / kg and $4 / kg respectively. If x kg of rice of brand A
and y kg of rice of brand B are mixed so that the cost of the mixture is $5 / kg , find x : y .
A. 1 : 2 B. 2 : 1 C. 1 : 3 D. 3 : 1
44. 2008/II/14
Suppose that y varies directly as x and inversely as z2 . If x and z are both decreased by 20% , then y
A. is decreased by 17% B. is decreased by 20%
C. is increased by 20% D. is increased by 25%
45. 2008/II/15
It is known that f(x) varies partly as x and partly as x2 . If f(1) = 5 and f(2) = 16 , then f(3) =
A. 21 B. 27 C. 33 D. 57
HKCEE MATHEMATICS | 7 Ratio, Ratio and Variation | P.6
54
1. 1990/II/36
If a < b < 0, which of the following must be true?
A. −a < −b B. b
a < 1 C. a
2 < b
2 D. 10
a < 10
b E. a
−1 < b
−1
2. 1991/II/37
If x < 0 < y, then which one of the following must be positive?
A. x + y B. x − y C. y − x D. xy E. x
y
3. 1992/II/37
How many integers x satisfy inequality 02076 2 ≤−− xx ?
A. 0 B. 1 C. 2 D. 3 E. 4
4. 1993/II/40
If the solution of the inequality 062 ≤+− axx is 3≤≤ xc , then
A. 2 ,5 == ca B. 2 ,5 =−= ca C. 2 ,5 −== ca D. 2 ,1 −== ca E. 2 ,1 =−= ca
5. 1994/II/6
If ( ) ( )151 +<+ xxx , then
A. 5<x B. 1or 5 >−< xx C. 5or 1 >−< xx D. 15 <<− x E. 51 <<− x
6. 1995/II/9
Find the values of x which satisfy both 4<− x and 23
162−>
−x
A. 54 <<− x B. 4−<x C. 4−>x D. 5<x E. 5>x
7. 1996/II/7
Solve 1 < –3x + 4 < 10 .
A. –2 < x < 1 B. –1 < x < 2 C. x < –2 or x > 1 D. x < –1 or x > 2 E. no solution
8. 1997/II/32
Find the values of x which satisfy both –2x < 3 and (x + 3)(x – 2) < 0 .
A. x < –3 B. x > 2 C. –3 < x < –3
2 D. –
3
2 < x < 2 E. x < –3 or x > –
3
2
9. 1997/II/33
If a < b < 0 , then which of the following must be true?
I. a2 < b
2 II. ab < a
2 III.
1 1
a b<
A. I only B. II only C. III only D. I and II only E. I and III only
HKCEE MATHEMATICS | 8.1 Inequality - Linear Inequality | P.1
55
10. 1998/II/3
Solve x2 + 5x – 6 ≤ 0 .
A. –6 ≤ x ≤ 1 B. –3 ≤ x ≤ –2 C. –1 ≤ x ≤ 6 D. x ≤ –6 or x ≥ 1 E. x ≤ –1 or x ≥ 6
11. 1999/II/7
Solve x2 + 10x – 24 > 0 .
A. x < –12 or x > 2 B. x < –6 or x > –4 C. x < –2 or x > 12
D. –12 < x < 2 E. –2 < x < 12
12. 2000/II/6
Find the values of x which satisfy both 03 >+x and 12 <− x .
A. 3−>x B. 2
1−>x C.
2
1>x D.
2
13 −<<− x E.
2
13 <<− x
13. 2001/II/38
If ba > , which of the following must be true?
I. ba −<− II. bba >+ III. 22ba >
A. I only B. II only C. III only D. I and II only E. I, II and III
14. 2002/II/9
Solve 03)12(2)12( 2 >−−+− xx .
A. 20 << x B. 11 <<− x C. 0<x or 2>x D. 1−<x or 1>x
15. 2003/II/8
The solution of 1>x and 252313 <−< x is
A. 1>x B. 51 << x C. 91 << x D. 95 << x
16. 2004/II/9
The solution of xx −<− 32 or 033 >+x is
A. 3−>x B. 1−>x C. 13 −<<− x D. 3−<x or 1−>x
17. 2005/II/9
The solution of 2(3 − x) > −4 is
A. x < 5 B. x > 5 C. x < 10 D. x > 10
18. 2007/II/6
The solution of 15 ≥ 4(x + 2) – 1 is
A. x ≤ –2 B. x ≤ 2 C. x ≥ –2 D. x ≥ 2
HKCEE MATHEMATICS | 8.1 Inequality - Linear Inequality | P.2
56
1. 1991/II/38
Which one of the following shaded regions represents the solution of
≤≤≤≤≤+≤
40
40
62
y
x
yx
?
2. 1993/II/6
Find the greatest value of yx 23 + if ( )yx, is a point lying in the region OABCD (including the boundary).
A. 15 B. 13 C. 12 D. 9 E. 8
3. 1994/II/5
In the figure, (x , y)is a point in the shaded region (including the boundary) and x , y are integers. Find the
greatest value of yx +3 .
A. 7 B. 8 C. 9.2 D. 10 E. 10.5
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.1
57
4. 1995/II/8
Which of the following shaded regions represents the solution of
≤+−≥−
≥
02
3
0
yx
yx
y
A. I B. II C. III D. IV E. V
5. 1996/II/9
In the figure, (x , y) is any point in the shaded region (including the boundary). Which of the following is/are
true?
I. x ≤ y II. x + y ≤ 4 III. x ≤ 6
A. I only B. II only C. III only D. I and III only E. II and III only
6. 1997/II/9
Which of the following systems of inequalities has its solution represented by the shaded region in the figure?
A.
6
6
x y
x y
x
+ ≥ ≥ ≤ B.
6
6
x y
x y
y
+ ≥ ≥ ≤ C.
6
6
x y
x y
x
+ ≥ ≤ ≤ D.
6
6
x y
x y
y
+ ≥ ≤ ≤ E.
6
6
x y
x y
x
+ ≤ ≥ ≤
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.2
58
7. 1998/II/41
If b < 0 and c < 0 , which of the following shaded regions may represent the solution of x + by + c ≥ 0 ?
8. 1999/II/43
In the figure, find the point (x, y) in the shaded region (including the boundary) at which bx – ay + 3 attains its
greatest value.
A. (0, 0) B. (–a, b) C. (a, b) D. (b, –a) E. (b, a)
9. 2000/II/42
According to the figure, which of the following represents the solution of
≤≤≥
≤≤
30
40
y
yx
x
?
A. Region I B. Region II C. Regions I and VI D. Regions II and III E. Regions II, III, IV
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.3
59
10. 2001/II/49
The shaded region in the figure represents the solution of one of the following systems of inequalities.
Which is it?
A.
≥≤+≤−
0
6
02
x
yx
yx
B.
≥≤+≤−
0
6
02
y
yx
yx
C.
≥≥+≤−
0
6
02
y
yx
yx
D.
≥≤+≥−
0
6
02
y
yx
yx
11. 2003/II/43
Which of the following systems of inequalities has its solution represented by the shaded region in the figure?
A.
≥≥+≤−
0
10
023
x
yx
yx
B.
≥≤+≥−
0
10
023
x
yx
yx
C.
≥≥+≤−
0
10
023
y
yx
yx
D.
≥≤+≥−
0
10
023
y
yx
yx
12. 2004/II/43
Which of the following shaded regions may represent the solution of 2−≤ yx ?
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.4
60
13 . 2005/II/41
Which of the regions in the figure may represent the solution of
2
2
0
x
x y
x y
≤ + ≥ − ≥ ?
A. Region I B. Region II C. Region III D. Region IV
14. 2006/II/41
In the figure, O is the origin. The equation of AB is 2x + y – 8 = 0 and the equation of BC is 2x + 3y – 12 = 0 .
If (x, y) is a point lying in the shaded region OABC (including the boundary), then the greatest value of
x + 3y + 4 is
A. 8 B. 13 C. 16 D. 28
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.5
61
15. 2007/II/43
Which of the regions in the figure may represent the solution of
4
8
2 8
y
x y
x y
≥ + ≤ + ≥
A. Region I B. Region II C. Region III D. Region IV
16. 2008/II/42
In the figure, the equation of PQ and QR are 3x + y = 36 and x + y = 20 respectively. If (x, y) is a point
lying in the shaded region OPQR (including the boundary), then the least value of 2x – 3y + 180 is
A. 72 B. 120 C. 160 D. 204
HKCEE MATHEMATICS | 8.2 Inequality - Linear Programming | P.6
62
1. 2006/II/39
Convert the decimal number 213
+ 24 + 3 to a binary number.
A. 100000000001112
B. 100000000010112
C. 100000000100112
D. 100000001000112
2. 2007/II/41
ABCDE7000016
A. 10(169) + 11(16
8) + 12(16
7) + 13(16
6) + 14(16
5) + 7(16
4)
B. 10(1610
) + 11(169) + 12(16
8) + 13(16
7) + 14(16
6) + 7(16
5)
C. 11(169) + 12(16
8) + 13(16
7) + 14(16
6) + 15(16
5) + 7(16
4)
D. 11(1610
) + 12(169) + 13(16
8) + 14(16
7) + 15(16
6) + 7(16
5)
3. 2008/II/40
110000110001112 =
A. 213
+ 212
+ 27 + 2
6 + 7 .
B. 213
+ 212
+ 27 + 2
6 + 14 .
C. 214
+ 213
+ 28 + 2
7 + 7 .
D. 214
+ 213
+ 28 + 2
7 + 14 .
HKCEE MATHEMATICS | 9 Decimal, Binary and Hexadecimal Number | P.1
63
1. 1990/II/11
In the figure, the circular cylinder and the circular cone have the same height. The radius of the base of the
cylinder is twice that of the cone. If the volume of the cone is 20 cm3, what is the volume of the cylinder,
in cm3?
A. 20 B. 80 C. 120 D. 240 E. 300
2. 1990/II/12
The length, width and height of a cuboid are in the ratios 3 : 2 : 1. If the total surface area of the cuboid is
88 cm2, find its volume in cm3
A. 6 B. 48 C. 48 2 D. 96 2 E. 384
3. 1992/II/13
The figure shows a solid platform with steps on one side and a slope on the other. Find its volume.
A. 0.75m 3 B. 0.84 m 3 C. 0.858 m 3 D. 1.008 m 3 E. 1.608 m 3
4. 1992/II/17
In the figure, a cone of height 3h is cut by a plane parallel to its base into a smaller cone of height h and a frustum.
Find the ratio of the volume of the smaller cone to the volume of the frustum.
A. 1 : 27 B. 1 : 26 C. 1 : 9 D. 1 : 8 E. 1 : 7
HKCEE MATHEMATICS | 10 Mensurations | P.1
64
5. 1993/II/14
The price of a cylindrical cake of radius r and height h varies directly as the volume. If r = 5 cm and h = 4 cm,
the price is $30. Find the price when r = 4 cm and h = 6 cm.
A. $25 B. $28.80 C. $31.50 D. $36 E. $54
6. 1993/II/16
In the figure, the base of the conical vessel is inscribed in the bottom of the cubical box. If the box and the
conical vessel have the same capacity, find rh : .
A. π:24 B. 1:3 C. π:6 D. π:3 E. π3:8
7. 1993/II/17
The figure shows a solid consisting of a cylinder of height h and a hemisphere of radius r. The area of the curved
surface of the cylinder is twice that of the hemisphere. Find the ratio volume of cylinder : volume of hemisphere.
A. 1 : 3 B. 2 : 3 C. 3 : 4 D. 3 : 2 E. 3 : 1
HKCEE MATHEMATICS | 10 Mensurations | P.2
65
8. 1994/II/13
In the figure, the paper cup in the form of a circular cone contains 10 ml of water. How many ml of water must be
added to fill up the paper cup?
A. 20 B. 80 C. 90 D. 260 E. 270
9. 1995/II/15
In the figure, the solid consists of a cylinder and a right circular cone with a common base which is a circle
of radius 3 cm. The height of the cylinder is 10 cm and the slant height of the cone is 5 cm. Find the total
surface area of the solid.
A. π75 cm2 B. π84 cm2 C. π93 cm2 D. π105 cm2 E. π114 cm2
10. 1995/II/48
In the figure, a solid wooden sphere of radius 3 cm is to be cut into a cube of side x cm. Find the largest
possible value of x.
A. 23 B. 32 C. 3 D. 22
3 E. 3
HKCEE MATHEMATICS | 10 Mensurations | P.3
66
11. 1996/II/18
In the figure, A and B are two right solid cylinders with the same base radius 1 . If the heights of A and B are
1 and 2 respectively, find The total surface area of A
The total surface area of B.
A. 1
8 B.
1
4 C.
1
2 D.
3
5 E.
2
3
12. 1996/II/27
The figure shows a right circular cylinder with AC being a diameter of its upper face. AB and CD are two vertical
lines on the curved surface. A curve is drawn on the surface of the cylinder from B to C. Find its shortest possible
length.
A. 2π cm B. 22 4π + cm C. 4 2 cm D. 24 1π + cm E. 24 4π + cm
13. 1996/II/45
The figure shows a frustum of a right circular cone. The radii of the upper face and the base are 1 cm and 2 cm
respectively. If the height is 3 cm , find the volume.
A. 3π cm3 B. 9
2π cm3 C.
11
2π cm3 D. 7π cm3 E.
15
2π cm3
HKCEE MATHEMATICS | 10 Mensurations | P.4
67
14. 1997/II/49
In the figure, the rocket model consists of three parts. Parts I and III can be joined together to form a right
circular cone. Part II is a right cylinder. Find the volume of the rocket model.
A. 260π cm3 B. 360π cm3 C. 620π cm3 D. 720π cm3 E. 900π cm3
15. 1998/II/20
The figure shows a right circular cone of base radius 4 cm and height 3 cm. Find the area of its curved surface.
A. 12π cm2 B. 16π cm2 C. 20π cm2 D. 24π cm2 E. 48π cm2
16. 1998/II/22
The figure shows a test tube consisting of a cylindrical upper part of radius 1 cm and a hemispherical lower part
of the same radius. If the height of the test tube is 12 cm, find its capacity.
A. 35
3π cm3 B.
37
3π cm3 C.
38
3π cm3 D.
40
3π cm3 E.
68
3π cm3
HKCEE MATHEMATICS | 10 Mensurations | P.5
68
17. 1998/II/42
In the figure, a right pyramid with a square base is divided into three parts A, B and C by two planes parallel
to the base such that the lengths of their slant edges are 1 cm , 2 cm and 3 cm respectively.
Find volume of A : volume of B : volume of C.
A. 1 : 2 : 3 B. 1 : 4 : 9 C. 1 : 8 : 27 D. 1 : 26 : 189 E. 1 : 27 : 216
18. 1999/II/22
The figure shows a right prism. Find its total surface area.
A. 104 cm2 B. 108 cm2 C. 114 cm2 D. 120 cm2 E. 140 cm2
19. 1999/II/23
In the figure, a cylindrical vessel of internal diameter 6 cm contains some water. A steel ball of radius 2 cm is
completely submerged in the water. Find the rise in the water level.
A. 32
27cm B.
8
27cm C.
16
9cm D.
4
9cm E.
8
3cm
HKCEE MATHEMATICS | 10 Mensurations | P.6
69
20. 1999/II/24
In the figure, the solid consists of a right circular cone and a hemisphere with a common base. Find the volume
of the solid.
A. 30π cm3 B. 33π cm3 C. 48π cm3 D. 54π cm3 E. 72π cm3
21. 1999/II/37
In the figure, a right circular cone is divided into two parts X and Y by a plane parallel to the base such that the
lengths of their slant edges are 4 cm and 3 cm respectively. Find the ratio of the curved surface areas of X and Y.
A. 16 : 9 B. 16 : 33 C. 16 : 49 D. 64 : 27 E. 64 : 279
22. 2000/II/33
In the figure, a solid wooden sphere of radius r cm is to be cut into a cube of side 3 cm. Find the smallest possible
value of r.
A. 2
33 B.
2
23 C.
2
3 D. 33 E. 23
HKCEE MATHEMATICS | 10 Mensurations | P.7
70
23. 2000/II/43
In the figure, ABCDEF is a right triangular prism. It is cut into two parts along the plane PQRS, which is
parallel to the face BCDF, and 5:2: =PBAP . Find ABCDEF
APQRES
prism theof volume
prism theof volume.
A. 7
2 B.
25
4 C.
49
4 D.
125
8 E.
343
8
24. 2001/II/8
In the figure, the solid consists of a cylinder and a hemisphere with a common base of radius 6 cm. Find the total
surface area of the solid.
A. 132π cm2 B. 168π cm2 C. 204π cm2 D. 240π cm2 E. 324π cm2
25. 2001/II/24
In the figure, a solid right circular cone of height 12 cm is put into a cylinder which has the same internal radius
as the base radius of the cone. Water is then poured into the cylinder until the water level just reaches the tip of
the cone. If the cone is removed, what is the height of the water in the cylinder?
A. 3 cm B. 4 cm C. 6 cm D. 8 cm E. 9 cm
HKCEE MATHEMATICS | 10 Mensurations | P.8
71
26. 2002/II/19
The figure shows a hemisphere, a right circular cone and a right cylinder with equal base radii. Their volumes
are a cm3, b cm3 and c cm3 respectively. Which of the following is true?
A. cba << B. bca << C. bac << D. abc <<
27. 2002/II/45
In the figure, P and Q are two right cylindrical vessels each containing some water. The two vessels are placed
on the same horizontal surface. The internal base radii of P and Q are in the ratio 1 : 3. A and B are two cubes
with sides in the ratio 1 : 2. A and B are put into P and Q respectively. Suppose both cubes are totally immersed in
water without any overflow. If the rise in water level in P is 1 cm, then the rise in water level in Q is
A. 3
2cm B.
8
9 cm C.
9
8 cm D.
27
8 cm
28. 2003/II/20
The figure shows a right circular cone of base radius 6 cm and height 8 cm. Find its volume.
A. 32π cm3 B. 60π cm3 C. 96π cm3 D. 288π cm3
HKCEE MATHEMATICS | 10 Mensurations | P.9
72
29. 2003/II/21
In the figure, the solid consists of a right circular cone and a hemisphere with a common base. Find the total
surface area of the solid.
A. 136π cm2 B. 248π cm2 C. 264π cm2 D. 392π cm2
30. 2005/II/17
The figure shows a solid right circular cone of height 5 cm and slant height 13 cm. Find the total surface area of
the cone.
A. 144π cm2 B. 156π cm2 C. 240π cm2 D. 300π cm2
31. 2005/II/18
The figure shows a right triangular prism. Find the volume of the prism.
A. 36 cm3 B. 72 cm3 C. 36 3 cm3 D. 72 3 cm3
32. 2006/II/17
In the figure, the area of the trapezium ABCD is
A. 345 cm2 B. 349 cm2 C. 690 cm2 D. 698 cm2
HKCEE MATHEMATICS | 10 Mensurations | P.10
73
33. 2006/II/18
In the figure, the solid consists of a hemisphere of radius 3 cm joined to the bottom of a right circular cylinder of
height 8 cm and base radius 3 cm . Find the volume of the solid.
A. 75π cm3 B. 90π cm3 C. 93π cm3 D. 108π cm3
34. 2006/II/20
In the figure, sector OXY is a thin metal sheet. By joining OX and OY together, which of the following right
circular cones can be folded?
35. 2007/II/16
In the figure, OAB is a sector with centre O . If the perimeter of the sector OAB is 12 cm , find OA correct to
the nearest 0.01 cm .
A. 3.36 cm B. 3.91 cm C. 4.31 cm D. 7.64 cm
HKCEE MATHEMATICS | 10 Mensurations | P.11
74
36. 2007/II/17
In the figure, the volume of the right prism is
A. 456 cm3 B. 540 cm3 C. 552 cm3 D. 636 cm3
37. 2007/II/18
If a solid metal hemisphere of radius r is melted and recast into 3 identical solid right circular cones of height
h and base radius r , then r : h =
A. 2 : 3 B. 3 : 2 C. 3 : 4 D. 4 : 3
38. 2008/II/16
If the radius of a sphere is measured as 8 cm correct to the nearest cm , then the least possible surface area of the
sphere is
A. 64π cm2 B. 225π cm2 C. 256π cm2 D. 1125
2
πcm2
39. 2008/II/18
The figure shows a solid right circular cone of height 12 cm . The circumference of the base is 18π cm .
Find the total surface area of the circular cone.
A. 81π cm2 B. 135π cm2 C. 216π cm2 D. 324π cm2
HKCEE MATHEMATICS | 10 Mensurations | P.12
75
40. 2008/II/19
In the figure, the volume of the right prism is
A. 128 cm3 B. 332 cm3 C. 384 cm3 D. 768 cm3
HKCEE MATHEMATICS | 10 Mensurations | P.13
76
1. 1990/II/13
In the figure, there are nine circles, each of radius 1. Find the shaded area.
A. 9 − 9π B. 36 − 9π C. 40 − 9π D. 10 − 10π E. 40 − 10π
2. 1990/II/20
In the figure, TQ is the tangent to the tangent to the circle at A. If arc AC = arc BC and ∠PAQ = 48o,
then ∠QAC =
A. 42o B. 48
o C. 66
o D. 71
o E. 84
o
3. 1990/II/21
In the figure, O is the centre of the circle. If OR // PQ and ∠ROQ = 42o, find ∠RMQ.
A. 21o B. 42
o C. 63
o D. 84
o E. 126
o
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.1
77
4. 1990/II/40
In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is greater
than the perimeter of the triangle by
A. 4π − 3 3 B. 4π −2
33 C. 2π − 3 D. 2π −
2
33 E. 2π − 3 3
5. 1990/II/41
Three equal circles of radii 1 touch each other as shown in the figure, shaded area =
A. 1−2
π B. 3 −
2
π C. 2 3 −
2
π D. 3 −
6
π E. 2 3 −
6
π
6. 1990/II/48
In the figure, AB is a diameter and ∠BAC = 30o. If the area of ∆ABC is 3 , then the radius of the circle is
A. 2
1 B. 1 C. 2 D. 3 E. 2
7. 1990/II/50
In the figure, PA and PC are tangents to the circle ABC. If ∠P = 48o, then ∠ABC =
A. 84o B. 96
o C. 106
o D. 114
o E. 132
o
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.2
78
8. 1990/II/51
In the figure, TA and TB are tangents to the circle ABC. If TA ⊥ TB and BD ⊥ AC, find ∠CBD.
A. 30o B. 40
o C. 45
o D. 50
o E. 60
o
9. 1990/II/53
In the figure AB, AC and BC are three tangents touching the circle at D, E and F respectively.
If AC = 24, BC = 18 and ∠ACB = 90o, find the radius of the circle.
A. 3 B. 4 C. 5 D. 6 E. 7
10. 1991/II/13
In the figure, TB touches the semi-circle at B. TA cuts the semi-circle at P such that TP = PA.
If the radius of the semi-circle is 2, find the area of the shaded region.
A. 12 − π B. 8 − π C. 6 − π D. 4 − π E. 2(4 − π )
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.3
79
11. 1991/II/19
In the figure, XPY and YQZ are semi-circles with areas A1 and A2 respectively.∠YXZ = 60
o and ∠YZX = 45
o.
The ratio A1 : A2 =
A. 2 : 3 B. 2 : 3 C. 2 : 3 . D. 2 : 3 E. 3 : 2
12. 1991/II/21
In the figure, O is the centre of the circle. Find a + c.
A. b B. 2b C. 180o − b D. 360
o − b E. 360
o − 2b
13. 1991/II/22
In the figure, O is the centre of the circle BCD. ABC and EDC are straight lines. BC = DC and ∠AED = 70o.
Find ∠BOD.
A. 40o B. 70
o C. 80
o D. 90
o E. 140
o
14. 1991/II/24
In the figure, TPA and TQB are tangents to the circle at P and Q respectively. If PQ = PR, which of the
following must be true?
I. ∠APR = ∠QRP
II. ∠QTP = ∠QPR
III. ∠QPR = ∠APR
A. I only B. II only C. III only D. I and II only E. I and III only
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.4
80
15. 1991/II/44
From a rectangular metal sheet of width 3 cm and length 40 cm, at most how many circles each of radius 1 cm
can be cut?
A. 20 B. 21 C. 22 D. 23 E. 24
16. 1991/II/52
In the figure, arc AB : arc BC : arc CD : arc DE : arc EA = 1 : 2 : 3 : 4 : 5. Find θ.
A. 30o B. 36
o C. 60
o D. 72
o E. 120
o
17. 1992/II/14
In the figure, TP and TQ are tangent to the circle of radius 3cm. Find the length of the minor arc PQ .
A. π3 cm B. π2 cm C. 2
3πcm D. π cm E.
2
πcm
18. 1992/II/16
In the figure, the equilateral triangle ACE of side 4 cm is inscribed in the circle. Find the area of the inscribed
regular hexagon ABCDEF.
A. 38 cm 2 B. 28 cm 2 C. 34 cm 2 D. 24 cm 2 E. 16 cm 2
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.5
81
19. 1992/II/24
In the figure, O is the center of the circle. findθ .
A. 42º B. 36º C. 24º D. 21º E. 18º
20. 1992/II/26
In the figure, the circle is inscribed in a regular pentagon. P, Q and R are points of contact. Find θ .
A. 30º B. 32º C. 35º D. 36º E. 45º
21. 1992/II/27
In the figure, ST is a tangent to the smaller circle. ABC is a straight line. If xTAD 2=∠ and xDPC 3=∠ , find x .
A. 30º B. 36º C. 40º D. 42º E. 45º
22. 1992/II/50
In the figure, the two circles touch each other at C. The diameter AB of the bigger circle is tangent to the smaller
circle at D. If DE bisects ADC∠ , find θ .
A. 24º B. 38º C. 45º D. 52º E. 66º
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.6
82
23. 1992/II/52
In the figure, O is the center of the circle. If the diameter AOB rotates about O, which of the following is/are
constant?
I. φθ + II. BDAC + III. BDAC ×
A. I only B. II only C. III only D. I and II only E. I and III only
24. 1993/II/26
In the figure, AB is a diameter. Find ∠ADC.
A. 100º B. 110º C. 120º D. 135º E. 140º
25. 1993/II/49
In the figure, if 3:2:1:: =arcABarcCAarcBC , which of the following is/are true?
I. 3:2:1:: =∠∠∠ CBA II. 3:2:1:: =cba III. 3:2:1sin:sin:sin =CBA
A. I only B. II only C. III only D. I and II only E. I, II and III
26. 1993/II/50
In the figure, TP and TQ are tangents to the circle at P and Q respectively. If M is a point on the minor arc PQ
and ∠PMQ= θ, then ∠PTQ =
A. 2
θ B. °− 90θ C. θ−°180 D. θ2180 −° E. °−1802θ
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.7
83
27. 1993/II/51
In the figure, O is the center of the circle. AB touches the circle at N. Which of the following is/are correct?
I. M, N, K, O are concyclic. II. ∆HNB ~ ∆NKB III. ∠OAN =∠NOB
A. I only B. II only C. III only D. I and II only E. I, II and III
28. 1993/II/54
In the figure, the three circles touch one another. XY is their common tangent. The two larger circles are equal.
If the radius of the smaller circle is 4 cm, find the radii of the larger circles.
A. 8 cm B. 10 cm C. 12 cm D. 14 cm E. 16 cm
29. 1994/II/19
In the figure, ABCD is a cyclic quadrilateral with AB = 5, BC = 2 and ADC∠ = 120º. Find AC.
A. 19 B. 21 C. 2 6 D. 34 E. 39
30. 1994/II/21
In the figure, O is the center of the circle. If AC = 3 and BAC∠ = ,6
π find the diameter AB.
A. 2
3 B. 6 C.
2
33 D. 2 3 E. 3 3
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.8
84
31. 1994/II/22
In the figure, PA is tangent to the circle at A, ∠CAP = 28º and BA = BC. Find x.
A. 28º B. 48º C. 56º D. 62º E. 76º
32. 1994/II/23
In the figure, O is the center of the inscribed circle of ∆ABC. If ∠OAC = 30º and ∠OCA = 25º, find ∠ABC.
A. 50º B. 55º C. 60º D. 62.5º E. 70º
33. 1994/II/44
In the figure, CDEF is a sector of a circle which touched AB at E. If AB = 25 and BC = 15, find the radius of the
sector.
A. 9 B. 10 C. 11.25 D. 12 E. 12.5
34. 1994/II/51
In the figure, ABCD is a semi-circle, CDE and BAE are straight lines. If ∠CBD = 30º and ∠DEA = 22º, find x.
A. 38º B. 41º C. 44º D. 52º E. 60º
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.9
85
35. 1994/II/52
In the figure, OABCD is a sector of a circle. If ∩∩∩
== CDBCAB , then x =
A. 105º B. 120º C. 135º D. 144º E. 150º
36. 1995/II/22
In the figure, ABCD is a semicircle. ∠CAD =
A. 25º B. 40º C. 45º D. 50º E. 65º
37. 1995/II/23
In the figure, O is the center of the circle, POQR is a straight line. TR is the tangent to the circle at T. =∠PRT
A. 20º B. 35º C. 45º D. 50º E. 70º
38. 1995/II/45
In the figure, O is the center of the circle. Find the area of the major segment ABC.
A. 2
4rπ
B. 2
4
3rπ
C. 2
2
1
4r
−π D. 2
2
1
4
3r
−π E. 2
2
1
4
3r
+π
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.10
86
39. 1995/II/46
In the figure, C1 and C2 are two circles. If area of region I : area of region II : area of region III = 2 : 1 : 3, then
radius of C1 : radius C2 =
A. 9 : 16 B. 2 : 3 C. 3 : 4 D. 2 : 3 E. 3 : 2
40. 1996/II/16
In the figure, O is the centre of the circle. AB and AC are tangents to the circle at B and C respectively. Area of
the shaded region =
A. (2 )6
π− cm
2 B. (2 )
3
π− cm
2 C. ( 3 )
6
π− cm
2 D. ( 3 )
3
π− cm
2 E.
3( )
2 6
π− cm
2
41. 1996/II/25
In the figure, O is the centre of the circle. Find x.
A. 20° B. 27.5° C. 35° D. 37.5° E. 40°
42. 1996/II/26
In the figure, O is the centre of the circle. PA is the tangent to the circle at A and CB // PA. Find x.
A. 21° B. 24° C. 42° D. 45° E. 48°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.11
87
43. 1996/II/50
In the figure, O is the centre of the circle. AP, AB and BR are tangents to the circle at P, Q and R respectively.
Which of the following must be true?
I. AP + BR = AB
II. OQ bisects ∠AOB
III. ∠AOB =1
2∠POR
A. I only B. II only C. I and II only D. I and III only E. I, II and III
44. 1997/II/16
In the figure, BEA is a semicircle. ABCD is a rectangle and DC touches the semicircle at E. Find the area of the
shaded region.
A. 9π B. 18π C. 36π D. 36 − 9π E. 36 + 9π
45. 1997/II/18
In the figure, BCA is a semicircle. If AC = 6 and CB = 4, find the area of the semicircle.
A. 5
2π B.
13
2π C. 10π D. 13π E. 26π
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.12
88
46. 1997/II/20
In the figure, EC is the tangent to the circle at C . Find ∠CBD .
A. 40° B. 50° C. 65° D. 70° E. 75°
47. 1997/II/48
In the figure, OXY is a sector with centre O. If Z is the mid point of YO, find area of triangle ∆OXZ : area of sector
∆OXY.
A. 1 : 2 B. 2 : 3π C. 2 : 3π D. 3 : 2π E. 3 3 : 2π
48. 1997/II/50
In the figure, AC is the angle bisector of ∠BAD . Which of the following statements must be true?
I. ∆BCE ~ ∆ADE II. ∆ABC ~ ∆AED I. ∆ABC ~ ∆BDA
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
49. 1997/II/51
In the figure, � 2AB = , � 3BC = , � 4CD = and � 6DA = , Find ∠BCD.
A. 72 ° B. 84° C. 90° D. 96° E. 144°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.13
89
50. 1998/II/19
In the figure, the radii of the two circles are 3 cm and 1 cm respectively. Find the ratio of the area of the shaded
part to that of the smaller circle.
A. 2 : 1 B. 3 : 1 C. 4 : 1 D. 8 : 1 E. 9 : 1
51. 1998/II/23
In the figure, OABC is a sector. Find the area of the shaded region.
A. (π – 2) cm2 B. (2π – 4) cm
2 C. (4π – 8) cm
2 D. (8π – 8) cm
2 E. (8π – 16) cm
2
52. 1998/II/28
In the figure, AB is a diameter of the circle and ABD is a straight line. ∠CBD =
A. 2θ B. 4θ C. 90° + θ D. 180° − θ E. 180° − 2θ
53. 1998/II/29
In the figure, AD is a diameter of the circle. If � � �: : 3 :5 : 7AB BC CD = , then ∠ADC =
A. 36° B. 45° C. 48° D. 49° E. 72°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.14
90
54. 1998/II/46
In the figure, ABC is a semicircle. Find the area of the shaded part.
A. 6π cm2
B. 15π cm2 C. (6 9 3)π − cm
2 D. (6 9 3)π + cm
2 E. (12 9 3)π − cm
2
55. 1998/II/49
In the figure, CE is tangent to the circle at C . Find ∠DCE .
A. 40° B. 42° C. 49° D. 54° E. 78°
56. 1999/II/21
In the figure, a square is inscribed in a circle with radius 1 cm. Find the area of the shaded region.
A. (π – 2) cm2
B. (π – 2 ) cm2
C.
(π – 1) cm2 D. (2π – 2) cm
2 E.
(2π – 1) cm
2
57. 1999/II/25
In the figure, ABCD is a semicircle. Find the area of the shaded region correct to the nearest 0.01 cm2.
A. 5.33 cm2
B. 2.87 cm2
C. 2.67 cm2 D. 1.33 cm
2 E. 0.17 cm
2
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.15
91
58. 1999/II/26
In the figure, O is the centre of the circle. Find x.
A. 12° B. 20° C. 24° D. 40° E. 60°
59. 1999/II/27
In the figure, AB is a diameter of the circle. Find x.
A. 26° B. 32° C. 38° D. 52° E. 64°
60. 1999/II/50
In the figure, AT is tangent to the circle at T and ABC is a straight line. Find AT.
A. 9 cm B. 12 cm C. 15 cm D. 16 cm E. 20 cm
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.16
92
61. 2000/II/20
In the figure, O is the center of the circle. EAOB and EDC are straight lines. Find x.
A. 40° B. 46° C. 57° D. 66° E. 68°
62. 2000/II/25
In the figure, PXQ, QYR and RZP are semicircles with areas A1 cm2, A2 cm
2 and A3 cm
2 respectively.
If 121 =A and 52 =A , find A3
.
A. 13 B. 17 C. 169 D. 13π E. 8
169π
63. 2000/II/31
In the figure, CAB is a semicircle and ABCD is a parallelogram. Find the area of ABCD.
A. 65 cm2 B. 60 cm
2 C. 52 cm
2 D. 32.5 cm
2 E. 30 cm
2
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.17
93
64. 2000/II/45
In the figure, AB is tangent to the circle at B. Find ∠DCE.
A. 70° B. 75° C. 90° D. 95° E. 105°
65. 2000/II/46
In the figure, : : 2 :1:3AB BC CD =� � �
. Find ∠ADC.
A. 56° B. 60° C. 63° D. 72° E. 84°
66. 2001/II/18
In the figure, AEC is a diameter and DEB is a straight line. Find x.
A. 54° B. 70° C. 74° D. 92° E. 94°
67. 2001/II/25
In the figure, OABC is a sector. Find the length of the arc ABC.
A. 2
3
πcm B. 4π cm C. 5π cm D. 6π cm E. 12π cm
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.18
94
68. 2001/II/26
In the figure, A, B and C are the centres of three equal circles, each of radius 1 cm. Find the area of the shaded
region.
A. 3
2 2
π − cm
2 B.
3 3
2 4
π − cm
2 C.
3
2 4
π + cm
2 D.
2
πcm
2 E.
3
2 4
π − cm
2
69. 2001/II/32
In the figure, ABCD is a semicircle, 3:4: =BDAB . Find AB correct to the nearest 0.1 cm.
A. 5.7 cm B. 7.6 cm C. 10.7 cm D. 13.0 cm E. 14.3 cm
70. 2001/II/45
In the figure, O is the center of the circle, AOB is a straight line and BCD is the tangent to the circle at C. Find x.
A. 50° B. 53° C. 56° D. 59° E. 62°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.19
95
71. 2001/II/46
In the figure, 1
2AB BC CD= =� � �
. Find ∠ABC.
A. 100° B. 105° C. 112.5° D. 130° E. 150°
72. 2002/II/17
The figure shows a rectangular inscribed in a circle. Find the area of the shaded region correct to the
nearest 0.1 cm2.
A. 60.0 cm2 B. 72.7 cm
2 C. 132.7 cm
2 D. 470.9 cm
2
73. 2002/II/20
In the figure, OCD and OAB are two sectors. The length of �AB is
A. 8
3π cm B.
10
3π cm C. (2π + 2) cm D. 4π cm
74. 2002/II/28
In the figure, O is the center of the semicircle ABCD and BC // AD. If °=∠ 42COD , then =x
A. 48° B. 63° C. 84° D. 90°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.20
96
75. 2002/II/29
In the figure, 1AED =�
and 4CFD =�
. If °=∠ 100ABC , then =∠ABD
A. 18° B. 20° C. 24° D. 25°
76. 2002/II/51
In the figure, EAF is a common tangent to the circles at the point A. Chords AC and BC of the smaller
circle are produced to meet the larger circle at G and D respectively. Which of the following must be true?
I. EAGADG ∠=∠
II. AGDABD ∠=∠
III. ADBBAE ∠=∠
A. I only B. II only C. I and III only D. II and III only
77. 2003/II/19
In the figure, OAB is a sector and�AB = π cm. Find the area of the sector.
A. 3
2π cm
2 B. 3π cm
2 C.
9
2π cm
2 D. 6π cm
2
78. 2003/II/25
In the figure, ABC is a semicircle with 7BC =�
and °=∠ 55ACB . Find AB�
.
A. 9 B. 10 C. 11 D. 14
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.21
97
79. 2003/II/50
The figure shows a circle with diameter AD. If CDBCAB == , find zyx ++ .
A. 315° B. 324° C. 330° D. 360°
80. 2003/II/51
In the figure, XAB and XDC are straight lines. If 5=DX , 6=AX and 4=AB , find CD.
A. 5 B. 7 C. 3
10 D.
5
24
81. 2003/II/52
In the figure, BE and BF are tangents to the circle at A and C respectively. If °=∠ 100ADC , then =∠ABC
A. 20° B. 30° C. 40° D. 50°
82. 2004/II/23
In the figure, O is the center of the circle ABCD. If EAB and EDOC are straight lines and AOEA = , find ∠AEO.
A. 18° B. 24° C. 27° D. 36°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.22
98
83. 2004/II/24
In the figure, O is the center of the circle ABC. Find x.
A. 17.5° B. 27.5° C. 35° D. 55°
84. 2004/II/25
In the figure, ABCD is a circle. AC and BD meet at E. If 4=AD , 2=AE , 5=EC and 4=BE , then =BC
A. 6 B. 7 C. 8 D. 10
85. 2004/II/26
In the figure, ABC is a circle. If °=∠ 30ABC and 4=�
AC , then the circumference of the circle is
A. 24 B. 48 C. 8π D. 16π
86. 2004/II/50
In the figure, ABCD is a circle. If ����
BCABDACD 222 === , then =x
A. 108° B. 112° C. 120° D. 144°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.23
99
87. 2004/II/51
In the figure, TS, SQ and QP are tangents to the circle at T, R and P respectively. If TS // PQ, 3=TS and
12=QP , then the radius of the circle is
A. 4.5 B. 6 C. 7.5 D. 9
88. 2005/II/19
In the figure, OAB is a sector of radius 2 cm. If the length of �AB is 3π cm, then the area of the sector OAB is
A. 3
2
πcm
2. B. 3π cm
2. C. 4π cm
2. D. 6π cm
2.
89. 2005/II/24
In the figure, ABCD is a circle. AB produced and DC produced meet at E. If AC and BD intersect at F,
then ∠ABD =
A. 41° B. 52° C. 56° D. 60°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.24
100
90. 2005/II/25
In the figure, ABCD is a circle. If AC is a diameter of the circle and AB = BD, then ∠CAD =
A. 18° B. 21° C. 27° D. 36°
91. 2005/II/49
In the figure, AB and AC are tangents to the circle at X and Y respectively. Z is a point lying on the circle.
If ∠BAC = 100°, then ∠XZY =
A. 40° B. 45° C. 50° D. 55°
92. 2005/II/50
In the figure, O is the centre of the circle and AOC is a straight line. If AB and BC are tangents to the circle such
that AB = 3, and BC = 4, then the radius of the circle is
A. 3
2 B.
12
7 C. 2 D.
5
2
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.25
101
93. 2005/II/51
In the figure, ABCD is a circle. If : : : 1: 2 :3 : 3AB BC CD DA =� � � �
and E is a point lying on BD, then ∠CAE =
A. 45° B. 50° C. 55° D. 60°
94. 2006/II/19
In the figure, O is the center of the circle. B and C are points lying on the circle. If OC = 2 cm and OA = 1 cm,
then the area of the shaded region OABC is
A. 2
πcm
2 B.
2
3
πcm
2 C.
3
2 3
π + cm
2 D.
23
3
π + cm2
95. 2006/II/46
In the figure, O is the centre of the circle ABC. If ∠OBC = 50° and ∠ACO = 20° , then ∠BOA =
A. 50° B. 60° C. 70° D. 80°
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.26
102
96. 2006/II/47
In the figure, O is the centre of the circle. A and B are points lying on the circle. If AOC is a straight line and
BC is a tangent to the circle, then the radius of the circle is
A. 3
2 B. 3 C. 2 3 D. 3 3
97. 2007/II/48
In the figure, A, B, C and D are points lying on the circle. If AB = 5, AD = 3 and BD = 7 , then ∠BCD =
A. 60° B. 85° C. 95° D. 120°
98. 2007/II/49
In the figure, A, B and C are points lying on the circle. AB is a diameter of the circle. DB is the tangent to
the circle at B. If ACD is a straight line with AC = 4 and CD = 2 , then AB =
A. 2 6 B. 4 3 C. 4 6 D. 8 3
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.27
103
99. 2008/II/20
In the figure, OAB and OCD are sectors with centre O . It is given that the area of the shaded region ABCD
is 54π cm2 . If AC = 6 cm , then OA =
A. 15 cm B. 21 cm C. 24 cm D. 30 cm
100. 2008/II/50
In the figure, O is the centre of the circle ABCD. If ∠ADC = 84° and ∠CBO = 38° , then ∠AOB =
A. 64° B. 88° C. 104° D. 168°
101. 2008/II/51
In the figure, AB is the tangent to the circle at B and ADC is a straight line. If AB : AD = 2 : 1 , then the
area of ∆ABD : the area of ∆BCD =
A. 1 : 2 B. 1 : 3 C. 1 : 4 D. 2 : 3
HKCEE MATHEMATICS | 11.1 Plane Geometry - Circles | P.28
104
1. 1991/II/54
In the figure, which of the pairs of triangles must be congruent?
A. I only B. II only C. I and III only D. II and III only E. I,II and III.
2. 2000/II/24
In the figure, CDAB = , ECDCAB ∠=∠ and CDEABC ∠=∠ . Which of the following must be true?
I. CDEABC ∆≅∆
II. EACABC ∆∆ ~
III. EAC is an isosceles triangle
A. I only B. III only C. I and II only D. I and III only E. I, II and III only
3. 2001/II/19
Which of the following pairs of triangles is/are similar?
A. II only B. III only C. I and II only D. I and III only E. I, II and III
HKCEE MATHEMATICS | 11.2 Plane Geometry - Similar and Congruent Triangles | P.1
105
4. 2002/II/26
In the figure, ABC and AFED are straight lines . CDEABF ∠=∠ and BE // CD. Which of the following triangles
are similar?
I. ∆ABF
II. ∆AEB
III. ∆ADC
A. I and II only B. I and III only C. II and III only D. I, II and III
5. 2003/II/27
Which of the following statements about the triangles in the figure must be true?
A. I and III are similar. B. I and IV are similar. C. II and III are similar. D. II and IV are similar.
HKCEE MATHEMATICS | 11.2 Plane Geometry - Similar and Congruent Triangles | P.2
106
1. 1990/II/23
In the figure, ABCDE is a regular pentagon. Find ∠AFD.
A. 120o B. 112
o C. 110
o D. 108
o E. 100
o
2. 1990/II/39
In the figure, AM = MB = MC = 5 and BC = 6. The area of triangle ABC =
A. 12 B. 16 C. 24 D. 30 E. 48
3. 1991/II/12
In the figure, ABCD is a square of side a and MNPQ is a square of side b. The four trapeziums are identical.
The area of the shaded region is:
A. 4
3 22ab +
B. 2
3 22ab −
C. 4
5 22ab +
D. 4
5 22ab −
E. 4
)( 2ba −
+ b2
4. 1991/II/14
An equilateral triangle and a square have equal perimeters. square theof Area
triangle theof Area =
A. 16
39 B.
4
3 C.
3
3 D.
9
34 E. 1
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.1
107
5. 1991/II/23
In the figure, ABCDE and ABXYZ are two identical regular pentagons. Find ∠AEZ.
A. 15o B. 18
o C. 24
o D. 30
o E. 36
o
6. 1991/II/25
In the figure, E and F are the mid-points of AB and AC respectively. G and H divide DB and DC respectively
in the ratio 1 : 3. If EF = 12, find GH.
A. 3 B. 4 C. 6 D. 8 E. 12
7. 1992/II/54
In the figure, ABCD is a square of side a and BDEF is a rhombus. CEF is a straight line. Find the length of
the perpendicular from B to DE.
A. a2
1 B.
3
2a C.
2
a D. a
2
3 E. a
8. 1993/II/38
In the figure, the rectangle has perimeter 16cm and area 15cm2. Find the length of its diagonal AC.
A. 32 cm B. 34 cm C. 7cm D. 226 cm E. 241 cm
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.2
108
9. 1993/II/41
In the figure, ABCD is a square and ABE is an equilateral triangle. =ABCD
ABE
of Area
of Area
A. 4
1 B.
3
1 C.
8
3 D.
4
3 E.
2
3
10. 1993/II/52
In the figure, ABCD and EFGH are two squares and ACH is an equilateral triangle. Find AB:EF.
A. 1 : 2 B. 1 : 3 C. 2:1 D. 3:1 E. 3:2
11. 1993/II/53
In the figure, a rectangular piece of paper ABCD is folded along EF so that C and A coincide. If 12=AB cm,
16=BC cm, find BE .
A. 3.5cm B. 4.5cm C. 5cm D. 8cm E. 12.5cm
12. 1994/II/14
In the figure, ABCD is a rectangular field of length p metres and width q metres. The path around the field is
of width 2 metres. Find the area of the path.
A. ( )qp 44 + m2 B. ( )422 ++ qp m
2 C. ( )1622 ++ qp m
2
D. ( )1644 ++ qp m2 E. ( )1644 +++ qppq m
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.3
109
13. 1994/II/53
In the figure, AB//DC and ∠DAB=∠DBC. Which of the following is/are true?
I. DC
BD
BD
AB=
II. BC
AD
BD
AB=
III. CD
BD
BD
AD=
A. I only B. II only C. III only D. I and II only E. II and III only
14. 1994/II/54
In the figure, ABCD is a trapezium with AB // DC, ∠ABC = 90º and MN is the perpendicular bisector of AD.
If AB = 9, BN = 2 and NC = 6, find CD.
A. 2
14 B. 6
4
3 C. 7 D. 41 E. 113
15. 1995/II/14
In the figure, ABCD is a trapezium. Find its area.
A. 36cm2
B. 45cm2 C. 48cm
2 D. 72cm
2 E. 90cm
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.4
110
16. 1995/II/24
In the figure, ABCD is a cyclic quadrilateral. If °=∠ 110DAB and BC = BD, find ∠DAC.
A. 20º B. 35º C. 40º D. 55º E. 70º
17. 1996/II/17
In the figure, the area of ABCD is
A. 36 B. 40 C. 44 D. 4 21 24+ E. 4 29 24+
18. 1996/II/28
In the figure, ABCDE is a regular pentagon and ABF is an equilateral triangle. Find x.
A. 120° B. 126° C. 144° D. 156° E. 168°
19. 1996/II/46
In the figure, if Area of triangle 1
Area of triangle 2
CDE
BCE= , find
Area of triangle
Area of trapezium
CDE
ABCD.
A. 1
10 B.
1
9 C.
1
8 D.
1
7 E.
1
6
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.5
111
20. 1996/II/51
In the figure, ABCD is a trapezium with AB // DC. AH bisects ∠BAD and DH bisects ∠ADC . Which of the
following must be true?
I. ∠AHD = 90°
II. ∠ADC = ∠BCD
III. ∠BAD + ∠BCD = 180°
A. I only B. II only C. III only D. I and III only E. II and III only
21. 1997/II/19
In the figure, ABCDE is a regular pentagon and ABF is an equilateral triangle. Find θ .
A. 66° B. 84° C. 90° D. 96° E. 108°
22. 1997/II/52
In the figure, ABCD is a parallelogram. PDC, PQRS and ABS are straight lines. If AQ = 4, QD = 2 and
BR = RC = 3, then PQ : QR : RS =
A. 1 : 1 : 1 B. 1 : 2 : 6 C. 2 : 1 : 3 D. 2 : 3 : 4 E. 8 : 12 : 9
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.6
112
23. 1997/II/53
In the figure, ABCD is a rectangle. CDE is a straight line and AE // BD. If the area of ABCD is 24 and F is a point
on BC such that BF : FC = 3 : 1, find the area of ∆DEF .
A. 2 B. 3 C. 4 D. 6 E. 8
24. 1997/II/54
In the figure, AB // DC. If the areas of ∆ABE and ∆CDE are 4 and 9 respectively, find the area of ∆BCE .
A. 4 B. 5 C. 6 D. 6.5 E. 9
25. 1998/II/21
In the figure, find the area of the pentagon ABCDE.
A. 16 cm2
B. 18 cm2 C. 20 cm
2 D. 24 cm
2 E. 32 cm
2
26. 1998/II/26
In the figure, PQRS is a trapezium. Find x correct to 3 significant figures.
A. 3.01 B. 5.57 C. 5.77 D. 6.00 E. 9.54
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.7
113
27. 1998/II/30
In the figure, AB = BC = CA = CD. Find ∠CBD .
A. 20° B. 25° C. 27.5° D. 30° E. 35°
28. 1998/II/38
In the figure, ABCD is a trapezium. Which of the following must be true?
I. AED is an equilateral triangle II. EBCD is a parallelogram III. AB = 2DC
A. I only B. II only C. I and II only D. I and III only E. I, II and III
29. 1999/II/20
In the figure, ABCD is a parallelogram. Find ∠ABC correct to the nearest degree.
A. 83° B. 97° C. 104° D. 124° E. 139°
30. 1999/II/29
In the figure, ABCDE is a regular pentagon and ABFG is a square. Find x.
A. 18° B. 27° C. 30° D. 36° E. 45°
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.8
114
31. 1999/II/54
In the figure, ABCD is a rectangle. M is the midpoint of BC and AC intersects MD at N.
Area of ∆NCD : area of ∆ABMN =
A. 1 : 2 B. 1 : 3 C. 2 : 3 D. 2 : 5 E. 4 : 7
32. 2000/II/7
In the figure, a square of side x cm is cut into 9 equal squares. If the total perimeter of the 9 small squares is
72 cm more than the perimeter of the original square, then =x
A. 6 B. 8 C. 9 D. 12 E. 18
33. 2000/II/8
The figure shows a trapezium of area 6 cm2. Find x.
A. 2 B. 3 C. 4 D. 6 E. 11
34. 2000/II/12
In the figure, ABCD is a rectangle formed by four squares each of area 1 cm2. DB is a diagonal. Find the area of
the shaded region.
A. 10
9 cm
2 B.
8
7 cm
2 C.
6
5 cm
2 D.
5
4 cm
2 E.
4
3 cm
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.9
115
35. 2000/II/19
In the figure, ABCD is a parallelogram. Find ∠BDE.
A. 30° B. 35° C. 40° D. 50° E. 55°
36. 2000/II/27
In the figure, find x correct to 3 significant figures.
A. 63.8 B. 78.5 C. 84.5 D. 87.3 E. 89.1
37. 2000/II/32
The figure shows a square, a triangle and a sector with areas a cm2 , b cm
2 and c cm
2 respectively.
Which of the following is true?
A. a > b > c B. a > c > b C. b > a > c D. b > c > a E. c > a > b
38. 2001/II/9
The figure shows a regular pentagon. Find its area correct to the nearest 0.01 cm
2.
A. 3.63 cm2 B. 5.88 cm
2 C. 6.18 cm
2 D. 6.88 cm
2 E. 8.51 cm
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.10
116
39. 2001/II/30
In the figure, find x correct to 3 significant figures.
A. 8.86 B. 9.34 C. 9.48 D. 10.7 E. 11.3
40. 2002/II/18
The figure shows a parallelogram ABCD with its diagonals meeting at E. If 3=AE cm and 2=BE cm, find
the area of the parallelogram correct to the nearest 0.1 cm2.
A. 2.3cm2 B. 7.7cm
2 C. 9.2cm
2 D. 18.3cm
2
41. 2002/II/27
In the figure, ABCDEFGH is a regular octagon. =++ zyx
A. 60° B. 67.5° C. 82.5° D. 90°
42. 2002/II/44
In the figure, ABCD is a parallelogram. E and F are points on AD and BC respectively such that AB // EF. EF
meets AC at G. If 2:1: =GCAG , then area of ABFG : area of EGCD =
A. 1: 2 B. 1: 4 C. 3: 4 D. 5: 8
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.11
117
43. 2002/II/50
In the figure, E and F are the mid-points of AB and AC respectively. G and H are points on BD and CD
respectively such that 5
3==
HC
DH
GB
DG. If 6=EF cm, then =GH
A. 3.6 cm B. 4.5 cm C. 7.2 cm D. 7.5 cm
44. 2003/II/16
The length of a side of a regular 8-sided polygon is 6 cm. Find its area, correct to 3 significant figures.
A. 27.6 cm2 B. 29.8 cm
2 C. 66.5 cm
2 D. 174 cm
2
45. 2003/II/18
In the figure, AEDC is a parallelogram. If AB : BC = 1: 2 and AF : FE = 2 : 1, then the area ∆ABF : area of ∆BCD
A. 1 : 2 B. 1 : 3. C. 1 : 4. D. 2 : 9.
46. 2003/II/44
In the figure, ABCD and PQRS are two rectangles of equal perimeter. If 2:3: =BCAB and 3:4: =QRPQ ,
then area of ABCD : area of PQRS =
A. 1 : 1 B. 1 : 2 C. 25 : 49 D. 49 : 50
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.12
118
47. 2003/II/53
In the figure, ABCD is a parallelogram and ADH, EBC and EFGH are straight lines. If 6=AD , 4=DH
and 4:3: =BCEB , then =GHEF :
A. 1 : 1 B. 3 : 4 C. 5 : 4 D. 9 : 8
48. 2004/II/17
In the figure, ABCD is a parallelogram and E is a point on AD such that 3:1: =EDAE . If the area of ∆ABE is
3 cm2, then the area of the shaded region is
A. 9 cm2 B. 15 cm
2 C. 21 cm
2 D. 24 cm
2
49. 2004/II/19
If the area of a regular 10 - sided polygon is 123 cm2, find the length of the side of the 10-sided polygon. Give
the answer correct to the nearest 0.l cm.
A. 3.9 cm B. 4.0 cm C. 6.8 cm D. 8.0 cm
50. 2005/II/16
In the figure, ABCD is a rhombus and CDE is an equilateral triangle. If ADE is a straight line, then the area of
the quadrilateral ABCE is
A. 2 3 cm2 B. 3 3 cm
2 C. 4 3 cm
2 D. 6 3 cm
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.13
119
51. 2005/II/26
If AC = BD and AB // DC, how many pairs of similar triangles are there in the figure?
A. 2 pairs B. 3 pairs C. 4 pairs D. 5 pairs
52. 2005/II/27
In the figure, ABCD is a square. If CEF is an equilateral triangle, then ∠CBF =
A. 45° B. 50° C. 60° D. 80°
53. 2005/II/28
In the figure, x =
A. 50° B. 60° C. 70° D. 90°
54. 2005/II/30
In the figure, the length of the line segment joining A and F is
A. 68 B. 77 C. 82 D. 85
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.14
120
55. 2005/II/43
In the figure, ACE and BDF are straight lines. If the areas of the quadrilaterals ABDC and CDFE are 16 cm2 and
5 cm2 respectively, then the length of AB is
A. 4.5 cm B. 5 cm C. 5.5 cm D. 6 cm
56. 2005/II/52
In the figure, ABCD is a parallelogram E, F and G are points lying on BC, CD and DA respectively. AE and
AF divide ∠BAD into three equal parts and BG bisects ∠ABC. If AE and AF intersect BG at H and I
respectively, then ∠GIF + ∠GHE =
A. 120° B. 150° C. 180° D. 210°
57. 2006/II/17
In the figure, the area of the trapezium ABCD is
A. 345 cm2 B. 349 cm
2 C. 690 cm
2 D. 698 cm
2
58. 2006/II/45
If the length of a side of a regular tetrahedron is 3 cm , then the height of the tetrahedron is
A. 3 cm B. 3 cm C. 6 cm D. 3 3
2cm
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.15
121
59. 2007/II/19
In the figure, ABCD is a parallelogram. E is a point lying on AB . If EC and BD intersect at F , then the ratio of
the area of ∆DEF to the area of ∆CBF is
A. 1 : 1 B. 1 : 2 C. 2 : 1 D. 2 : 3
60. 2007/II/27
If the sum of the interior angles of a convex n-sided polygon is 4 times the sum of the exterior angles of the
polygon, then n =
A. 4 B. 6 C. 8 D. 10
61. 2007/II/28
In the figure, AY and CY are the angle bisectors of ∠BAX and ∠DCX respectively. If ∠AXC = 100° , then
∠AYC =
A. 40° B. 50° C. 60° D. 80°
62. 2008/II/7
In the figure, the rectangle ABCD is divided into eight identical rectangles. Find the area of the rectangle ABCD.
A. 40 cm2 B. 80 cm
2 C. 96 cm
2 D. 112 cm
2
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.16
122
63. 2008/II/21
In the figure, ABCD is a parallelogram. M is a point lying on BC such that BM : MC = 1 : 2 . If BD and AM
intersect at G and the area of ∆BGM is 1 cm2 , then the area of the parallelogram ABCD is
A. 9 cm2 B. 11 cm
2 C. 12 cm
2 D. 24 cm
2
64. 2008/II/27
In the figure, AB // CD and AC = BD . If ∠CAD = 20° and ∠ADB = 80° , then ∠ADC =
A. 30° B. 40° C. 50° D. 60°
65. 2008/II/28
According to the figure, which of the following must be true?
A. a + b = c B. a + b = c + 90° C. a + c = b + 540° D. a + b + c = 720°
66. 2008/II/48
In the figure, AB // CD , AB = 150 cm and CD = 80 cm . Find BD correct to the nearest cm.
A. 60 cm B. 62 cm C. 64 cm D. 65 cm
HKCEE MATHEMATICS | 11.3 Plane Geometry - Polygons | P.17
123
1. 2006/II/48
Let O be the origin. If the coordinates of the points A and B are (6, 0) and (0, 6) respectively, then the
coordinates of the in-centre of ∆ABO are
A. (0, 0)
B. (2, 2)
C. (3, 3)
D. ( 6 3 2,6 3 2− − )
2. 2006/II/49
In the figure, ABC is an acute-angled triangle, AB = AC and D is a point lying on BC such that AD is
perpendicular to BC. Which of the following must be true?
I. The circumcentre of ∆ABC lies on AD.
II. The orthocentre of ∆ABC lies on AD.
IIII. The centroid of ∆ABC lies on AD.
A. I and II only B. I and III only C. II and III only D. I, II and III
3. 2007/II/50
If ∆ABC is an obtuse-angled triangle, which of the following points must lie outside ∆ABC?
I. The centroid of ∆ABC
II. The circumcentre of ∆ABC
III. The orthocentre of ∆ABC
A. I and II only B. I and III only C. II and III only D. I, II and III
4. 2008/II/52
Let O be the origin. If the coordinates of the points A and B are (48, 0) and (24, 18) respectively, then the
y-coordinate of the orthocentre of ∆ABO is
A. –7 B. 6 C. 8 D. 32
HKCEE MATHEMATICS | 11.4 Plane Geometry - Centers in a triangle | P.1
124
1. 2006/II/25
If the plane figure above is rotated anticlockwise about the point O through 90o , which of the following is its
image?
2. 2007/II/25
Which of the following plane figures have rotational symmetry?
A. I and II only B. I and III only C. II and III only D. I, II and III
3. 2007/II/26
In the figure, the square ABCD is divided into nine identical squares and four of them are shaded. The number
of axes of reflectional symmetry of the square ABCD is
A. 2 B. 4 C. 5 D. 8
HKCEE MATHEMATICS | 11.5 Plane Geometry - Rotational and Reflectional Symmetries | P.1
125
4. 2008/II/25
If the plane figure above is rotated anticlockwise about the point O through 135o , which of the following is its
image?
5. 2008/II/26
Which of the following triangles have reflectional symmetry but do not have rotational symmetry?
A. I and III only B. I and IV only C. II and III only D. II and IV only
HKCEE MATHEMATICS | 11.5 Plane Geometry - Rotational and Reflectional Symmetries | P.2
126
1. 1990/II/16
sin(180o + θ ) + sin(θ − 90
o) =
A. sinθ + cosθ B. sinθ − cosθ C. cosθ − sinθ D. −cosθ − sinθ E. 2sin θ
2. 1990/II/17
If 0o ≤ x < 360
o, which of the following equations has only one root?
A. sin x = 0 B. sin x = 2
1 C. sin x = 2 D. cos x = 0 E. cos x = −1
3. 1990/II/18
If tan θ =3
4− and θ lies in the second quadrant, then sin θ − 2 cos θ =
A. 2 B. – 2 C. 5
11 D.
5
2 E.
5
2−
4. 1990/II/44
If sin θ and cos θ are the roots of the equation x2 + k = 0, then k =
A. −1 B. 2
1− C.
4
1− D.
4
1 E.
2
1
5. 1991/II/16
+ θθ
tancos
1(1 − sin θ ) =
A. sin θ B. cos θ C. cos2 θ D. 1 + sin θ E. sin θ tan θ
6. 1991/II/17
)180tan(
)90sin(o
o
+
−
θθ
=
A. cos θ B. −cos θ C. θθ
sin
cos2
D. θθ
sin
cos2
E. θsin
1
7. 1991/II/18
For 0 ≤ θ < 2π, how many roots does the equation tan θ + 2 sin θ = 0 have?
A. 1 B. 2 C. 3 D. 4 E. 5
8. 1991/II/47
cos2
π + cos π + cos
2
3π+ cos 2π + …+cos 10π = ? (Given π = 180 degree)
A. 0 B. 1 C. −1 D. 10 E. −10
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.1
127
9. 1992/II/18
The greatest value of θsin21− is
A. 5 B. 3 C. 1 D. 0 E. −1
10. 1992/II/20
In which two quadrants will the solution(s) of 0cossin <θθ lie?
A. In quadrants I and II only B. In quadrants I and III only C. In quadrants II and III only
D. In quadrants II and IV only E. In quadrants III and IV only
11. 1992/II/21
If °=++ 180CBA , then ( ) =++ CBAcoscos1
A. 0 B. A2sin C. A
2cos1+ D. AAcossin1+ E. AAcossin1−
12. 1992/II/23
Which of the following equations has/have solutions?
I. 1sincos2 22=− θθ
II. 2sincos2 22=− θθ
III. 3sincos2 22=− θθ
A. I only B. II only C. III only D. I and II only E. II and III only
13. 1993/II/19
=−
×− θ
θθ
θsin
cos1
sin1
cos 2
2
A. θsin B. θcos C. θtan D. θsin
1 E.
θcos
1
14. 1993/II/20
=+− θθθ 244 sin2sincos
A. 0 B. 1 C. ( )22sin1 θ− D. ( )22cos1 θ− E. ( )222 sincos θθ −
15. 1993/II/22
The largest value of 1cos2sin3 22 −+ θθ is
A. 1 B. 2
3 C. 2 D. 3 E. 4
16. 1993/II/45
Solve 03tan2tan 24 =−+ θθ for °<≤° 3600 θ .
A. 45º, 135º only B. 45º, 225º only C. 45º, 60º, 225º, 240º
D. 45º, 120º, 225º, 300º E. 45º, 135º, 225º, 315º
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.2
128
17. 1994/II/16
=−
−+ 1sin
cos
1sin
cos
θθ
θθ
A. θcos
2 B. −
θcos
2 C. 0 D. 2 tanθ E. −2 tanθ
18. 1994/II/18
( )( ) =
−°+°θθ
90cos
180sin
A. tanθ B. −tanθ C. θtan
1 D. 1 E. −1
19. 1994/II/47
For °≤≤° 3600 x , how many roots does the equation ( )2cossin +xx = 0 have?
A. 0 B. 1 C. 2 D. 3 E. 4
20. 1994/II/48
The largest value of ( ) 112cos32 +−θ is
A. 2 B. 5 C. 17 D. 26 E. 50
21. 1995/II/16
=−+
1sin1
cos2
θθ
A. θsin− B. θsin C. 2sin −θ D. ( )
θθθ
sin1
sin1sin
+
−− E.
( )θθθ
sin1
sin1sin
+
−
22. 1995/II/17
If π20 << x , solve 3
1sin =x correct to 3 significant figures.
A. 0.327 or 2.81 B. 0.327 or 3.47 C. 0.340 or 2.80 D. 0.340 or 3.48 E. 0.340 or 5.94
23. 1995/II/18
The greatest value of xsin12
1−
is
A. 2
1 B.
4
1 C. 1 D. 2 E. 4
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.3
129
24. 1996/II/19
If 0° ≤ θ ≤ 360° , solve 2 sinθ = − 3 .
A. 120° or 240° B. 120° or 300° C. 150° or 330° D. 210° or 330° E. 240° or 300°
25. 1996/II/20
2
1cos
cos
tan
θθ
θ
−
=
A. sinθ B. cosθ C. cos2θ D.
1
cosθ E.
1
tanθ
26. 1996/II/22
If 0 ≤ x ≤ π , solve 2 sinx + 3 cosx = 0 correct to 3 significant figures.
A. 0.588 B. 0.983 C. 2.16 D. 2.55 E. no solution
27. 1997/II/40
cos(90 )sin(180 )
tan(360 )
A A
A
− −=
−
� �
�
A. sin cosA A− B. sin cosA A C. 2cos A− D. 2cos A E. 2sin A
28. 1997/II/43
For 0 ≤ θ ≤ 2π , how many roots does the equation tanθ (tanθ − 2) = 0 have?
A. 1 B. 2 C. 3 D. 4 E. 5
29. 1998/II/44
1 sin cos
cos 1 sin
θ θθ θ
++ =+
A. 1 B. 2(1 sin )θ+ C. 2
cosθ D.
2
cos (1 sin )θ θ+ E.
1 sin cos
cos (1 sin )
θ θθ θ
+ +
+
30. 1998/II/47
For 0° ≤ x ≤ 360° , how many roots does the equation 3 sin2
x + 2 sin x − 1 = 0 have?
A. 0 B. 1 C. 2 D. 3 E. 4
31. 1999/II/46
cos(90 )cos( )
sin(360 )
A A
A
− −=
−
�
�
A. cos A− B. cos A C. sin A D. 2cos
sin
A
A− E.
2cos
sin
A
A
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.4
130
32. 1999/II/47
If 0 ≤ θ ≤ 2π , solve (cosθ − 3)(3 sinθ − 2) = 0 correct to 3 significant figures.
A. 0.730 or 1.23 B. 0.730 or 2.41 C. 0.730 or 3.87
D. 0.730 or 6.21 E. 0.730 or 2.41
33. 2000/II/51
If 1
cosk
θ = and 0° < θ ≤ 90° , then tan( 270 )θ − =�
A. 21
k
k−
−
B. 2
1
1k−
−
C. 2
1
1k −
D. 2 1k− − E. 2 1k −
34. 2001/II/17
If °<<<° 900 yx , which of the following must be true?
I. yx sinsin <
II. yx coscos <
III. yx cossin <
A. I only B. II only C. I and II only D. I and III only E. II and III only
35. 2001/II/42
For °≤≤° 3600 x , how many roots does the equation xx coscos3 = have?
A. 2 B. 3 C. 4 D. 5 E. 6
36. 2001/II/43
If 2)90tan( =−° θ , then =+
θθθθ
cos
cossinsin 23
A. 2 B. 2
1 C.
2
1 D.
2
1− E. −2
37. 2002/II/21
For 0° ≤ θ ≤ 90°, the maximum value of θ2sin3
2
+ is
A. 5
2 B.
2
1 C.
3
2 D. 1
38. 2002/II/22
If °<<° 9045 θ , which of the following must be true ?
I. θθ sintan >
II. θθ costan >
III. θθ sincos >
A. I and II only B. I and III only C. II and III only D. I, II and III
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.5
131
39. 2002/II/46
If 5
3sin =θ and θ lies in the first quadrant, then =+°+−° )180sin()90sin( θθ
A. 5
1 B.
5
1− C.
5
7 D.
5
7−
40. 2002/II/47
=−−++ )]cos(1)][cos(1[ θπθπ
A. θ2sin B. 2)cos1( θ− C. 2)cos1( θ+ D. )sin1)(cos1( θθ −−
41. 2002/II/48
For °≤≤° 3600 x , how many roots does the equation xx sin2tan = have?
A. 2 B. 3 C. 4 D. 5
42. 2003/II/22
If θ is an acute angle and θθ cossin = , then =θcos
A. 2
1 B.
2
2 C.
2
3 D. 1
43. 2003/II/45
For °≤≤° 3600 θ , how many roots does the equation 04sin5cos2 2=−− θθ have?
A. 1 B. 2 C. 3 D. 4
44. 2003/II/46
=−°−°
)90cos(
)180tan(
θθ
A. θcos
1 B.
θcos
1− C.
θθ2cos
sin D.
θθ
2cos
sin−
45. 2003/II/47
1 degree =
A. 180
π radian B.
180
π radians C.
1
180π radian D. 180π radians
46. 2004/II/20
For °≤≤° 900 x , the least value of xcos2
4
−
is
A. 0 B. 1 C. 2 D. 4
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.6
132
47. 2004/II/46
=
−
θθ
θ
sin
cos
1cos
A. θtan− B. θtan C. θθ
cos
sin 3−
D. θθ
θcossin
1cos −
48. 2004/II/47
If π=+ BA , which of the following must be true?
I. BA sinsin =
II. BA sincos =
III. BA coscos =
A. I only B. II only C. I and III only D. II and III only
49. 2005/II/20
For 0 90θ° ≤ ≤ ° , the greatest value of 5 sin
4 sin
θθ
−
+ is
A. 4
5 B. 1 C.
5
4 D. 2
50. 2005/II/44
For 0° ≤ x ≤ 360°, how many distinct roots does the equation cos (sin 1)x x − = 0 have?
A. 2 B. 3 C. 4 D. 5
51. 2005/II/45
sin(90°−x) + cos(x+180°) =
A. 0 B. −2cosx C. sinx + cosx D. sinx − cosx
52. 2005/II/46
sin2
1° + sin2
3° + sin2
5° + …… + sin2
87° + sin2
89° =
A. 22 B. 22.5 C. 44.5 D. 45
53. 2006/II/21
2sin(90 )sin 60 cos 0 cosθ θ− − =� � �
A. sinθ B. 3 sinθ C. 3 cosθ D. ( 3 1)cosθ−
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.7
133
54. 2006/II/22
If 0° ≤ θ ≤ 45° , which of the following must be true?
I. tanθ < cosθ II. I. sinθ < tanθ III. I. sinθ < cosθ
A. I only B. III only C. I and II only D. II and III only
55. 2006/II/44
For 0° ≤ x ≤ 360° , how many roots does the equation 23cos 4cos 1 0x x− + = have?
A. 2 B. 3 C. 4 D. 5
56. 2007/II/20
If x and y are acute angles such that x + y = 90° , which of the following must be true?
I. sinx = cosy II. sin(90° – x) = cos(90° – y) III. tanx tany = 1
A. I and II only B. I and III only C. II and III only D. I, II and III
57. 2007/II/21
cos sin
sin cos
A A
A A+ =
A. 1 B. 21 tan A+ C. sin cosA A D. 1
sin cosA A
58. 2007/II/22
Solve the equation sin 3 cosθ θ= , where 0 90θ≤ ≤� � .
A. 0θ = � B. 30θ = � C. 45θ = � D. 60θ = �
59. 2008/II/45
For 0 360θ≤ <� � , how many roots does the equation 23sin 2sin 1 0θ θ+ − = have?
A. 2 B. 3 C. 4 D. 5
60. 2008/II/47
For 0 360θ≤ ≤� � , the least value of 2 sin
2 sin
θθ
+
− is
A. –1 B. 1
3 C. 1 D. 3
HKCEE MATHEMATICS | 12.1 Trigonometry - Trigonometric Equations | P.8
134
1. 1990/II/45
The figure shows the graph of y = 3 sin2x. The point P is
A. (3
4π, −3) B. (
4
3π, −3) C. (
3
4π, −1) D. (
4
3π, −1) E. (
2
3π, −1)
2. 1991/II/48
The figure shows the graph of the function
A. y = 2 cos x . B. y = 2 − sin x C. 2 + sin x D. y = 2 − cos x E. y = 2 + cos x
3. 1992/II/22
The figure shows the graph of the function
A. ( )π+xtan B. ( )π−xtan C. xtanπ D. xtan+π E. xtan−π
4. 1993/II/46
The figure shows the graph of the function
A. ( )xy −°= 350sin B. ( )°+= 10sin xy C. ( )°+= 10cos xy
D. ( )°−= 10sin xy E. ( )°−= 10cos xy
HKCEE MATHEMATICS | 12.2 Trigonometry - Trigonometric Graph | P.1
135
5. 1994/II/17
Which of the following figures shows the graph of xy sin1+= ?
6. 1995/II/49
If °≤≤° 3600 x , the number of points of intersection of the graphs xy sin= and xy tan= is
A. 1 B. 2 C. 3 D. 4 E. 5
7. 1995/II/50
In the figure shows the graph of the function
A. 2
cos°= x
y B. °= xy cos2
1 C. °= xy cos D. °= xy cos2 E. °= xy 2cos
8. 1996/II/21
The figure shows the graph of y = 1
2cos2x The point P is
A. ( , 2)2
π B.
1( , )
2π C. ( ,1)π D.
1(2 , )
2π E. (2 ,1)π
HKCEE MATHEMATICS | 12.2 Trigonometry - Trigonometric Graph | P.2
136
9. 1997/II/44
In the figure, f(x) =
A. sin1
2 2
x+ B. sin2x +
1
2 C.
1
2sin
1
2 2
x+ D.
1
2sinx+
1
2 E.
1
2sin2x +
1
2
10. 1998/II/45
The figure shows the graph of the function
A. cosy x= B. cos( )y x= − C. cos( )2
y xπ
= − D. cos( )2
y xπ
= + E. cos( )y xπ= −
11. 1999/II/16
Which of the following may represent the graph of y = cos x° for 0 ≤ x ≤ 90?
12. 2000/II/11
Which of the following may represent the graph of °= xy tan for 900 ≤≤ x ?
HKCEE MATHEMATICS | 12.2 Trigonometry - Trigonometric Graph | P.3
137
13. 2000/II/53
In the figure, the area of ∆ABC is
A. 3
π B.
2
3
π C. π D.
4
3
π E. 2π
14. 2001/II/44
The figure shows the graph of the function
A. 2
sinx
y = B. xy sin2= C. xy sin1+= D. xy cos1+= E. xy cos1−=
15. 2007/II/46
Let k be a constant and –90° < θ < 90°. If the figure shows the graph of y = k sin(x° + θ), then
A. k = –2 and θ = –30° B. k = –2 and θ = 30° C. k = 2 and θ = –30° D. k = 2 and θ = 30°
16. 2008/II/46
Let a and b be constants. If the figure shows the graph of cos(2 120 )y a x b= + +� � , then
A. a = 1 and b = 3 B. a = 2 and b = 2 C. a = 3 and b = 1 D. a = 4 and b = 0
HKCEE MATHEMATICS | 12.2 Trigonometry - Trigonometric Graph | P.4
138
1. 1990/II/22
In the figure, AC // DE, FG // BC and AD : DF : FB = 1 : 2 : 3. If BE = 10, find FG.
A. 5 B. 6 C. 8 D. 9 E. 10
2. 1990/II/46
In the figure, ABCD is a parallelogram. BD =
A. 5 B. 7 C. 13 D. 27 E. 37
3. 1990/II/49
In the figure, AC = CD, ∠ABC = 30o and ∠CED = 120
o.
DE
AB=
A. 2
1 B.
3
1 C. 2 D. 3 E. 2
4. 1990/II/52
In the figure, if CD = CF, CE = BE and DA = DB, then ∠C =
A. 30 B. 36 C. 40 D. 45 E. 60
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.1
139
5. 1991/II/20
In the figure, ∠A = 30o and ∠B = 120
o. The ratio of the altitudes of the triangle ABC from A and from B is
A. 2:1 B. 3 : 1 C. 2 : 1 D. 1 : 2 E. 1 : 3
6. 1991/II/29
In the figure, A and B are the positions of two boats. The bearing of B from A is
A. N55oE B. N70
oE C. N20
oE D. S35
oE E. S75
oE
7. 1991/II/49
In the figure, the height of the vertical pole PO is
A. 7.5 m B. 15 m C. 15 2 m D. 15 3 m E. 45 m
8. 1991/II/50
In the figure, find the length of AB, correct to the nearest cm.
A. 14 B. 15 C. 16 D. 17 E. 18
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.2
140
9. 1991/II/51
In the figure, ABC and CDE are equilateral triangles. Find ∠ADE.
A. 15o B. 35
o C. 40
o D. 45
o E. 50
o
10. 1991/II/53
In the figure, M is the mid-point of BC and AD = 2DB. AM and CD intersect at K. Find AKC
ADK
∆∆ of area
of area.
A. 2
1 B.
3
2 C.
4
3 D.
5
4 E. 1
11. 1992/II/19
In the figure, find θcos .
A. 4
1− B. 16
11 C.
4
3 D.
8
7 E.
9
77
12. 1992/II/25
In the figure, ABCD is a square with side 6. If BE=CE=5, find AE.
A. 61 B. 9 C. 10 D. 36 E. 109
13. 1992/II/46
In the figure, find θtan .
A. 3
1 B.
8
1 C.
8
3 D.
7
2 E.
2
1
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.3
141
14. 1992/II/49
In ∆ABC, °=∠ 30A , 6=c . If it is possible to draw two distinct triangles as shown in the figure, find the
range of values of a .
A. 30 << a B. 60 << a C. 63 << a D. 3>a E. 6>a
15. 1992/II/51
In the figure, EB and EC are the angle bisectors of ∠ABC and ∠ACD respectively. If °=∠ 40A , find ∠E .
A. 20º B. 25º C. 30º D. 35º E. 40º
16. 1992/II/53
In the figure, 16=AB , 8=CD , 9=BF , 4=GD , 2=EG . Find GC .
A. 4.5 B. 5 C. 6 D. 8 E. 10
17. 1993/II/15
Find the perimeter of the sector in the figure.(1 rad =180 degree)
A. 2.25 cm B. 3 cm C.
+ 360
π cm D. 4.5 cm E. 6 cm
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.4
142
18. 1993/II/21
In the figure, 5
4cos −=A . Find a .
A. 153 B. 137 C. 89 D. 41 E. 25
19. 1993/II/23
In the figure, AB = BC, BP = CP and CPBP⊥ . Find θtan .
A. 4
1 B.
3
1 C.
2
1 D.
3
1 E.
2
3
20. 1993/II/24
In the figure, points A, B, C and D are concyclic. Find x.
A. 20º B. 22.5º C. 25º D. 27.5º E. 30º
21. 1993/II/25
In the figure, BA//DE and AC=AD. Find θ .
A. 34º B. 54º C. 70º D. 72º E. 76º
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.5
143
22. 1993/II/42
In the figure, the radii of the sectors OPQ and ORS are 5 cm and 3 cm respectively, =OPQsector of Area
region shaded of Area
A. 25
4 B.
5
2 C.
25
9 D.
25
16 E.
25
21
23. 1993/II/47
In the figure, ABC is an equilateral triangle and the radii of the three circles are each equal to 1. Find the
perimeter of the triangle.
A. 12 B. ( )°+ 30tan13 C. ( )°+ 30tan16 D.
°+
30tan
113 E.
°+30tan
116
24. 1994/II/11
The bearing of A from B is 075º. What is the bearing of B from A?
A. 015º B. 075º C. 105º D. 195º E. 255º
25. 1994/II/15
In the figure, OACB is a sector of radius r. If ∠AOB=3
π, find the area of the shaded part.
A. 2
4
3
6r
−π B. 2
4
1
6r
−π C. 2
2
3
3r
−π D. 2
2
1
3r
−π E. 2
4
3
3rr −
π
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.6
144
26. 1994/II/24
In the figure, AB=AD and BC=CD. If ∠BAD=80º and ∠ADC=65º, then ∠BCD=
A. 100º B. 130º C. 145º D. 150º E. 160º
27. 1994/II/25
In the figure, x , y and z are the exterior angles of ∆ABC. If zyx :: = 6:5:4 , then ∠BAC=
A. 48° B. 84° C. 96° D. 120° E. 132°
28. 1994/II/45
In the figure, AD : DB = 1 : 2, AE : EC = 3 : 2. Area of ∆BDE : Area of ∆ABC =
A. 1 : 3 B. 2 : 5 C. 3 : 4 D. 4 : 25 E. 36 : 65
29. 1994/II/46
In the figure, area of ∆ABC : area of square BCDE = 2 : 1. Find PQ : BC
A. 1 : 2 B. 1 : 3 C. 1 : 4 D. 2 : 3 E. 3 : 4
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.7
145
30. 1994/II/49
In the figure, sinA : sinB : sin C = 4 : 5 : 6. If AB = 8, find AC.
A. 53
1 B. 6
3
2 C. 9
5
3 D. 10 E. 12
31. 1994/II/50
In the figure, AB = p, ∠ACB =θ . Find CD.
A. θsinp B. θcosp C. θθ
2cos
sinp D.
θθ
cos
sin 2p
E. θθ
sin
cos2p
32. 1995/II/19
According to the figure, which of the following must be true?
A. bccba 3222 −+= B. bccba −+= 222 C. bccba2
3222 ++=
D. bccba ++= 222 E. bccba 3222 ++=
33. 1995/II/20
In the figure, the bearing of B from A is
A. 015º B. 045º C. 075º D. 165º E. 345º
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.8
146
34. 1995/II/21
In the figure, BDC is a straight line. Arrange AD, BD and DC in ascending order of magnitude.
A. DCBDAD << B. BDDCAD << C. BDADDC <<
D. ADBDDC << E. DCADBD <<
35. 1995/II/25
In the figure, AB = AC and AD = AE. ∠DAC =
A. 45º B. 50º C. 55º D. 60º E. 65º
36. 1995/II/26
In the figure, ∠ADE = ∠ACB. Find x.
A. 4 B. 8 C. 10 D. 12 E. 16
37. 1995/II/47
In the figure, DE = DB, AC = 13 and BC = 5. Area of ADE∆ : Area of =∆ACB
A. 64 : 169 B. 5 : 13 C. 4 : 9 D. 8 : 13 E. 2 : 3
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.9
147
38. 1995/II/52
In the figure, PB touches the semicircle ADB at B. PD =
A. θcos2
d B. θθ tansind C.
θθ tansin
d D.
θθ
tan
cosd E.
θθ
cos
tand
39. 1995/II/53
In the figure, a + b + c + d + e + f =
A. 270º B. 360º C. 450º D. 540º E. 720º
40. 1995/II/54
According to the figure, which of the following must be true?
A. dcba +=+ B. cbda +=+ C. °=+++ 360dcba
D. °=+++ 540dcba E. °=−−+ 72022 dcba
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.10
148
41. 1996/II/15
In the figure, area of ∆ACD : area of ∆BCD =
A. 1 : 1 B. a : b C. b : a D. a2 : b
2 E. b
2 : a
2
42. 1996/II/24
In the figure, find x correct to 3 significant figures.
A. 2.71 B. 2.98 C. 3.31 D. 3.88 E. 4.14
43. 1996/II/47
In the figure, find θ correct to the nearest degree.
A. 16º B. 19º C. 26º D. 35º E. 36º
44. 1996/II/48
In the figure, the bearing of two ships A and B from a lighthouse L are 020º and 080º respectively. B is 400 m
and at bearing of 130º from A. Find the distance of B from L.
A. 400 m B. 400
sin 60�
m C. 400sin 50
sin 60
�
�
m D. 400sin 70
sin 60
�
�
m E. 400sin 70
sin 80
�
�
m
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.11
149
45. 1996/II/52
In the figure, DE : EF =
A. 1 : 1 B. 2 : 1 C. 3 : 1 D. 3 : 2 E. 4 : 1
46. 1997/II/12
In the figure, sinθ + tanθ =
A. a a
c b+ B.
a b
c a+ C.
b a
c b+ D.
b b
c a+ E.
c a
a b+
47. 1997/II/13
In the figure, find θ correct to the nearest degree.
A. 78º B. 91º C. 102º D. 114º E. 125º
48. 1997/II/14
In the figure, the square sandwich ABCD is cut into two equal halves along EF so that AE : ED = 2 : 1. Find θ
correct to the nearest degree.
A. 56º B. 63º C. 64º D. 71º E. 72º
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.12
150
49. 1997/II/15
In the figure, the area of ∆ABC is 18. Find ∠ABC correct to the nearest degree.
A. 30º B. 44º C. 46º D. 60º E. 69º
50. 1997/II/17
In the figure, find x.
A. 52° B. 58° C. 61° D. 70° E. 81°
51. 1997/II/42
In the figure, CD =
A. sin
sin sin
r βα γ B.
sin
cos sin
r βα γ C.
sin sin
sin
r α βγ D.
cos sin
sin
r α βγ E.
sin
sin
r βα
52. 1998/II/18
In the figure, OAB is an equilateral triangle. Find the bearing of B from A.
A. 10° B. 80° C. 170° D. 260° E. 350°
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.13
151
53. 1998/II/24
In the figure, find CD.
A. 6 cm B. 4 cm C. 4 3 cm D. 2 3 cm E. 2 3
3cm
54. 1998/II/25
In the figure, find x correct to 3 significant figures.
A. 48.2 B. 55.1 C. 58.4 D. 67.5 E. 73.4
55. 1998/II/27
In the figure, PQ and RS are two vertical poles standing on the horizontal ground. The angle of elevation of R
from P is 20° and the angle of depression of S from P is 40°. If RS = 5 m, then PR =
A. 5sin 40
sin 70
�
�
m B. 5sin 50
sin 60
�
�
m C. 5sin 60
sin 50
�
�
m D. 5sin 70
sin 40
�
�
m E. 5
sin 50 sin 60� �
m
56. 1998/II/31
In the figure, AB = 2BC. Find BC correct to 3 significant figures.
A. 0.775 cm B. 1.00 cm C. 1.34 cm D. 1.73 cm E. 1.80 cm
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.14
152
57. 1998/II/50
In the figure, ABCD, AFE, CGE and FGD are straight lines. If AB = BC = 2CD, then CG : GE =
A. 1 : 2 B. 1 : 3 C. 1 : 4 D. 1 : 5 E. 1 : 6
58. 1999/II/15
In the figure, the bearing of B from C is
A. N5°E B. N65°E C. N85°E D. S5°W E. S85°W
59. 1999/II/17
In the figure, find x correct to 3 significant figures.
A. 1.28 B. 1.29 C. 1.35 D. 1.53 E. 1.65
60. 1999/II/18
In the figure, AC
AB=
A. 4
3 B.
5
4 C.
2
2 D.
6
2 E.
6
3
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.15
153
61. 1999/II/19
In the figure, find x correct to 1 decimal place.
A. 15.0 B. 18.4 C. 22.5 D. 24.1 E. 26.6
62. 1999/II/28
In the figure, ACD and ECB are straight lines. If ∠EAC = ∠CAB and EA = EB, find x.
A. 22° B. 34° C. 44° D. 46° E. 68°
63. 1999/II/30
In the figure, AEB and ADC are straight lines. Find ED.
A. 10
3cm B.
40
13cm C. 3 cm D. 40 cm E. 80 cm
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.16
154
64. 1999/II/48
The figure shows five sectors. Which of the marked angles measures 2 radians?
A. a B. b C. c D. d E. e
65. 2000/II/13
In the figure, the areas of the two triangles are equal. Find θ.
A. 7.2° (correct to the nearest 0.1°) B. 7.5° (correct to the nearest 0.1°)
C. 14.5° (correct to the nearest 0.1°) D. 15° E. 30°
66. 2000/II/26
In the figure, find the area of the triangle correct to the nearest 0.1 cm2.
A. 7.3 cm2 B. 10.7 cm
2 C. 12.7 cm
2 D. 15.0 cm
2 E. 19.1 cm
2
67. 2000/II/28
In the figure, ABCD is a rectangle. Find CF.
A. θsin)( ba + cm B. θcos)( ba + cm C. )cossin( θθ ba + cm
D. )sincos( θθ ba + cm E. θ2sin22ba + cm
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.17
155
68. 2000/II/29
In the figure, DAB is a straight line. =θtan
A. °20tan2 B. °20tan2
1 C.
°20tan
2 D.
°20tan2
1 E. °40tan
69. 2000/II/30
According to the figure, the bearing of B from C is
A. 050° B. 130° C. 140° D. 310° E. 320°
70. 2000/II/44
π degrees =
A. 2
180
πradian B.
2
180
πradian C.
180
πradian D. 180 radians E. 1 radian
71. 2000/II/54
In the figure, AEC and BED are straight lines. If the area of 4=∆ABE cm2 and the area of 5=∆BCE cm
2,
find the area of ∆CDE.
A. 4.5 cm2 B. 5 cm
2 C. 6 cm
2 D. 6.25 cm
2 E. 9 cm
2
72. 2001/II/4
The figure shows a right-angled triangle where 4:3: =BCAB . Find θsin .
A. 3
5 B.
4
3 C.
4
5 D.
5
3 E.
5
4
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.18
156
73. 2001/II/7
In the figure, find x correct to 3 significant figures.
A. 2.65 B. 2.79 C. 3.16 D. 4.00 E. 4.36
74. 2001/II/20
In the figure, =x
A. 50° B. 55° C. 60° D. 65° E. 70°
75. 2001/II/31
Ship A is 8 km due north of a light house L and ship B is 6 km due east of L. Find the bearing of B from A.
A. N53.1°W (correct to the nearest 0.1°) B. N36.9°W (correct to the nearest 0.1°) C. N36.9°E (correct to the nearest 0.1°) D. S53.1°E (correct to the nearest 0.1°) E. S36.9°E (correct to the nearest 0.1°)
76. 2001/II/50
In the figure, ADB, BEC and CFA are straight lines. If the area of ∆ABC is 225 cm
2, find the area of the
parallelogram DECF.
A. 81 cm2 B. 108 cm
2 C. 126 cm
2 D. 135 cm
2 E. 162 cm
2
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.19
157
77. 2001/II/52
In the figure, ABCD and AGFE are straight lines. Find CF.
A. 4 cm B. 3 cm C. 2
7cm D.
2
5cm E.
3
7cm
78. 2002/II/16
In the figure, =AC
A. °°
48sin
77sinx B.
°°
48sin
55sinx C.
°°
77sin
48sinx D.
°°
55sin
77sinx
79. 2002/II/23
In the figure, find x correct to 3 significant figures.
A. 0.963 B. 1.05 C. 1.10 D. 1.57
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.20
158
80. 2002/II/24
In the figure, AB and CD are the heights of two buildings on the same level ground. If 9=AB m, 20=AC m
and the angle of depression of A from D is 50°, find the angle of elevation of D from B correct to the nearest 0.1°.
A. 21.3° B. 24.2° C. 36.6° D. 53.4°
81. 2002/II/25
In the figure, ABAC 3= . Find AB correct to 3 significant figures.
A. 1.26 cm B. 1.41 cm C. 1.79 cm D. 2.83 cm
82. 2002/II/42
How many different triangles can be constructed so that the lengths of the three sides are x cm, 2x cm and 12 cm,
where x is an integer?
A. 5 B. 7 C. 9 D. 11
83. 2003/II/17
In the figure, ABDF and ACEG are straight lines. If the area of ∆ABC is 16 cm2 and the area of quadrilateral
BDEC is 20 cm2, then the area of quadrilateral DFGE is
A. 24 cm2
B. 28 cm2
C. 36 cm2
D. 44 cm2
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.21
159
84. 2003/II/23
In the figure, the bearing of A from B is
A. N38°W B. N52°W C. S38°E D. S52°E
85. 2003/II/24
In the figure, =θcos
A. 16
15 B.
20
13 C.
52
25 D.
65
23
86. 2003/II/26
In the figure, =AB
A. 2
x B. x
2
2 C. x
2
3 D. x2
87. 2003/II/28
In the figure, ABD and ACE are straight lines. If 4:3: =CEAC , then =DEBC :
A. 1 : 2 B. 3 : 4 C. 3 : 7 D. 4 : 7
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.22
160
88. 2003/II/47
1 degree =
A. 180
πradian B.
180
πradians C.
1
180πradian D. 180π radians
89. 2003/II/49
In the figure, xPQ = cm and ySR = cm. Find PS.
A. αcos2
xy − cm B.
)cos(2 βα +y
cm C. αβ
sin
sinx cm D.
)sin(
sin)(
βαβ
+
− xy cm
90. 2004/II/18
In the figure, AD and BC meet at E. If CE : EB = 3 : 1, then area of ∆ABD : area of ∆ CDE =
A. 1 : 1 B. 1 : 3 C. 2 : 3 D. 4 : 9
91. 2004/II/21
In the figure, find AC correct to 2 decimal places.
A. 5.04 B. 9.17 C. 11.14 D. 15.62
92. 2004/II/22
In the figure, =xsin
A. 3
4 B.
4
3 C.
5
3 D.
5
4
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.23
161
93. 2004/II/27
In the figure, ABC is a straight line. If BD // CE, then ∠DCE =
A. 56° B. 68° C. 112° D. 124°
94. 2004/II/28
In the figure, ABCD and AGFE are straight lines. If 2=BC cm, 3=CD cm, 6=BG cm and 10=CF cm,
then =DE
A. 12 cm B. 14 cm C. 15 cm D. 16 cm
95. 2004/II/45
In the figure, OAB is a sector. The perimeter and the area of the sector are x cm and y cm2 respectively. If
x = y , then �AB =
A. 5 cm B. 10 cm C. 5
3
πcm D.
10
3
πcm
96. 2005/II/15
In the figure, the bearing of P from Q is
A. N27°W B. S27°E C. N63°W D. S63°E
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.24
162
97. 2005/II/21
In the figure, θ is an acute angle. Find θ correct to the nearest degree.
A. 35° B. 50° C. 56° D. 57°
98. 2005/II/22
In the figure, cosθ =
A. 1
8 B.
1
4 C.
7
8 D.
7
4
99. 2005/II/23
In the figure, ABCD is a rectangle. If BED is a straight line, then the area of ∆ABE is
A. 3
6cm
2 B.
3
2cm
2 C.
2 3
3cm
2 D. 3 cm
2
100. 2005/II/29
In the figure, OABC and OFED are straight lines. If AB : BC = 2 : 3 and FA : DC = 1 : 5, then OA : AB =
A. 1 : 1 B. 1 : 2 C. 5 : 8 D. 5 : 13
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.25
163
101. 2006/II/16
In the figure, PA = QA . If the bearings of P and Q from A are N42°E and S28°E respectively, then the bearing
of P from Q is
A. N7°E B. N27°E C. N35°E D. N55°E
102. 2006/II/23
In the figure, sin x =
A. 3
7 B.
3
5 C.
4
5 D.
4
3
103. 2006/II/26
In the figure, ABC and AED are straight lines. If AB = 8 cm , BC = 4 cm and CD = 9 cm , then BE =
A. 32
9cm B.
9
2cm C. 5 cm D. 6 cm
104. 2007/II/15
A and B are two points on a map. If the bearing of A from B is 110° , then the bearing of B from A is
A. 070° B. 250° C. 290° D. 340°
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.26
164
105. 2007/II/23
In the figure, ABC is a right-angled triangle. BD is the angle bisector of ∠ABC . If AB = c , then CD =
A. 3
c B.
2 3
c C.
3
2
c D.
3
4
c
106. 2007/II/47
In the figure, find x correct to the nearest integer.
A. 14 B. 15 C. 16 D. 17
107. 2008/II/22
In the figure, D is a point lying on AC such that BD is perpendicular to AC . Find AD : DC.
A. 1: 2 B. 2 :1 C. 3 :1 D. 3 : 2
108. 2008/II/24
In the figure, tanθ =
A. 5
12 B.
5
13 C.
12
13 D.
13
12
HKCEE MATHEMATICS | 12.3 Trigonometry - 2D | P.27
165
1. 1990/II/19
The figure shows a right pyramid with a square base. VAB, VBC, VCD and VDA are equilateral triangles.
Find sin ∠VAH.
A. 2
1 B.
4
1 C.
2
1 D.
3
1 E.
2
3
2. 1990/II/47
In the figure, A, B and C are three points on the same horizontal plane. A is due north of B, C is due east of B
and H is a point vertically above A. Which of the following angles is/are 90o?
I. ∠HAC
II. ∠ABC
III. ∠HBC
A. I only B. II only C. I and II only D. I and III only E. I ,II and III
3. 1990/II/54
In the figure, ∆PTQ, ∆SQR and ∆RUT are equilateral triangles. Which of the following is/are true?
I. ∆UPT ≅ ∆RQT II. PU = QS III. PQSU is a parallelogram
A. All of them B. None of them C. I and II only D. I and III only E. II and III only.
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.1
166
4. 1991/II/45
Question 4 and 5 refer to the figure below, which shows a cuboid ABCDEFGH with AE = 2a, EF = 2b and
FG = 2c. AC and BD intersect at X. XE =
A. 222cba ++ B. 222 )2( cba ++ C. 222 )2( cba ++ D. 222)2( cba ++ E. 2 222
cba ++
5. 1991/II/46
If the angle between XE and the plane EFGH is θ, then tan θ =
A. b
a B.
b
a2 C.
b
ca22)2( +
D. 22
cb
a
+ E.
22
2
cb
a
+
6. 1992/II/15
Find the ratio of the volume of the tetrahedron ACHD to the volume of the cube ABCDEFGH in the figure.
A. 1 : 8 B. 1 : 6 C. 1 : 4 D. 1 : 3 E. 1 : 2
7. 1992/II/47
In the figure, if θ is the angle between the diagonals AG and BH of the cuboid, then
A. 3
2
2sin =
θ B.
4
3
2sin =
θ C.
3
1
2sin =
θ D.
3
2sin =θ E.
4
3sin =θ
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.2
167
8. 1992/II/48
In the figure, OA is perpendicular to the plane ABC. cm2=== ACABOA and cm22=BC . If M and N
are the mid-points of OB and OC respectively, find the area of ∆AMN.
A. 2cm2
1 B. 21cm C. 2cm2 D. 2cm
2
3 E. 2cm3
9. 1993/II/48
In the figure, ABCDEFGH is a cuboid. The diagonal AH makes an angle θ with the base ABCD. Find θtan .
A. 5
3 B.
12
3 C.
13
3 D.
178
3 E.
5
153
10. 1994/II/20
In the figure, PC is a vertical pole standing on the horizontal plane ABC. If ABC∠ =90º, BAC∠ =30º, AC =6
and PC =5, find tanθ .
A. 5
3 B.
6
5 C.
3
5 D.
5
33 E.
9
35
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.3
168
11. 1995/II/51
In the figure, ABCDEFGH is a cuboid. =θtan
A. 3
1 B.
3
1 C. 1 D. 3 E. 3
12. 1996/II/23
The figure shows a cube. Which of the following is/ are equal to ∠AGE?
I. ∠AGF
II. ∠BDF
III. ∠DEG
A. I only B. II only C. III only D. I and II only E. II and III only
13. 1996/II/49
The figure shows a right prism with a right-angled triangle as the cross-section. Find the angle between the line
BF and the plane ABCD correct to the nearest degree.
A. 22° B. 34° C. 37° D. 42° E. 56°
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.4
169
14. 1997/II/41
In the figure, ABCD is a rectangle inclined at an angle of 45° to the horizontal plane BCEF. Find the inclination
of AC to the horizontal plane correct to the nearest degree.
A. 27° B. 30° C. 35° D. 45° E. 55°
15. 1998/II/48
The figure shows a right pyramid with a square base ABCD. Let θ be the angle between the planes VAB and VCD.
Find sin2
θ.
A. 1
2 B.
3
2 C.
1
3 D.
1
5 E.
2
5
16. 1999/II/49
In the figure, ABCDEFGH is a rectangular block. Find the inclination of EM to the plane ABCD correct to the
nearest degree.
A. 23° B. 25° C. 65° D. 71° E. 75°
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.5
170
17. 2000/II/52
The figure shows a right triangular prism. Find its volume.
A. ββαα cossincossin3
1 2 m3 B. ββαα cossincossin
3
1 2 m3
C. ββαα cossincossin2
1 m
3
D. ββαα cossincossin2
1 2 m3 E. ββαα cossincossin
2
1 2 m3
18. 2001/II/51
In the figure, PC is a vertical pole standing on the horizontal ground ABC. D is a point on line AB. If
°=∠=∠ 90CDBBCA , 3=AC m, 4=BC m and 5=PC m, find θtan .
A. 25
12 B.
25
16 C.
16
25 D.
12
25 E.
9
25
19. 2002/II/49
In the figure, ABCDEFGH is a rectangular block with a square base ABCD. Find ∠FBH correct to the
nearest degree.
A. 21° B. 41° C. 45° D. 60°
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.6
171
20. 2003/II/48
The figure shows a cuboid. Which of the following are right angles?
I. ∠CAF II. ∠DHG III. ∠AGC
A. I and II only B. I and III only C. II and III only D. I, II and III
21. 2004/II/48
The figure shows the cube ABCDEFGH of side 2 cm. X and Y are the mid-points of AB and GH respectively.
Find XY.
A. 3 B. 22 C. 5 D. 6
22. 2004/II/49
In the figure, ABCDEFGH is a rectangular block. EG and FH meet at X. M is the mid-point of EH. Which of the
following makes the greatest angle with the plane ABCD?
A. AG B. AH C. AM D. AX
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.7
172
23. 2005/II/47
In the figure, B, C and D are three points on a horizontal plane such that ∠CBD = 90°. If AB is a vertical pole,
then ∠BCD =
A. 15° B. 30° C. 45° D. 60°
24. 2005/II/48
In the figure, VABCD is a right pyramid with a square base. If the angle between VA and the base is 45°, then
∠AVB =
A. 45°. B. 60°. C. 75°. D. 90°.
25. 2006/II/24
The figure shows a right prism ABCDEF with a right-angled triangle as the cross-section. The angle
between BD and the plane CDEF is
A. ∠BDE B. ∠BDF C. ∠DBE D. ∠DBF
HKCEE MATHEMATICS | 12.4 Trigonometry - 3D | P.8
173
26. 2007/II/24
In the figure, ABCDE is a right pyramid with the square base BCDE. F is a point lying on AC such that BF
and DF are perpendicular to AC. The angle between the plane ABC and the plane ACD is
A. ∠ACB B. ∠BAD C. ∠BCD D. ∠BFD
27. 2008/II/49
The figure shows a right prism ABCDEF with a right-angled triangle as the cross-section. A , B , E and F lie
on the horizontal ground. G and H are two points on the horizontal ground so that G , A , B and H are collinear.
It is given that AB = 6 m , AG = 3 m and BH = 2 m . If ∠DAE = a , ∠CBF = b , ∠CHF = c and ∠DGE = d ,
Which of the following must be true?
A. a < d < c B. c < a < d C. c < d < b D. d < c < b
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174
1. 1990/II/27
ABCD is a line segment. AB : BC : CD = 3 : 2 : 1. If A = (4, 5), D = (10, 11), find C.
A. (5, 6) B. (6, 7) C. (7, 8) D. (8, 9) E. (9, 10)
2. 1990/II/28
If the line y = mx + b and b
y
a
x+ = 1 are perpendicular, find m.
A. b
a B.
a
b C. ab D.
b
a− E.
a
b−
3. 1990/II/29
In the figure, the slopes of the straight lines l1, l2, l3, and l4 are m1, m2, m3, m4 respectively. Which of the following
is true?
A. m1 > m2 > m3 > m4 B. m2 > m1 > m3 > m4 C. m1 > m2 > m4 > m3
D. m2 > m1 > m4 > m3 E. m4 > m3 > m2 > m1
4. 1991/II/27
Let A and B be the points (4, −7) and (−6, 5) respectively. The equation of the line passing through the
mid-point of AB and perpendicular to 3x − 4y + 14 = 0 is
A. 3x − 4y − 1 = 0 B. 3x + 4y + 7 = 0 C. 4x − 3y + 1 = 0 D. 4x + 3y − 7 = 0 E. 4x + 3y + 7 = 0
5. 1991/II/28
PQRS is a parallelogram with vertices P = (0, 0), Q = (a, b) and S = (−b, a). Find R.
A. (−a, −b) B. (a, −b) C. (a − b, a − b) D. (a − b, a + b) E. (a + b, a + b)
6. 1992/II/28
If the two lines 012 =+− yx and 013 =−+ yax do not intersect, then =a
A. −6 B. −2 C. 2 D. 3 E. 6
7. 1992/II/31
The mid-points of the sides of a triangle are ( )4,3 , ( )0,2 and ( )2,4 . Which of the following points is a
vertex of the triangle?
A. ( )3,5.3 B. ( )2,3 C. ( )1,3 D. ( )2,5.1 E. ( )2,1
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8. 1993/II/27
If the point (1, 1), (3, 2) and (7, k) are on the same straight line, then k =
A. 3 B. 4 C. 6 D. 7 E. 10
9. 1993/II/28
A(0, 0), B(5, 0) and C(2, 6) are the vertices of a triangle. P(9, 5), Q(6, 6) and R(2, –9) are three points. Which
of the following triangles has/have area(s) greater than the area of ∆ABC?
I. ∆ABP II. ∆ABQ III. ∆ABR
A. I only B. II only C. III only D. I and II only E. II and III only
10. 1993/II/29
A circle of radius 1 touches both the positive x-axis and the positive y-axis. Which of the following is/are true?
I. Its center is in the first quadrant.
II. Its center lies on the line 0=− yx .
III. Its center lies on the line 1=+ yx .
A. I only B. II only C. III only D. I and II only E. I and III only
11. 1994/II/26
The points A(4,−1), B(−2, 3) and C (x, 5) lie on a straight line. Find x.
A. −5 B. −4 C. 0 D. 2 E. 3
12. 1994/II/27
In the figure, the shaded part is bounded by the axes, the lines x = 3 and x + y =5. Find its area.
A. 10.5 B. 12 C. 15 D. 19.5 E. 21
13. 1995/II/27
In the figure, the equation of the straight line L is
A. 03 =−x B. 03 =−− yx C. 03 =+− yx D. 03 =−+ yx E. 03 =++ yx
HKCEE MATHEMATICS | 13.1 Coordination Geometry - Straight lines | P.2
176
14. 1995/II/28
In the figure, OA=AB. If the slope of AB is m, find the slope of OA.
A. −1 B. m
1 C.
m
1− D. m E. m−
15. 1996/II/29
If a , b and c are all positive, which of the following may represent the graph of ax + by + c = 0?
16. 1996/II/31
Find the equation of the straight line which passes through (3, –1) and is perpendicular to 2x – y + 1 = 0.
A. x + 2y – 1 = 0 B. x + 2y + 1 = 0 C. x – 2y – 5 = 0
D. 2x + y – 5 = 0 E. 2x – y – 7 = 0
17. 1996/II/53
A(–3, 2) and B(1, 3) are two points. C is a point on the AB produced such that AB : BC = 1 : 2 . Find the
coordinates of C.
A. 5 7
( , )3 3
− B. 1 8
( , )3 3
− C. 7
(3, )2
D. (5, 4) E. (9, 5)
18. 1997/II/21
In the figure, find the area of ∆ABC.
A. 6 B. 7.5 C. 14 D. 17.5 E. 28
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177
19. 1997/II/22
Which of the following lines is perpendicular to the line 12 3
x y+ = ?
A. 3x + 2y = 1 B. 3x – 2y = 1 C. 2x + 3y = 1 D. 2x – 3y = 1 E. 12 3
x y− =
20. 1997/II/47
In the figure, AEB and ADC are straight lines. ED // BC and ED : BC = 2 : 3 . If the coordinates of A and B are
(4, 7) and (0, 1) respectively, find the coordinates of E.
A. (4
,33
) B. (8
,53
) C. (8 5
,5 17
) D. (12 23
,5 5
) E. (8 19
,7 7
)
21. 1998/II/32
Find the equation of the straight line passing through (–1 ,1) and parallel to 5x + 4y = 0.
A. 4x – 5y + 9 = 0 B. 4x + 5y + 1 = 0 C. 5x – 4y + 9 = 0
D. 5x + 4y – 1 = 0 E. 5x + 4y + 1 = 0
22. 1998/II/33
In the figure, PQRS is a parallelogram. Find the slope of PR.
A. 13
15 B.
15
13 C.
9
11 D.
11
9 E. –5
23. 1998/II/54
A(7, 14) and B(1, 2) are two points. C is a point on AB produced such that AB : BC = 2 : 1 . Find the
coordinates of C.
A. (–5, –10) B. (–2, –4) C. (3, 6) D. (5, 10) E. (10, 20)
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178
24. 1999/II/31
A(–4, 2) and B(1, –3) are two points. C is a point on the y-axis such that AC = CB . Find the coordinates of C.
A. (3 1
,2 2
− − ) B. (–1, 0) C. (1, 0) D. (0, –1) E. (0, 1)
25. 1999/II/32
In the figure, OABC is a parallelogram. If the equation of OC is 2x – y = 0 and the length of CB is 3 , find
the equation of AB.
A. x – 2y – 3 = 0 B. 2x – y – 3 = 0 C. 2x – y + 3 = 0 D. 2x – y – 6 = 0 E. 2x – y + 6 = 0
26. 2000/II/17
In the figure, find the area of ∆ABC.
A. 12 B. 15 C. 16 D. 20 E. 25
27. 2000/II/18
Consider the three straight lines
.0346:and 42
3:,0346: 321 =+−+−==−+ yxLxyLyxL
Which of the following is/are true?
I. L1 // L2 II. L2 // L3 III. L1 ⊥ L3
A. I only B. II only C. III only D. I and III only E. II and III only
28. 2000/II/50
A(−1, −4) and B(3, 4) are two points. The line 0=− yx cuts AB at P so that 1:: rPBAP = . Find r.
A. 3 B. 2 C. 1 D. 2
1 E.
3
1
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179
29. 2001/II/6
If the straight lines 0132 =+− yx and 015 =−+ kyx are perpendicular to each other, find k.
A. 2
15− B.
3
10− C.
10
3 D.
3
10 E.
2
15
30. 2001/II/33
If the straight lines 052 =+− yx and 01 =+− yax intersect at (1, b), find a and b.
A. 4−=a , 3−=b B. 1−=a , 0=b C. 1=a , 3=b D. 2=a , 3−=b E. 2=a , 3=b
31. 2001/II/34
In the figure, A, B and C are points on a rectangular coordinate plane. AC and BC are parallel to the x-axis
and y-axis respectively. If the coordinates of C are (2, 1) and the equation of the straight line AB is 32 += xy ,
find the distance between A and B.
A. 5 B. 2
53 C. 37 D. 53 E. 65
32. 2002/II/4
If 0<a and 0>b , which of the following may represent the graph of baxy += ?
33. 2002/II/30
If the length of the line segment joining the points (2, 3) and (k, 1−k) is 4, then k =
A. 2 B. 4 C. 0 or 4 D. −2 or 2
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180
34. 2002/II/31
In the figure, the equation of the straight line L is
A. 02 =++ yx B. 02 =−+ yx C. 02 =+− yx D. 02 =−− yx
35. 2002/II/32
In the figure, the area of ∆ABC is
A. 3 B. 8 C. 9 D. 18
36. 2003/II/29
In the figure, the straight lines L1 and L2 intersect at (2, 4). Find the equation of L1.
A. 102 =+ yx B. 62 −=− yx C. 82 =+ yx D. 02 =− yx
37. 2003/II/30
In the straight line 02 =++ kyx passes through the point of intersection of the two straight lines 03 =−+ yx
and 01 =+− yx , find k.
A. −4 B. −2 C. 2 D. 4
38. 2003/II/31
P(−10, −8) and Q(4, 6) are two points. If R is a point on the x-axis such that RQPR = , then the coordinates
of R are
A. (−4,0) B. (−3, −1) C. (−3,0) D. (−2,0)
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181
39. 2004/II/29
If 0>a , 0>b and 0<c , which of the following may represent the graph of the straight line 0=++ cbyax ?
40. 2004/II/30
In the figure, L1 and L2 are two straight lines intersecting at a point on the y−axis. If the equation of L1 is
022 =−+ yx , then the equation of L2 is
A. 012 =+− yx B. 022 =−− yx C. 012 =++ yx D. 022 =−+ yx
41. 2004/II/31
If (−2, 3) is the mid−point of (a, −1) and (4, b), then b =
A. −7 B. 7 C. −8 D. 8
42. 2005/II/31
A(2 , 5) and B(6 , −3) are two points. If P is a point lying on the straight line x = y such that AP = PB, then the
coordinates of P are
A. (−2 , −2) B. (−2 , 4) C. (1 , 1) D. (4 , 1)
43. 2005/II/32
In the figure, ABCD is a parallelogram. The coordinates of B are
A. (3 , 2) B. (3 , 5) C. (4 , 5) D. (4 , 6)
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182
44. 2005/II/33
If the equation of the straight line L is 2 3 0x y− + = , then the equation of the straight line passing through the
point (2 , −1) and perpendicular to L is
A. 2 3 0x y+ + = B. 2 3 0x y+ − = C. 2 3 0x y+ + = D. 2 3 0x y+ − =
45. 2006/II/28
If k < 0 , which of the following may represent the graph of the straight line x – y = k ?
46. 2006/II/29
The straight line 4x + y – 2 = 0 is perpendicular to the straight line
A. 4x + y – 9 = 0 B. 4x – y + 9 = 0 C. x + 4y – 9 = 0 D. x – 4y + 9 = 0
47. 2006/II/30
If the straight line 5x – 3y = 30 cuts the x-axis and the y-axis at A and B respectively, then the coordinates of
the mid-point of AB are
A. (3, –5) B. (–3, 5) C. (5, –3) D. (–5, 3)
48. 2006/II/31
If the points (0, 0) , (2, 0) and (1, b) are the vertices of an equilateral triangle, then b =
A. 1 B. 3 C. 1 or –1 D. 3 or – 3
49. 2007/II/29
If the point (3, –2) is rotated clockwise about the origin through 90° , then the coordinates of its image are
A. (2, 3) B. (3, 2) C. (–2, –3) D. (–3, –2)
50. 2007/II/31
Find the equation of the straight line which is perpendicular to the straight line x + 2y + 3 = 0 and passes through
the point (1, 3).
A. x + 2y – 7 = 0 B. x – 2y + 5 = 0 C. 2x + y – 5 = 0 D. 2x – y + 1 = 0
HKCEE MATHEMATICS | 13.1 Coordination Geometry - Straight lines | P.9
183
51. 2007/II/32
The figure shows the graph of the straight line ax + by + 1 = 0 . Which of the following is true?
A. a > 0 and b > 0 B. a > 0 and b < 0 C. a < 0 and b > 0 D. a < 0 and b < 0
52. 2008/II/29
The coordinates of the points A and B are (–2, a) and (b, 7) respectively. If the coordinates of the mid-point of
AB are (1, 5) , then a =
A. 0 B. 3 C. 4 D. 17
53. 2008/II/31
In the figure, the straight line L1 and L2 are perpendicular to each other. Find the equation of L2.
A. 3 0x y− = B. 3 0x y+ = C. 3 0x y− = D. 3 0x y+ =
54. 2008/II/32
In the figure, L1 , L2 , L3 and L4 are straight lines. If m1 , m2 , m3 and m4 are the slopes of L1 , L2 , L3 and
L4 respectively, which of the following must be true?
A. m1 < m2 < m3 < m4 B. m1 < m2 < m4 < m3 C. m2 < m1 < m3 < m4 D. m2 < m1 < m4 < m3
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184
1. 1990/II/30
In the figure, a circle cuts the x-axis at 2 points 6 units apart. If the circle has centre (4, 5), then its equation is :
A. (x − 4)2 + (y − 5)
2 = 25 B. (x − 4)
2 + (y − 5)
2 = 34 C. (x − 4)
2 + (y − 5)
2 = 52
D. (x + 4)2 + (y + 5)
2 = 34 E. (x + 4)
2 + (y + 5)
2 = 25
2. 1991/II/26
The circle x2 + y
2 + 4x + ky + 4 = 0 passes through the point (1, 3). The radius of the circle is
A. 68 B. 48 C. 17 D. 6 E. 3
3. 1992/II/29
If hk <<0 , which of the following circles intersect(s) the y-axis?
I. ( ) ( ) 222kkyhx =−+− II. ( ) ( ) 222
hkyhx =−+− III. ( ) ( ) 2222khkyhx +=−+−
A. I only B. II only C. III only D. I and II only E. II and III only
4. 1992/II/30
If the line 3+= mxy divides the circle 052422 =−−−+ yxyx into two equal parts, find m .
A. 4
1− B. −1 C. 0 D.
4
5 E. 2
5. 1993/II/30
What is the area of the circle 0261022 =−+−+ yxyx ?
A. π32 B. π34 C. π36 D. π134 E. π138
6. 1994/II/28
AB is a diameter of the circle .0182222 =−−−+ yxyx If A is (3,5), then B is
A. (2 , 3) B. (1 , −1) C. (−1 , −3) D. (−5 , −7) E. (−7 , −9)
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185
7. 1994/II/29
The equations of two circles are =+++
=−−+064
064
22
22
yxyx
yxyx
Which of the following is/are true?
I. The two circles have the same center.
II. The two circles have equal radii.
III. The two circles pass through the origin.
A. I only B. II only C. III only D. I and III only E. II and III only
8. 1995/II/29
The table below shows the centres and the radii of two circles C1 and C2.
Which of the following may represent the relative positions of C1 and C2?
9. 1995/II/30
In the figure, the equation of the circle is
A. 0522 =−+ yx B. 0222 =+−+ yxyx C. 0222 =−++ yxyx
D. 02422 =+−+ yxyx E. 02422 =−++ yxyx
10. 1996/II/30
The equation of the circle centred at (a, b) and tangential to the x-axis is
A. x2 + y
2 – 2ax – 2by + a
2 = 0 B. x
2 + y
2 – 2ax – 2by + b
2 = 0 C. x
2 + y
2 – 2ax – 2by + a
2 + b
2 = 0
D. x2 + y
2 + 2ax + 2by + a
2 = 0 E. x
2 + y
2 + 2ax + 2by + b
2 = 0
centre radius
C1 (2, 2) 3
C2 (5, –2) 2
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186
11. 1996/II/54
C1 : x2 + y
2 = 4 and C2 : x
2 + y
2 = 9 are two circles. A chord AB of C2 touches C1. Find the length of AB.
A. 5 B. 2 5 C. 65 D. 2 65 E. 10
12. 1997/II/45
The equation of a circle is given by x2 + y
2 – 4x + 6y – 3 = 0 . Which of the following statements is/are true?
I. The centre of the circle is (–2, 3)
II. The radius of the circle is 4
III. The origin is inside the circle.
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
13. 1997/II/46
A circle has (a, 0) and (0, b) as the end points of a diameter. Which of the following points lie(s) on this circle?
I. (–a, –b) II. (0, 0) III. (a, b)
A. II only B. III only C. I and II only D. II and III only E. I, II and III
14. 1998/II/52
The circle x2 + y
2 – 2x – 7y – 8 = 0 intersects the x-axis at A and B . Find the length of AB.
A. 2 B. 6 C. 7 D. 9 E. 85
15. 1998/II/53
The equations of two circles are x2 + y
2 + ax – by = 0 and x
2 + y
2 – ax + by = 0 . Which of the following must
be true?
I. The two circles have the same centre.
II. The two circles have equal radii.
III. The line joining the centres of the two circles passing through the origin.
A. I only B. II only C. III only D. I and II only E. II and III only
16. 1999/II/51
In the figure, find the equation of the circle with AB as a diameter.
A. x2
+ y2 – 2x + 2y – 23 = 0 B. x
2 + y
2 – 2x + 2y – 3 = 0 C. x
2 + y
2 + 2x – 2y – 23 = 0
D. x2
+ y2 + 2x – 2y – 3 = 0 E. x
2 + y
2 – 25 = 0
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187
17. 1999/II/52
The figure shows a circle centred at C and passing through O(0, 0) , A(6, 0) and B(0, 8) . Which of the following
must be true?
I. C lies on the line 16 8
x y+ =
II. The radius of the circle is 10
III. OC is perpendicular to AB
A. I only B. II only C. I and II only D. I and III only E. I, II and III only
18. 1999/II/53
Two circles with equations (x + 1)2 + (y + 1)
2 = 25 and (x – 11)
2 + (y – 8)
2 = 100 touch each other externally
at a point P . Find the coordinates of P .
A. (–3, –2) B. (7 4
,5 5
) C. (3, 2) D. (7
5,2
) E. (7, 5)
19. 2000/II/48
If the center of the circle 03)1(22 =−++++ ykkxyx lies on 01 =++ yx , find k.
A. 2
3 B.
2
1 C. 0 D. −1 E. −
2
3
20. 2000/II/49
If the straight line 1+= mxy is tangent to the circle 1)2( 22 =+− yx , then =m
A. 3
4− B. 0 C.
3
4 D. 0 or
3
4− E. 0 or
3
4
21. 2001/II/53
Consider the circle 0216822 =+−−+ yxyx . Find the equation of the chord whose mid-point is (5, 2).
A. 05559 =−+ yx B. 02343 =−+ yx C. 07 =−+ yx D. 03 =+− yx E. 03 =−− yx
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188
22. 2001/II/54
In the figure, the inscribed circle of ∆OPQ touches PQ at R. Find the coordinates of R.
A.
2,
2
3 B.
5
12,
5
6 C.
5
8,
5
9 D.
7
16,
7
9 E.
7
12,
7
12
23. 2002/II/52
In the figure, 0=x , 0=− ay and 0=+ ay are tangents to the circle. The equation of the circle is
A. 222 ayx =+ B. 0222 =−+ axyx C. 0222 =−+ ayyx D. 022 222 =++++ aayaxyx
24. 2002/II/53
The equation of a circle is given 2222 )()( babyax +=++− , where 0>a and 0>b . Which of the following
must be true?
I. The center of the circle is (a, −b).
II. The circle passes through the origin.
III. The circle cuts the x-axis at two distinct points.
A. I and II only B. I and III only C. II and III only D. I, II and III
25. 2003/II/54
The circle 36)4( 22 =+− yx intersects the positive x-axis and positive y-axis at A and B respectively. Find AB.
A. 30 B. 302 C. 34 D. 342
26. 2004/II/52
If the straight line 03 =−+ yx divides the circle 04222 =−−++ kyxyx into two equal parts, then =k
A. −4 B. 4 C. −8 D. 8
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189
27. 2004/II/53
The equation of a circle is 012422 =++−+ yxyx . Which of the following is/are true?
I. The circle touches the y−axis.
II. The origin lies outside the circle.
III. The center of the circle lies in the second quadrant.
A. II only B. III only C. I and II only D. I and III only
28. 2005/II/53
In the figure, O is the origin. If the equation of the circle passing through O, A, B and C is (x + 3)2
+ (y − 4)2 = 25,
then the area of the rectangle OABC is
A. 36 B. 48 C. 50 D. 64
29. 2005/II/54
In the figure, the circle passing through A(0,8) and B(0,2) touches the positive x-axis. The equation of the circle is
A. x2 + y
2 − 8x − 10y + 16 = 0 B. x
2 + y
2 + 8x + 10y + 16 = 0
C. x2 + y
2 − 10x − 10y + 16 = 0 D. x
2 + y
2 + 10x + 10y + 16 = 0
30. 2006/II/50
Consider the circle x2 + y
2 – 4x + 6y – 40 = 0 . Find the slope of the diameter passing through the point (1, 2).
A. –5 B. –3 C. 1
3
− D.
1
5
−
31. 2006/II/51
A circle C cuts the y-axis at A and B . If AB = 8 and the coordinates of the centre of C are (–3, 5) , then the
equation of C is
A. x2 + y
2 + 6x – 10y = 0 B. x
2 + y
2 – 6x + 10y = 0
C. x2 + y
2 + 6x – 10y + 9 = 0 D. x
2 + y
2 – 6x + 10y + 9 = 0
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32. 2007/II/51
A circle C touches the y-axis. If the coordinates of the centre of C are (–3, 4) , then the equation of C is
A. (x – 3)2 + (y + 4)
2 = 9 B. (x – 3)
2 + (y + 4)
2 = 16
C. (x + 3)2 + (y – 4)
2 = 9 D. (x + 3)
2 + (y – 4)
2 = 16
33. 2007/II/52
Let a be a constant. If the circle x2 + y
2 + ax – 6y – 3 = 0 passes through the point (–2, 3) , then the area of the
circle is
A. 8π B. 10π C. 16π D. 55π
34. 2008/II/53
The equation of a circle is x2 + y
2 – 4x – 8y + 11 = 0 . Which of the following are true?
I. The coordinates of the centre of the circle are (2, 4) .
II. The radius of the circle is 3.
III. The origin lies outside the circle.
A. I and II only B. I and III only C. II and III only D. I, II and III
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1. 2006/II/27
If the polar coordinates of the points A and B are (5, 45°) and (12, 135°) respectively, then the distance between
A and B is
A. 3 B. 7 C. 13 D. 17
2. 2007/II/30
If the rectangular coordinates of the point A are (–1, 1), then the polar coordinates of A are
A. (1, 135°) B. (1, 225°) C. ( 2 , 135°) D. ( 2 , 225°)
3. 2008/II/30
If the polar coordinates of the point P are (2, 300°) , then the rectangular coordinates of P are
A. ( 3− ,1) B. (–1, 3 ) C. (1, 3− ) D. ( 3 ,–1)
HKCEE MATHEMATICS | 13.3 Coordination Geometry - Polar Coordinates | P.1
192
1. 1990/II/6
Let a > b > 0. If a and b are respectively the 1st and 2
nd terms of a geometric progression, the sum to infinity
of the progression is
A. ba −
1 B.
b
a
−1 C.
ab
ab
−
D. ba
a
+
2
E. ba
a
−
2
2. 1990/II/38
Let a, x1, x2, b and a, y1, y2, y3, b be two arithmetic progressions. =−
−
23
12
yy
xx
A. 4
3 B.
4
3 C. 1 D.
5
4 E.
4
5
3. 1991/II/40
If the sum to n terms of an A.P. is n2 + 3n, find the 7
th term of the A.P.
A. 16 B. 18 C. 54 D. 70 E. It cannot be found
4. 1991/II/41
If x, y, z are in G.P, which of the following must be true?
I. x + 3, y + 3, z + 3 are in G.P. II. 3x, 3y, 3z are in G.P III. x2, y
2, z
2 are in G.P.
A. I only B. II only C. III only D. I and II only E. II and III only
5. 1992/II/35
If the quadratic equation 022 =+− cbxax has two equal roots, which of the following is/are true?
I. cba , , form an arithmetic progression.
II. cba , , form an geometric progression.
III. Both roots are a
b.
A. I only B. II only C. III only D. I and II only E. II and III only
6. 1992/II/42
Find the ( )th2n term of the G.P. ,...4 ,2 ,1 ,2
1−− .
A. n22 B. n22− C. 322 −
−n
D. 222 −n E. 222 −
−n
7. 1993/II/10
If 3, a, b, c, 23 are in A.P., then a + b + c =
A. 13 B. 26 C. 33 D. 39 E. 65
HKCEE MATHEMATICS | 14 Sequence | P.1
193
8. 1993/II/37
Given that the positive numbers p, q, r, s are in G.P., which of the following must be true?
I. kp, kq, kr, ks are in G.P., where k is a non-zero constant.
II. srqp aaaa , , , are in G.P., where a is a positive constant.
III. log ,log ,log ,log srqp are in A.P.
A. I only B. II only C. I and II only D. I and III only E. I, II and III only
9. 1994/II/12
If the sum to infinity of a G.P. is4
81 and its second term is –9, the common ratio is
A. −3
1 B.
3
1 C. −
3
4 D.
3
4 E. −
9
4
10. 1994/II/41
If the product of the first n terms of the sequence 10, 10 2 , 10 3 , … , 10 n exceeds 10 ,55 find the minimum value
of n.
A. 9 B. 10 C. 11 D. 12 E. 56
11. 1995/II/42
In an A.P., the sum of the first 2 terms is 3 and the sum of the first 3 terms is 2. The common difference is
A. 3
5− B. −1 C. 1 D.
3
5 E.
3
7
12. 1996/II/12
Find the n-th term of the A.P. 4, 2, 0, –2, … .
A. 2 + 2n B. 4 – 2n C. 4 + 2n D. 6 – 2n E. (5 – n) n
13. 1996/II/13
The sum to infinity of a G.P. is 2 . If the first term is 3
2 , find the common ratio.
A. 1
2− B.
1
4− C.
1
4 D.
1
2 E.
3
2
14. 1996/II/42
If the common difference of the A.P. a1 , a2 , a3 , … is d , then the common difference of the A.P.
2a1 + 3 , 2a2 + 3 , 2a3 + 3 , … is
A. 2 B. 3 C. d D. 2d E. 2d + 3
15. 1997/II/35
The n-th term of an arithmetic sequence is 3 + 2n . Find the sum of the first 50 terms of the sequence.
A. 103 B. 2575 C. 2700 D. 2750 E. 5400
HKCEE MATHEMATICS | 14 Sequence | P.2
194
16. 1997/II/36
The first term of a geometric sequence is a . If the sum to infinity of the sequence is 3
4a , then its common
ratio is
A. 1
3− B.
1
4− C.
1
4 D.
1
3 E.
3
4
17. 1997/II/37
a , b , c , d are 4 consecutive terms of a geometric sequence. Which of the following must be true?
I. b2 = ac II.
b d
a c= III.
3d c
a b
=
A. II only B. I and II only C. I and III only D. II and III only E. I, II and III
18. 1998/II/13
If a , b , c , d are consecutive terms of an arithmetic sequence, which of the following must be true?
I. b – a = d – c
II. d , c , b , a are consecutive terms of an arithmetic sequence
III. a < b < c < d
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
19. 1998/II/43
Find the sum to infinity of the geometric sequence –1, 1
x,
2
1
x− ,
3
1
x, … , where x > 1 .
A. 1
1x
−
− B.
1
1x
−
+ C.
1
x
x
−
− D.
1
x
x
−
+ E.
1
x
x +
20. 1999/II/10
The n-th term of an arithmetic sequence is 2 + 5n . Find the sum of the first 100 terms of the sequence.
A. 502 B. 12450 C. 25200 D. 25450 E. 25700
21. 1999/II/44
The sum of the first two terms of a geometric sequence is 3 and the sum to infinity of the sequence is 4. Find
the common ratio of the sequence.
A. 1
7− B.
1
7 C.
1
4 D.
1
2− E.
1
2− or
1
2
22. 2000/II/15
The 1st and 10
th terms of an arithmetic sequence are 2 and 29 respectively. The 20
th term of the sequence is
A. 56 B. 58 C. 59 D. 60 E. 62
HKCEE MATHEMATICS | 14 Sequence | P.3
195
23. 2000/II/16
Which of the following could be a geometric sequence/geometric sequences?
I. 3, 33, 3
5, 3
7, …
II. 9, 99, 999, 9999, …
III. 10, −100, 1000, −10000, …
A. III only B. I and II only C. I and III only D. II and III only E. I, II and III
24. 2001/II/14
The sum of the first n terms of an arithmetic sequence is n2. Find the 10
th term of the sequence.
A. 19 B. 21 C. 28 D. 31 E. 100
25. 2001/II/15
The nth
term of a geometric sequence is n2
1− . Find the first term and the common ratio.
first term common ratio
A. −1 1
2
B. −1
2 −
1
2
C. −1
2
1
2
D. −1
2 1
E. 1 −1
2
26. 2002/II/11
The 10th
term of an arithmetic sequence is 29 and the sum of the first 10 terms is 155. The 2nd
term of the
sequence is
A. 2 B. 4.7 C. 5 D. 43
27. 2002/II/43
If the geometric mean of two positive numbers a and b is 100, then the arithmetic mean of alog and blog is
A. 2
1 B. 1 C. 2 D. 10
28. 2003/II/10
The sum of the 4th
term and the 5th
term of a geometric sequence is −4. If the sum of the first two terms is 32,
find the first term of the sequence.
A. −6 B. −2
1 C. 19 D. 64
HKCEE MATHEMATICS | 14 Sequence | P.4
196
29. 2004/II/11
Let na be the nth term of an arithmetic sequence. If 101 =a and 132 =a , then =+++ 302221 aaa �
A. 765 B. 835 C. 865 D. 1605
30. 2004/II/44
If 81, a, b, 3 is a geometric sequence, then =− ab
A. −18 B. 18 C. −26 D. 26
31. 2005/II/11
If the 2nd term and the 5th term of a geometric sequence are −3 and 192 respectively, then the common ratio of
the sequence is
A. −8 B. −4 C. 4 D. 8
32. 2005/II/42
If four arithmetic means are inserted between 12 and 27, then the sum of the four arithmetic means is
A. 78 B. 90 C. 105 D. 117
33. 2006/II/12
In the figure, the 1st pattern consists of 3 dots. For any positive integer n , the (n + 1) th pattern is formed by
adding (2n + 3) dots to the nth pattern. Find the number of dots in the 6th pattern.
A. 35 B. 37 C. 48 D. 50
34. 2006/II/42
The first negative term in the arithmetic sequence 2006 , 1998 , 1990 , … is
A. –8 B. –6 C. –4 D. –2
35. 2006/II/43
Let a , b and c be positive integers. If b ac= , which of the following must be true?
I. loga2 , logb
2 , logc
2 is an arithmetic sequence.
II. a3 , b
3 , c
3 is a geometric sequence.
III. 4a , 4
b , 4
c is a geometric sequence.
A. I and II only B. I and III only C. II and III only D. I, II and III
…
HKCEE MATHEMATICS | 14 Sequence | P.5
197
36. 2007/II/9
In the figure, the 1st pattern consists of 4 dots. For any positive integer n , the (n + 1) th pattern is formed by
adding 4 dots to the nth pattern. Find the number of dots in the 9th pattern.
A. 36 B. 40 C. 81 D. 100
37. 2007/II/44
Let an be the nth term of an arithmetic sequence. If a1 = a2 – 6 and a1 + a2 + … + a28 = 1624 , then a1 =
A. –52 B. –26 C. –23 D. 139
38. 2007/II/45
The sum of all the positive terms in the geometric sequence 4 , –2 , 1 , … is
A. 8 B. 8
3 C.
16
3 D.
16
5
39. 2008/II/11
In the figure, the 1st pattern consists of 10 dots. For any positive integer n , the (n + 1) th pattern is formed by
adding (2n + 5) dots to the nth pattern. Find the number of dots in the 7th pattern.
A. 50 B. 65 C. 82 D. 101
40. 2008/II/43
Let a , b and c be positive integers. Which of the following must be arithmetic sequences?
I. a + 10 , 2a + 7 , 3a + 4 , 4a + 1
II. 8b – 1 , 82b
– 2 , 83b
– 3 , 84b
– 4
III. logc3 , logc
8 , logc
13 , logc
18
A. I and II only B. I and III only C. II and III only D. I, II and III
41. 2008/II/44
If a – 6 , a , a + 5 is q geometric sequence, then the common ratio of the sequence is
A. –30 B. 5
6 C.
6
5 D. 6
…
…
HKCEE MATHEMATICS | 14 Sequence | P.6
198
1. 1990/II/26
There are 7 bags, 3 of which are empty and the remaining 4 each contains a ball. An additional ball is now put
into one of the bags at random. After that a bag is randomly selected. Find the probability of selecting an
empty bag.
A. 7
2 B.
7
3 C.
49
6 D.
49
12 E.
49
18
2. 1991/II/32
A fair die is thrown 3 times. The probability that “6” occurs exactly once is
A. 3
1 B.
3
6
1
C.
6
1
3
1× D.
2
6
5
6
1
E.
2
6
5
6
1
3. 1992/II/33
Two cards are drawn randomly from five cards D , , , CBA and E . Find the probability that card A is drawn
while card C is not.
A. 25
3 B.
20
3 C.
25
4 D.
25
6 E.
10
3
4. 1993/II/31
Two fair dice are thrown. What is the probability of getting a total of 5 or 10?
A. 9
1 B.
36
5 C.
6
1 D.
36
7 E.
9
2
5. 1994/II/31
A box contains 5 eggs, 2 of which are rotten. If 2 eggs are chosen at random, find the probability that exactly
one of them is rotten.
A. 5
2 B.
5
3 C.
10
3 D.
25
6 E.
25
12
6. 1995/II/31
In a shooting game, the probability that A will hit a target is5
3 and the probability that B will hit it is
3
2. If each
fires once, what is the probability that they will both miss the target?
A. 3
1 B.
4
1 C.
5
2 D.
15
2 E.
15
11
HKCEE MATHEMATICS | 15 Probability | P.1
199
7. 1995/II/32
The figure shows that Mr. Chan has 3 ways to leave town X and Mr. Lee has 2 ways to leave town Y. Mr. Chan
and Mr. Lee leave town X and town Y respectively at the same time. If they select their ways randomly, find
the probability that they will meet on their way.
A. 2
1 B.
3
1 C.
3
2 D.
6
1 E.
6
5
8. 1996/II/34
There are 10 parcels. Two of them contain one pen each. If a man opens the parcels at random, what is the
probability that he can find the two pens by opening two parcels only?
A. 1
25 B.
1
45 C.
1
50 D.
1
90 E.
1
100
9. 1996/II/35
In a certain game, the probability that John will win is 0.3 . If he plays the game 3 times, find the probability
that he will win at least once.
A. 0.147 B. 0.441 C. 0.657 D. 0.9 E. 0.973
10. 1997/II/25
Two fair dice are thrown. Find the probability that the sum of the two numbers shown is 8.
A. 1
4 B.
1
6 C.
1
11 D.
1
12 E.
5
36
11. 1997/II/26
In a test, there are 3 questions. For each question, the probability that John correctly answers it is 2
5. Find
the probability that he gets exactly 2 questions correct.
A. 2
3 B.
4
25 C.
12
25 D.
12
125 E.
36
125
12. 1998/II/35
Two cars are drawn randomly from five cards numbered 2 , 2 , 3 , 5 and 5 respectively. Find the probability that
the sum of the numbers on the cards drawn is 5.
A. 1
5 B.
2
5 C.
1
10 D.
2
25 E.
4
25
HKCEE MATHEMATICS | 15 Probability | P.2
200
13. 1998/II/36
In a shooting game, the probability that Mr. Tung will hit the target is 2
3. If he shoots twice, find the probability
that he will hit the target at least once.
A. 1
9 B.
2
9 C.
4
9 D.
2
3 E.
8
9
14. 1999/II/35
Two cards are drawn randomly from four cards numbered 1 , 2 , 3 and 4 respectively. Find the probability that
the sum of the numbers drawn is odd.
A. 1
6 B.
1
4 C.
1
3 D.
1
2 E.
2
3
15. 1999/II/36
Tom and Mary each throws a dart. The probability of Tom’s dart hitting the target is 1
3 while that of Mary’s is
2
5. Find the probability of only one dart hitting the target.
A. 2
15 B.
3
15 C.
7
15 D.
11
15 E.
13
15
16. 2000/II/21
Two fair dice are thrown. Find the probability that at least one “6” occurs.
A. 3
1 B.
6
1 C.
18
5 D.
36
7 E.
36
11
17. 2000/II/22
A bag contains six balls which are marked with the numbers −3, −2, −1, 1, 2 and 3 respectively. Two balls are
drawn randomly from the bag. Find the probability that the sum of the numbers drawn is zero.
A. 30
1 B.
10
1 C.
5
1 D.
3
1 E.
2
1
18. 2001/II/35
Two cards are drawn randomly from five cards numbered 1, 2, 3, 4 and 4 respectively. Find the probability that
the sum of the two numbers drawn is even.
A. 2
1 B.
5
2 C.
10
3 D.
10
7 E.
25
13
19. 2001/II/36
A bag contains 2 black balls and 3 white balls. A boy randomly draws balls from the bag one at a time (without
replacement) until a white ball appears. Find the probability that he will make at least 2 draws.
A. 5
2 B.
5
3 C.
10
1 D.
10
3 E.
10
7
HKCEE MATHEMATICS | 15 Probability | P.3
201
20. 2002/II/35
Two numbers are drawn randomly from five cards numbered 3, 4, 5, 6 and 7 respectively. Find the probability that
the product of the numbers drawn is even.
A. 5
3 B.
10
1 C.
10
7 D.
25
16
21. 2002/II/36
In a test, there are two questions. The probability that Mary answers the first question correctly is 0.3 and the
probability that Mary answers the second question correctly is 0.4. The probability that she answers at least one
question correctly is
A. 0.42 B. 0.46 C. 0.58 D. 0.88
22. 2003/II/34
A bag contains 2 black balls, 2 green balls and 2 yellow balls. Peter repeats drawing one ball at a time randomly
from the bag without replacement until a green ball is drawn. Find the probability that he needs at most 4 draws.
A. 15
1 B.
15
2 C.
15
14 D.
81
65
23. 2003/II/35
1232� is a 5-dight number, where � is an integer from 0 to 9 inclusive. The probability that the 5-dight number
is divisible by 4 is
A. 3
1 B.
4
1 C.
5
1 D.
10
3
24. 2004/II/33
A bag contains 3 red balls and 4 green balls. If two balls are drawn randomly from the bag one by one without
replacement, then the probability that the two balls are of different colors is
A. 7
2 B.
7
4 C.
49
12 D.
49
24
25. 2004/II/34
Peter and May each throws a dart. The probability of Peter’s hitting the target is 0.2. The probability of May’s
hitting the target is 0.3. Find the probability of at least one dart hitting the target.
A. 0.38 B. 0.44 C. 0.5 D. 0.56
26. 2005/II/35
Bag X contains 1 white ball and 3 red balls while bag Y contains 3 yellow balls and 6 red balls. A ball is
randomly drawn from bag X and put into bag Y. If a ball is now randomly drawn from bag Y, then the
probability that the ball drawn is red is
A. 1
2 B.
2
3 C.
21
40 D.
27
40
HKCEE MATHEMATICS | 15 Probability | P.4
202
27. 2005/II/36
If a fair die is thrown three times, then the probability that the three numbers thrown are all different is
A. 5
9 B.
17
18 C.
125
216 D.
215
216
28. 2006/II/32
Which of the following could be the probability of an event?
A. 3
π B.
2005
2006 C. –0.2006 D. 1.2006
29. 2006/II/33
Two fair dice are thrown. Find the probability that the sum of the two numbers thrown is a prime number.
A. 1
2 B.
5
11 C.
5
12 D.
7
18
30. 2006/II/52
One letter is chosen randomly from each of the two words ‘FORTY’ and ‘FIFTY’ . Find the probability that
the two letters chosen are the same.
A. 0.08 B. 0.16 C. 0.32 D. 0.48
31. 2006/II/53
There are two questions in a test. The probability that David answers the first question correctly is 1
4 and the
probability that David answers the second question correctly is 1
3. Given that David answers at least one
question correctly in the test, find the probability that he answers the second question correctly.
A. 1
2 B.
2
3 C.
3
5 D.
4
5
32. 2007/II/33
Two numbers are randomly drawn at the same time from five cards numbered 1 , 2 , 3 , 4 and 5 respectively. Find
the probability that the sum of the numbers drawn is a multiple of 3.
A. 2
5 B.
3
10 C.
9
20 D.
9
25
33. 2007/II/53
A bag contains 8 black balls and 5 white balls. If two balls are drawn randomly from the bag one by one
without replacement, then the probability that the two balls are of the same colour is
A. 14
39 B.
19
39 C.
89
156 D.
89
169
HKCEE MATHEMATICS | 15 Probability | P.5
203
34. 2007/II/54
One letter is chosen randomly from each of the two words ‘CUBE’ and ‘CONE’ . Find the probability that
the two letters chosen are different.
A. 1
4 B.
3
4 C.
1
8 D.
7
8
35. 2008/II/33
4�is a 2-digit number, where � is an integer from 0 to 0 inclusive. Find the probability that the 2-digit number
is a prime number.
A. 0.2 B. 0.3 C. 0.4 D. 0.5
HKCEE MATHEMATICS | 15 Probability | P.6
204
1. 1990/II/24
If the mean of the numbers 3, 3, 3, 3, 4, 4, 5, 5, 6, x is also x, which of the following is/are true?
I. Mean = Median II. Mode = Range III. Median = Mode
A. I and II only B. I and III only C. II and III only D. None of them E. All of them
2. 1990/II/25
Ten years ago, the mean age of a band of 11 musicians was 30. One of them is now leaving the band at the
age of 40. What is the present mean age of the remaining 10 musician?
A. 40 B. 39 C. 37 D. 30 E. 29
3. 1991/II/30
The mean and standard deviation of a distribution of test scores are m and s respectively. If 4 marks are added
to each score of the distribution, what are the mean and standard deviation of the new distribution?
Mean Standard Deviation
A. m + 4 s
B. m + 4 s + 2
C. m + 4 s + 4
D. M s + 2
E. M s + 4
4. 1991/II/31
The graph shows the frequency curves of two symmetric distributions P and Q. Which of the following is /are
true?
I. The mean of P < the mean of Q.
II. The mode of P > the mode of Q.
III. The inter-quartile range of P < the inter-quartile range of Q.
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
HKCEE MATHEMATICS | 16 Statistics | P.1
205
5. 1992/II/32
The table shows the mean marks of two classes of students in a mathematics test.
Number of students Mean mark
Class A 38 72
Class B 42 54
A student in Class A has scored 91 marks. It is found that his score was wrongly recorded as 19 in the
calculation of the mean mark for Class A in the above table. Find the correct mean mark of the 80 students
in the two classes
A. 61.65 B. 62.55 C. 63 D. 63.45 E. 63.9
6. 1992/II/34
The figure shows the cumulative frequency curves of three distributions. Arrange the three distributions in the
order of their standard deviations, from the smallest to the largest.
A. I, II, III B. I, III, II C. II, I , III D. II, III , I E. III, I , II
7. 1993/II/32
A group of n numbers has mean m. If the numbers 1, 2 and 6 are removed from the group, the mean of
the remaining n–3 numbers remains unchanged. Find m.
A. 1 B. 2 C. 3 D. 6 E. 3−n
8. 1993/II/33
The figure shows the frequency polygons of two symmetric distributions A and B with the same mean. Which of
the following is/are true?
I. Interquartile range of A < Interquartile range of B
II. Standard deviation of A > Standard deviation of B
III. Mode of A > Mode of B
A. I only B. II only C. III only D. I and III only E. II and III only
HKCEE MATHEMATICS | 16 Statistics | P.2
206
9. 1994/II/30
In the figure, the pie chart shows the monthly expenditure of a family. If the family spends $4800 monthly on
rent, what is the monthly expenditure on entertainment?
A. $240 B. $600 C. $720 D. $1 800 E. $12 000
10. 1994/II/32
The mean, standard deviation and interquartile range of n numbers are m, s and q respectively. If 3 is added to
each of the n numbers, what will be their new mean, standard deviation and interquartile range?
Mean Standard deviation Interquartile range
A. m s q
B. m s + 3 q + 3
C. m + 3 s q
D. m + 3 s q + 3
E. m + 3 s + 3 q + 3
11. 1995/II/33
The mean of a set of 9 numbers is 12. If the mean of the first 5 numbers is 8, the mean of the other four numbers
is
A. 4 B. 10 C. 16 D. 17 E. 25
HKCEE MATHEMATICS | 16 Statistics | P.3
207
12. 1995/II/34
The figure shows the frequency curves of two symmetric distributions A and B. Which of the following is/are
true?
I. The mean of A = the mean of B.
II. The inter-quartile range of A > the inter-quartile range of B.
III. The standard deviation of A > the standard deviation of B.
A. I only. B. I and II only C. I and III only D. II and III only E. I, II and III
13. 1996/II/32
The bar chart below shows the number of electronic dictionaries sold in a shop last week:
Of those electronic dictionaries sold last week, what percentage were sold on Sunday?
A. 16% B. 18% C. 20% D. 22.5% E. 25%
14. 1996/II/33
Which of the following cannot be read directly from a cumulative frequency curve?
I. Mean II. Median III. Mode
A. I only B. II only C. I and II only D. I and III only E. II and III only
HKCEE MATHEMATICS | 16 Statistics | P.4
208
15. 1997/II/23
In the pie chart, if x : y : z = 75 : 106 : 119 , find x .
A. 25 B. 45 C. 75 D. 90 E. 120
16. 1997/II/24
The histogram below shows the distribution of the weights of 30 students. Find the mean weight of these students.
A. 36.5 kg B. 38.5 kg C. 39 kg D. 39.5 kg E. 41.5 kg
17. 1998/II/34
In the figure, CA and CB are the cumulative frequency curves of two distributions of weights A and B
respectively. Which of the following is/are true?
I. median of A > median of B
II. range of A > range of B
III. inter-quartile range of A > inter-quartile range of B
18. 1998/II/51
A. I only B. I and II only C. I and III only D. II and III only E. I, II and III
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19. 1999/II/33
Find the median and mode of the ten numbers
6 , 8 , 3 , 3 , 5 , 5 , 5 , 7 , 7 , 11 .
A. median = 5 , mode = 5 B. median = 5 , mode = 5.5 C. median = 5.5 , mode = 5
D. median = 5.5 , mode = 6 E. median = 6 , mode = 5
20. 2000/II/23
}8,6,4,2,{ ++++ xxxxx and }9,7,5,3,1{ +++++ xxxxx are two groups of numbers. Which of the
following is/are true?
I. The two groups of numbers have the same range.
II. The two groups of numbers have the same standard deviation.
III. The two groups of numbers have the same mean.
A. I only B. II only C. III only D. I and II only E. I and III only
21. 2001/II/5
The bar chart below shows the distribution of scores in a test. Find the percentage of scores which are less than 3.
A. 35% B. 40% C. 50% D. 65% E. 70%
22. 2001/II/21
If the mean of the ten numbers 8, 6, 6, 6, 7, 4, 10, 9, 9, x is 7, find the median of the ten numbers.
A. 5.5 B. 6 C. 6.5 D. 7 E. 7.5
23. 2002/II/33
The pie chart below shows the expenditure of a family in January 2002. The percentage of the expenditure on
Rent was
A. 12.5% B. 22.5% C. 25% D. 45%
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24. 2002/II/34
For the five numbers x, 1−x , 2−x , x, 8+x , which of the following must be true?
I. The median is 2−x
II. The mean is 1+x
III. The mode is 2
A. I only B. II only C. I and III only D. II and III only
25. 2002/II/54
The standard deviation of the four numbers 7−m , 1−m , 1+m and 7+m is
A. 2.5 B. 4 C. 5 D. 10
26. 2003/II/33
The median of the five numbers 15, 1−x , 3−x , 4−x and 17+x is 8. Find the mean of the five numbers.
A. 8 B. 12 C. 13.6 D. 14.4
27. 2003/II/36
x is the mean of the group of numbers {a, b, c, d, e}. Which of the following statements about the two groups of
numbers {a, b, c, d, e} and { a, b, c, d, e, x} must be true?
I. The two groups of numbers have the same mean.
II. The two groups of numbers have the same range.
III. The two groups of numbers have the same standard deviation.
A. I only B. III only C. I and II only D. II and III only
28. 2004/II/32
The mean weight of 36 boys and 32 girls is 46 kg. If the mean weight of the boys is 52 kg, then the mean weight
of the girls is
A. 39.25 kg B. 40 kg C. 40.67 kg D. 49 kg
29. 2004/II/35
The pie chart below shows the expenditure of a student in March 2004. If the student spent $ 520 on meals, then
the student’s total expenditure on entertainment and clothing was
A. $ 780 B. $ 1 092 C. $ 1 352 D. $ 1 872
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30. 2005/II/34
If the mean of five numbers 15, x + 4, x + 1, 2x − 7 and x − 3 is 6, then the mode of the five numbers is
A. 1 B. 4 C. 5 D. 15
31. 2006/II/34
{ }6 , 3 , 4 , 5x x x x− − + + and { }8 , 1 , 2 , 9x x x x− − + + are two groups of numbers. Which of the
following is/are true?
I. The two groups of numbers have the same mean.
II. The two groups of numbers have the same median.
III. The two groups of numbers have the same range.
A. I only B. II only C. I and III only D. II and III only
32. 2006/II/35
The box-and-whisker diagram below shows the distribution of the weights (in kg) of some students. Find the
inter-quartile range of their weights.
A. 5 kg B. 10 kg C. 15 kg D. 30 kg
33. 2006/II/36
The scatter diagram below shows the relation between x and y . Which of the following may represent the
relation between x and y?
A. y varies directly as x2. B. y decreases when x increases.
C. x increases when y increases. D. x remains unchanged when y increases.
34. 2006/II/54
The standard deviation of the five numbers 10a + 1 , 10a + 3 , 10a + 5 , 10a + 7 and 10a + 9 is
A. 8 B. 12
5 C. 10 D. 2 2
35. 2007/II/34
If the mode of the seven numbers 8 , 7 , 1 , 3 , 7 , a and b is 8 , then the median of the seven numbers is
A. 3 B. 6 C. 7 D. 8
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36. 2007/II/35
In the figure, BX and BY are the box-and-whisker diagrams for the distributions X and Y respectively. Let
1µ , q1 and r1 be the mean, the interquartile range and the range of X respectively while 2µ , q2 and r2 be
the mean, the interquartile range and the range of Y respectively. Which of the following must be true?
I. 1µ < 2µ II. q1 < q2 III. r1 < r2
A. I only B. II only C. I and III only D. II and III only
37. 2007/II/36
The stem-and-leaf diagram below shows the distribution of the weights (in kg) of some students.
Which of the following frequency curves may represent the distribution of their weights?
38. 2008/II/34
Let A be a group of numbers { } , , ,α β γ δ and B be another group of numbers { } , , , ,α β γ δ µ , where
< <α β γ δ µ< < . Which of the following must be true?
I. The range of A is smaller than that of B.
II. The mean of A is smaller than that of B.
III. The median of A is smaller than that of B.
A. I and II only B. I and III only C. II and III only D. I, II and III
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39. 2008/II/35
The frequency curve below shows the distribution of the heights (in cm) of the students in a school.
Which of the following box-and-whisker diagrams may represent the distribution of their heights?
40. 2008/II/36
If y increases when x increases, which of the following scatter diagrams may represent the relation between
x and y?
41. 2008/II/54
The bar chart below shows the numbers of cars sold for brand A , brand B and brand C in a certain month.
A sales representative makes the following claims:
I. In that month, the number of cars sold for brand C is two times that for brand B.
II. In that month, the total number of cars sold for brand A and brand B is less than the number of cars sold
for brand C.
III. In that month, the number of cars sold for brand A is 50% less than that for brand C.
Which of the above claims are false?
A. I and II only B. I and III only C. II and III only D. I, II and III
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HKCEE Mathematics Paper II Answers 1990-2008 1 Percentages 01. D 02. C 03. A 04. C 05. A 06. D 07. C 08. C 09. A 10. D 11. D 12. C 13. B 14. E 15. B 16. E 17. E 18. C 19. C 20. C 21. C 22. A 23. C 24. D 25. C 26. A 27. B 28. D 29. B 30. D 31. B 32. C 33. B 34. D 35. A 2 Estimation and Error 01. D 02. D 03. B 04. C 05. D 06. C 3.1 Algebraic Expression - LCM and HCF 01. D 02. C 03. B 04. B 05. C 06. C 07. C 08. C 09. C 10. D 11. A 3.2 Algebraic Expression – Factorization 01. E 02. B 03. A 04. E 05. D 06. B 07. B 08. E 09. A 10. E 11. C 12. A 13. B 14. D 15. D 16. A 17. A 18. D 19. A 20. A 21. B 22. C 3.3 Algebraic Expression - Algebraic Simplification 01. A 02. A 03. E 04. E 05. A 06. A 07. A 08. C 09. A 10. C 11. D 12. D 13. A 14. C 15. C 16. D 17. A 18. D 19. E 20. C 21. A 22. D 23. B 24. E 25. A 26. E 27. E 28. D 29. D 30. A 31. A 32. D 33. C 34. A 35. A 36. B 4.1 Polynomials - Function and Graph 01. E 02. D 03. C 04. B 05. D 06. E 07. B 08. A 09. E 10. B 11. C 12. C 13. A 14. A 15. B 16. D 17. D 18. C 19. D 20. C 4.2 Polynomials - Remainder and Factor Theorem 01. A 02. A 03. A 04. A 05. C 06. D 07. A 08. D 09. B 10. B 11. E 12. D 13. E 14. D 15. A 16. A 17. D 18. D 19. C 20. B 21. A 22. B 23. A 4.3 Polynomials – Identities 01. C 02. A 03. A 04. B 05. E 06. B 07. E 08. C 09. A 10. B 11. A 12. A 13. B 14. D 15. D 16. B
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5.1 Equations - Simultaneous Equations 01. C 02. E 03. B 04. E 05. A 06. C 07. D 08. D 09. C 10. D 11. B 12. B 13. C 14. D 15. D 16. D 17. C 18. D 19. B 20. D 5.2 Equations - Quadratic Equations and Graphs 01. B 02. C 03. E 04. E 05. E 06. E 07. B 08. A 09. E 10. E 11. E 12. A 13. C 14. A 15. A 16. B 17. C 18. A 19. D 20. E 21. D 22. B 23. E 24. A 25. B 26. A 27. E 28. C 29. B 30. B 31. A 32. C 33. C 34. C 35. C 36. C 37. D 38. B 39. D 40. C 41. A 42. D 43. A 44. D 45. C 46. A 47. A 6.1 Indices 01. A 02. A 03. A 04. E 05. A 06. C 07. A 08. B 09. E 10. A 11. B 12. B 13. C 14. E 15. A 16. A 17. C 18. B 19. B 20. A 21. B 22. B 6.2 Logarithms 01. A 02. D 03. A 04. B 05. B 06. C 07. A 08. B 09. A 10. E 11. D 12. E 13. C 14. A 15. A 16. D 17. A 6.3 Surds 01. E 02. E 03. D 04. D 05. E 06. B 07. E 08. E 09. B 10. B 11. B 12. A 13. D 14. C 15. A 7 Ratio, Ratio and Variation 01. C 02. C 03. E 04. D 05. A 06. B 07. D 08. E 09. C 10. D 11. D 12. C 13. C 14. C 15. C 16. B 17. C 18. E 19. B 20. C 21. C 22. D 23. D 24. C 25. C 26. B 27. D 28. A 29. D 30. A 31. D 32. C 33. C 34. B 35. A 36. C 37. C 38. D 39. D 40. C 41. A 42. B 43. C 44. D 45. C 8.1 Inequality - Linear Inequality 01. D 02. C 03. E 04. A 05. E 06. E 07. A 08. D 09. B 10. A 11. A 12. B 13. A 14. D 15. D 16. A 17. A 18. B 8.2 Inequality - Linear Programming 01. E 02. B 03. D 04. D 05. C 06. A 07. C 08. D 09. D 10. A 11. D 12. D 13. C 14. C 15. B 16. B
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9 Decimal, Binary and Hexadecimal Number 01. C 02. A 03. A 10 Mensurations 01. D 02. B 03. C 04. B 05. B 06. A 07. E 08. D 09. B 10. B 11. E 12. B 13. D 14. A 15. C 16. A 17. D 18. E 19. A 20. A 21. B 22. A 23. C 24. B 25. D 26. D 27. C 28. C 29. C 30. D 31. C 32. A 33. B 34. A 35. A 36. A 37. B 38. B 39. C 40. C 11.1 Plane Geometry – Circles 01. B 02. C 03. C 04. E 05. B 06. C 07. D 08. C 09. D 10. C 11. C 12. A 13. C 14. D 15. C 16. C 17. B 18. A 19. C 20. D 21. B 22. B 23. D 24. B 25. A 26. E 27. E 28. E 29. A 30. D 31. B 32. E 33. D 34. B 35. E 36. B 37. D 38. E 39. E 40. D 41. A 42. A 43. D 44. A 45. B 46. E 47. D 48. B 49. D 50. D 51. E 52. C 53. C 54. E 55. D 56. A 57. D 58. D 59. C 60. C 61. E 62. B 63. B 64. E 65. C 66. C 67. B 68. A 69. B 70. D 71. D 72. B 73. A 74. B 75. B 76. A 77. A 78. C 79. C 80. B 81. A 82. B 83. D 84. C 85. A 86. A 87. B 88. B 89. C 90. A 91. A 92. B 93. B 94. C 95. B 96. B 97. A 98. A 99. C 100. A 101. B 11.2 Plane Geometry - Similar and Congruent Triangles 01. A 02. D 03. E 04. D 05. A 11.3 Plane Geometry – Polygons 01. D 02. C 03. A 04. D 05. B 06. C 07. C 08. B 09. D 10. D 11. A 12. D 13. D 14. C 15. A 16. C 17. D 18. E 19. B 20. A 21. D 22. C 23. B 24. C 25. C 26. B 27. A 28. A 29. B 30. B 31. D 32. C 33. B 34. B 35. A 36. D 37. B 38. D 39. B 40. C 41. B 42. D 43. B 44. D 45. B 46. D 47. D 48. C 49. B 50. B 51. C 52. B 53. C 54. D 55. B 56. C 57. C 58. C 59. A 60. D 61. B 62. C 63. D 64. B 65. D 66. D 11.4 Plane Geometry - Centers in a triangle 01. D 02. D 03. C 04. D 11.5 Plane Geometry - Rotational and Reflectional Symmetries 01. D 02. C 03. B 04. A 05. D
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12.1 Trigonometry - Trigonometric Equations 01. D 02. E 03. A 04. B 05. B 06. D 07. D 08. A 09. B 10. D 11. B 12. D 13. C 14. B 15. C 16. E 17. A 18. E 19. D 20. C 21. A 22. C 23. C 24. E 25. B 26. C 27. A 28. E 29. C 30. D 31. A 32. B 33. B 34. A 35. D 36. B 37. C 38. A 39. A 40. A 41. D 42. B 43. B 44. B 45. A 46. C 47. A 48. A 49. C 50. A 51. A 52. B 53. D 54. D 55. A 56. D 57. D 58. D 59. B 60. B 12.2 Trigonometry - Trigonometric Graph 01. B 02. B 03. D 04. B 05. A 06. C 07. A 08. B 09. E 10. E 11. A 12. E 13. C 14. E 15. D 16. B 12.3 Trigonometry - 2D 01. B 02. E 03. D 04. B 05. B 06. B 07. B 08. B 09. B 10. B 11. D 12. E 13. B 14. C 15. A 16. C 17. E 18. A 19. C 20. C 21. C 22. D 23. E 24. E 25. A 26. D 27. B 28. B 29. E 30. B 31. E 32. E 33. D 34. C 35. B 36. C 37. C 38. B 39. B 40. A 41. A 42. B 43. B 44. D 45. C 46. A 47. C 48. E 49. C 50. E 51. A 52. B 53. A 54. E 55. B 56. C 57. D 58. E 59. B 60. D 61. B 62. B 63. A 64. E 65. A 66. C 67. D 68. A 69. D 70. A 71. D 72. E 73. A 74. C 75. E 76. B 77. E 78. A 79. B 80. C 81. B 82. B 83. B 84. B 85. B 86. B 87. C 88. A 89. D 90. D 91. B 92. D 93. B 94. D 95. A 96. D 97. C 98. C 99. B 100. C 101. A 102. C 103. D 104. C 105. B 106. B 107. C 108. A 12.4 Trigonometry - 3D 01. C 02. E 03. A 04. D 05. E 06. B 07. A 08. D 09. C 10. C 11. A 12. B 13. A 14. C 15. C 16. C 17. E 18. D 19. B 20. A 21. D 22. D 23. B 24. B 25. B 26. D 27. D 13.1 Coordination Geometry - Straight lines 01. E 02. A 03. D 04. E 05. D 06. A 07. E 08. B 09. C 10. * 11. A 12. A 13. B 14. E 15. E 16. A 17. E 18. C 19. D 20. A 21. E 22. A 23. B 24. E 25. D 26. E 27. A 28. A 29. D 30. E 31. D 32. D 33. D 34. A 35. C 36. A 37. A 38. A 39. D 40. A 41. B 42. A 43. C 44. D 45. C 46. D 47. A 48. D 49. C 50. D 51. A 52. B 53. D 54. D
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13.2 Coordination Geometry – Circles 01. B 02. E 03. E 04. B 05. C 06. C 07. E 08. D 09. E 10. A 11. B 12. D 13. D 14. B 15. E 16. A 17. A 18. C 19. B 20. D 21. E 22. C 23. B 24. D 25. B 26. D 27. C 28. B 29. A 30. A 31. C 32. C 33. C 34. D 13.3 Coordination Geometry - Polar Coordinates 01. C 02. C 03. C 14 Sequence 01. E 02. B 03. A 04. E 05. E 06. D 07. D 08. D 09. A 10. C 11. A 12. D 13. C 14. D 15. C 16. A 17. E 18. B 19. D 20. D 21. E 22. C 23. C 24. A 25. C 26. C 27. C 28. D 29. B 30. A 31. B 32. A 33. C 34. D 35. A 36. A 37. C 38. C 39. C 40. B 41. B 15 Probability 01. E 02. E 03. E 04. D 05. B 06. D 07. B 08. B 09. C 10. E 11. E 12. A 13. E 14. E 15. C 16. E 17. C 18. B 19. A 20. C 21. C 22. C 23. D 24. B 25. B 26. D 27. A 28. B 29. C 30. B 31. B 32. A 33. B 34. D 35. B 16 Statistics 01. A 02. A 03. A 04. C 05. D 06. C 07. C 08. A 09. B 10. C 11. D 12. E 13. C 14. D 15. D 16. C 17. A 18. C 19. C 20. D 21. B 22. C 23. A 24. B 25. C 26. B 27. C 28. A 29. A 30. A 31. B 32. C 33. B 34. D 35. C 36. B 37. C 38. D 39. A 40. A 41. A
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HKCEE Mathematics Paper II Answers 1990-2008
1990 1991 1992 1993
01. AAAEA EEBCC 01. AEAEE CBEDA 01. AAEDA AAAED 01. CABEA BBBCD
11. DBBDC DEACC 11. CACDA BDDCB 11. ECCBB ABBDD 11. BABBE AEACB
21. CBDAA EEADB 21. ACBDC EEDBA 21. BDDCE DBAEB 21. ACCCC BBC*C
31. CCEAC DDBCE 31. CEDDD CCEEA 31. EDECE EEEBB 31. DCAAD EDBDA
41. BAEBB EECDD 41. EBDCD EABBB 41. ADCCC BADCB 41. DDDDE BECAE
51. CBDA 51. BCBA 51. ADCC 51. EDAE
1994 1995 1996 1997
01. BCCBD EAECB 01. DDDAE CCDEB 01. DECCB BAECD 01. BADDB EBDAC
11. EADDA AAEAC 11. CCEAB ACCED 11. CDCEA DDEEB 11. BACEC AEBDE
21. DBEDB AACEB 21. CBDCB CBEDE 21. BCBBA ABEEA 21. CDDCE EEABB
31. BCCBA DAEAC 31. DBDED EABEE 31. ACDBC CCAEA 31. CDBAC AECCA
41. CDCDB EDCBE 41. AACBE ECBCA 41. ADECD BBDAD 41. CAEED DADAB
51. BEDC 51. ABBA 51. ACEB 51. DCBC
1998 1999 2000 2001
01. DBACE DBBDE 01. ABDBE CADAD 01. EACEB BCBDA 01. ACAEB DABDE
11. DBBCE EBBDC 11. CCCBE ABDBB 11. EBAAC CEAAE 11. BBEAC CACEC
21. CAEAE BBCCA 21. AEAAD DCBBA 21. ECDDB CDDAD 21. CACDB ADECB
31. CEAAA ECAAA 31. EDCEE CBEEE 31. BBAAD CEDEB 31. EBEDB AEACB
41. CDDCE EDCDD 41. CBDED ABECC 41. CDCAE CBBDA 41. DDBED DADAB
51. CBEB 51. AACD 51. BECD 51. DEEC
2002 2003 2004 2005
01. DBADB AADDB 01. CCAAC ADDCD 01. CACAC CDDAB 01. BDADB DDCAD
11. CCDAA ABCDA 11. BDDAD DBBAC 11. BBCCB ACDBC 11. BDCCD BDCBC
21. CABCB DBBBD 21. CBBBC BACAA 21. BDBDC ABDDA 21. CCBCA CBCCD
31. ACABC CDDBC 31. ABBCD CBCDA 31. BABBA CDDAB 31. ACDAD ADADA
41. CBCDC AADBB 41. CCDDB BAADC 41. CCDAA AADDA 41. CABAA BBBAB
51. ABDC 51. BADB 51. BDCC 51. BCBA
2006 2007 2008
01. BCAAA BADAB 01. ADAAD BBDAB 01. BAABC CCDAA
11. CCDDC AABCA 11. DDABC AABAD 11. CACDC BCCCC
21. DDCBD DCCDA 21. DDBDC BDBCC 21. DCAAA DBDBC
31. DBCBC BBACB 31. DAACB CCDBA 31. DDBDA ADCAA
41. CDAAC BBDDA 41. ACBCC DBAAC 41. BBBBB BBDDA
51. CBBD 51. CCBD 51. BDDA