1 Unit 1: Arithmetic 1.1 Proportion Demonstrate understanding of primary ideas of proportion Solve problems involving direct and inverse proportions (with 2 or more variables) A typical 3 variable problem: Find the number of people required to complete a certain number of jobs in a certain number of days e.g. (J04 Q17)
1. J95 Q8 What is the smaller angle between the minute and hour hands of a 12-hour clock at 3.40 pm? (A)
150
(B)
160
(C)
130
(D)
120
(E)
180
1.2 Ratio Demonstrate understanding of primary ideas of ratio 1. J97 Q3 In the diagram, calculate the ratio of the area of the shaded region to the total area of the two identical smaller circles.
(A)
1:1
(B)
1:2
(C)
1:3
(D)
2:3
(E)
3:4
2. J97 Q25 Three boys, Tom, John and Ken, agreed to share some marbles in the ratio of 9 : 8 : 7 respectively. John then suggested that they should share the marbles in the ratio 8 : 7 : 6 instead. Who would then get more marbles than before and who would get less than before if the ratio was changed?
2 3. J99 Q6 The diagram below shows three semi-circles whose centers all lie on the same straight line ABC. Suppose AB 2BC .
A
B
C
The ratio of the shaded area to the area of the largest semi-circle is (A)
1:2
(B)
4:9
(C)
1:3
(D)
2:3
(E)
2:5
1.3 Rates Demonstrate understanding of primary ideas of rate distance Use the formula – speed time total distance Use the formula – average speed total time Convert units (e.g. km/h to m/s and vice versa) 1. J95 Q25 Two pipes can be used to fill a swimming pool. The first can fill the pool in three hours, and the second can fill the pool in four hours. There is also a drain that can empty the pool in six hours. Both pipes were being used to fill the pool. After an hour, a careless maintenance man accidentally opened the drain. How long more will it take for the pool to fill?
2. J96 Q4 Town A and Town B are linked by a straight road. A factory is sited along the road such that it is twice as far away from Town A as its distance from Town B. A truck left Town B at 9.00 am and reached the factory 1 hour later. A car which travels three times faster than the truck need to reach the factory at the same time as the truck. What time must the car leave Town A? (A) (E)
8.20 am 10.00 am
(B)
8.40 am
(C)
9.20 am
(D)
9.40 am
3. J99 Q14 If John walks from home to school at the speed of 4 km per hour, and walks back at the speed of 3.5 km per hour, find the average speed in km per hour for the whole trip. (A)
3.75
(B)
3.6
(C)
11 3 15
(D)
3 54
(E)
3 23
3 1.4 Percentage Calculate percentage (including percentage increase and decrease) 1. J96 Q1 If the side of a square is increased by 30%, the area of the square will increase by (A)
30%
(B)
60%
(C)
69%
(D)
900% (E)
None of the above
2. J97 Q7 (percentage) Originally 23 of the students in a class failed in an examination. After taking a re-examination, 40% of the failed students passed. What is the total pass percentage of the class?
26 23 % (B)
(A)
33 13 % (C)
40% (D)
60%
(E)
73 13 %
3. J97 Q8 (percentage) A company’s sales increase by 20% in 1993 followed by another 25% in 1994. The sales decreased by 25% in 1995, however. This was followed by another decrease of 20% in 1996. By what percentage did the company’s sales increase or decrease over this four-year period? (A) (D)
Increased by 5% Decreased by 10%
(B) (E)
Decreased by 5% (C) No increase or decrease
Increased by 10%
1.5 Statistics Use the formulae – arithmetic mean =
x1 x2 xn n
1. J96 Q21 (arithmetic, average, inequality, reasoning) Class A, with 15 students in the class, scored an average of 94 marks in a mathematics test. The maximum possible score of the test is 100 marks. What is the lowest possible score that any of the 15 students could have scored?
1.6 Relative velocity Use primary ideas of relative velocity to solve problems
VP VP VP / Q
: True (actual) velocity of a moving object P relative to the Earth. : True speed (ground speed) of P : Relative (apparent) velocity of a moving object P relative to a moving object Q (observer) VQ / P : Relative (apparent) velocity of a moving object Q relative to a moving object P (observer) Relative velocity equation
4
V P = VP / Q + VQ or VP / Q = V P + (– VQ )
1. J95 Q16 Two trains are each traveling towards each other at 180 km/h. A passenger in one train notices that it takes 5 seconds for the other train to pass him. How long is the second train? (A)
100 m (B)
200 m (C)
250 m (D)
400 m (E)
500 m
2. J95 Q17 In a river with a steady current, it takes Bionic Woman 6 minutes to swim a certain distance upstream, but it takes her only 3 minutes to swim back. How many minutes would it take a doll of the Bionic Woman to float this same distance downstream? (A) (E)
8 minutes 12 minutes
(B)
9 minutes
(C)
10 minutes
(D)
11 minutes
3. J99 Q30 Two men are walking at different steady paces upstream along the bank of a river. A ship moving downstream at constant speed takes 15 seconds to pass the first man. Five minutes later it reaches the second man and takes 10 seconds to pass him. Starting then, how long will it take for the two men to meet? (Give you answer in terms of seconds).
1.7 Binary Operation Perform binary operation A binary operator in mathematics is defined as an operator defined on a set that takes two elements of the set and returns a single element. An example would be integer multiplication "" where a, b are both integers and a b returns an integer. 1. J97 Q9 The operation is defined by: a b = a2 – b2. Evaluate (1997 1996) (1996 1995). (A)
3991
(B)
3993
(C)
7984
(D)
15968 (E)
None of the above
2. J99 Q16 Let be the binary operator on positive integers defined by a b = ab. Consider the following identities: (i) (ii) (iii)
ab=ab (a b) c = a (b c) a (b + c) = (a b)+(a c)
5 (iv)
(a + b) c = (a c)+(b c)
(A) All are true (D) (iii) is true
(B) (ii) and (iii) are true (E) None is true.
(C) (iii) and (iv) are true
Unit 2: Mathematical Reasoning 2.1 Logic and paradoxes Use strategies of making suppositions, eliminating possibilities and making logical deductions to evaluate the truth of statements 1. J00 Q20 Four people, A, B, C and D are accused in a trial. It is known that if A is guilty, then B is guilty; if B is guilty, then C is guilty or A is not guilty; if D is guilty, then A is guilty and C is not guilty; if D is guilty, then A is guilty. How many of the accused are guilty? (A) (E)
1 (B) 2 (C) 3 (D) Insufficient information to determine
4
6 Unit 3: Algebra 3.1 Algebraic representation and formulae Use letters to express generalized numbers and express arithmetic processes algebraically Use the strategy: Students should note some questions need not be solved algebraically. They can consider specific cases by substituting appropriate numbers, reducing the problem to an arithmetic one 3.2 Algebraic Manipulation Manipulate/Simplify algebraic expressions (including algebraic fractions). (Students should be able to use tricks like adding “new” terms while still maintaining integrity of question to solve problems e.g. J98 Q26) (Partial fractions) Express an (algebraic) fraction as a difference of two fractions 1 1 1 (Classic example: ) n(n 1) n n 1 Manipulate algebraic fractions/expressions in an equation (usually to substitute the result into another expression/equation to solve a given problem e.g. J04 Q12) 3.3 Algebraic Manipulation (Expansion and Factorisation) Factorise expressions of the form ax ay Know and use the identity – x 2 y 2 ( x y)( x y) and other equivalent forms e.g. x y , ( x y )( x y ) x y … x y x y Know and use the identity - ( x y) 2 x 2 2 xy y 2 and other equivalent forms e.g. 1 1 ( x y) 2 ( x y) 2 4 xy , ( x) 2 2 2 x 2 … (e.g. J04 Q23, J01 Q21) (Note: x x to solve some questions, repeated use of this identity is necessary (e.g. J04 23, J01 Q21)) Factorise trinomials Factorisation by grouping (students should be comfortable with atypical scenarios involving more than 4 terms e.g. J03 Q18 involves factorization of n4 2n3 2n2 2n 1 ) Know and use the identity – x 3 y 3 ( x y)( x 2 xy y 2 ) Know and use the technique of completing the square (e.g. J00 Q9: to determine the minimum or maximum value of expression)
Expansion and Factorisation Some useful identities
7
(a b) 2 a 2 2ab b 2 (a b) 2 a 2 2ab b 2 (a b)(a b) a 2 b 2 (a b) 3 a 3 3a 2 b 3ab 2 b 3 (a b) 3 a 3 3a 2 b 3ab 2 b 3 a 3 b 3 (a b)(a 2 ab b 2 ) a 3 b 3 (a b)(a 2 ab b 2 ) The absolute value of function (e.g. square root of a square) The absolute value (or modulus) of x means the numerical value of x, not considering its sign, and is denoted by a .
a 2 a for all real number a?
Is
Consider: When a = 2, When a = –2,
a 2 22 4 2 a a 2 (2) 2 4 2 a
a if a 0 (or a2 a if a 0
a2 a )
1. J95 Q6 The sum of two positive numbers equals the sum of the reciprocals of the same two numbers. What is the product of these two numbers? (A)
1
(B)
2
(C)
4
(D)
1 2
(E)
1 4
2. J95 Q24 If x and y satisfy x 2 y 2 7 , find the maximum value of x2 2 y 2 2 x 4 .
3. J96 Q15 If the value of 76x – 19y is 114, the value of 36x – 9y is (A)
54
(B)
60
(C)
88
(D)
92
(E)
108
8 4.
J96 Q16
Let a < 0. Find (A)
1
a 2 (1 a) 2 in terms of a.
(B)
–1
(C)
2a – 1 (D)
1 – 2a
(E)
None of the above
5.
J96 Q18 1 1 If x 5 , find the value of x is________. x x
6. J97 Q2 Given that 1995x 3 1996 x 2 1997 x 1998 1005x 3 1006 x 2 1007 x 1962 , the value of x3 x2 x 1 _______________. (A)
–4
(B)
3
(C)
–3
(D)
1
(E)
–1
7. J97 Q14 Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles (A) (C)
is 8 or 38 is 20 or 26
(B) (D)
cannot be determined from the given data is 14 or 32 (E) is none of these
8. J97 Q15 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
9. J98 Q26 Let m and n be two integers such that 8m 9n mn 6 . Find the largest possible value of m.
10. J98 Q27 Find the largest value of x which satisfies the equation (4 x 5) 2 (4 x 5)( x 2 x 2) ( x 2 x 2) 2 ( x 2 3x 7) 2 .
9 11. J99 Q2 If we increase the length and the width of a rectangle by 10 cm each, the area of the rectangle will increase by 300 cm2. The perimeter of the original rectangle in cm is (A)
28
(B)
30
(C)
36
(D)
40
(E)
50
(E)
63
12. J99 Q9 Given that 2 x 4 x 11 1, the value of x2 x 1 is (A)
71
(B)
81
(C)
91
(D)
47
13.
J99 Q26 1 1 1 Let . What is the value of x y 4 2 y 3xy 2 x ? y x 2 xy
14. J00 Q9 For any real numbers a, b and c, find the smallest possible value the following expression can take: 3a 2 27b 2 5c 2 18ab 30c 237 . (A)
190
(B)
192
(C)
200
(D)
237
(E)
239
15. J00 Q13 In the following diagram, ABCD is a rectangle and ADEF, CDHG, BCLM and ABNO are square. Suppose the perimeter of ABCD is 16 cm and the total area of the four squares is 68 cm2. Find the area of ABCD in cm2. F E
H
O A
D
B
C G
N
(A)
15
(B)
20
(C)
M 25
(D)
L 30
(E)
40
10 3.4 Solutions of Equations Construct equations from given situations Solve linear equations in one unknown Manipulate and/or solve simultaneous equations (Students should be comfortable with atypical questions e.g. 2 equations but many unknown (J04 Q1). Such questions can usually be simplified further through cancellation of “excess variables” (J04 Q1), clever manipulation (J04 Q27) Solve quadratic equations by factorization Solve quadratic equations by completing the square Solve quadratic equations by the use of formula Solve complex equations (e.g. surds, polynomials of higher degrees) through nonroutine methods (e.g. by using a suitable substitution to simplify equation (e.g. from polynomial of high degree to trigonometric function S01 Q25, from surds to quadratic J98 Q15, from exponential to quadratic J96 Q11) 1. J95 Q13 When some people sat down to lunch, they found there was one person too many for each to sit at a separate table, so they sat two to a table and one table was left free. How many tables were there? (A)
2
(B)
3
(C)
4
(D)
5
(E)
6
2. J95 Q21 John is now twice as old as Peter. If their combined age is 54 years, what is their combined age when Peter is as old as John is now?
3.
J95 Q30
On a plane, two men had a total of 135 kilograms of luggage. The first paid $12 for his excess luggage and the second paid $24 for his excess luggage. Had all the luggage belonged to one person, the excess luggage charge would have been $72. At most how many kilograms of luggage is each person permitted to bring on the plane free of additional charge?
4. J96 Q11 If x and y satisfy the following simultaneous equations 16 x 16 y 64512 and 4 x 4 y 224 , find the sum x y . (A)
5.5
(B)
6.5
(C)
8.5
(D)
12.5
(E)
16.5
11 5. J98 Q1 A bag contains 28 marbles which are coloured either red, white, blue or green. There are 4 more red marbles than white ones, 3 more white marbles than blue ones and 2 more blue marbles than green ones. Find the number of white marbles.
6. J98 Q15 It is given that a, b are two positive numbers satisfying a( a b) 3 b( a 5 b) . Find the value of
3a 2 ab 2b . a ab b
7. J99 Q4 Let x denote the absolute value of a number x defined by x
if x 0
x –x x < 0. The number of solutions of the equation x 1 2 2 is (A)
0
(B)
1
(C)
2
(D)
3
(E)
4
8. J99 Q24 When I am as old as my father is now, my son will be seven years older than I am now. At present, the sum of the ages of my father, my son and myself is 100. How old am I?
9. J00 Q16 A student has taken n examinations and 1 more examination is upcoming. If he scores 100 in the upcoming examination, his overall average (of the n + 1 examinations) will be 90; and if he scores 60 in the upcoming examination, his overall average will be 85. Find the number n. (A)
5
(B)
6
(C)
7
(D)
8
(E)
9
12 3.5 Functions (Absolute Function) Know and use the properties of absolute function The absolute value of function (e.g. square root of a square) The absolute value (or modulus) of x means the numerical value of x, not considering its sign, and is denoted by x .
x if x 0 x x if x 0
3.6 Roots Use the formulae for the product and sum of roots Use the condition for a quadratic equation to have two real roots, two equal roots and no real roots Determine the existence of integral/rational roots for quadratic equations through computation of the discriminant Sum of roots and Product of roots If and are two roots of the quadratic equation ax 2 bx c 0 , then b + = a c = a 1. J99 Q29 Let a and b the two real roots of the quadratic equation
x 2 (k 1) x k 2 3k 4 0 where k is some real number. What is the largest possible value of a 2 b2 ?
2. J00 Q24 Suppose the equation
x 2 (4 a) x a 1 0 has two solutions which differ by 5. Find all possible values of a.
13
3.7 Indices
1.
Apply the laws of indices Perform operations with indices Determine the nth root of a number Solve indicial equations, solve equations of the form a x b
J95 Q1
The simplest expression for (A)
1
(B)
4
2 40 is 4 20 (C)
1 2
20
(D)
2 20
(E)
218
2. J95 Q7 What is the value of x which satisfies
21995 21995 21995 21995 21995 21995 21995 21995 = 2 x ? (A)
1996
(B)
1997
(C)
1998
(D)
1999
(E)
2000
3. J96 Q9 Suppose 19961996 19961996 19961996 = 1996 x , what is the value of x? 1996 terms (A)
1996
(B)
4.
J96 Q10
Solve the equation (A)
5
(B)
1997
(C)
1998
(D)
1999
(E)
2000
( x 1) ( x 1) ( x 1) 2 2 . 7
6
(C)
8
(D)
12
(E)
15
5. J97 Q1 Given that 19981998 19981997 = 1998x 1997 , find the value of x.
(A)
0
(B)
1
(C)
1996
(D)
1997
(E)
1998
14 6.
J97 Q11
If a 41357, b 5357 and c 2 2357 , find the sum of all the digits in
ab . c
(A)
None of the above
1
(B)
10
(C)
15
(C)
54
(D)
357
(D)
55
(E)
7. J00 Q2 5 The fifth root of 5 5 is (A)
55
(B)
5 (5
5
1)
5
4
(E)
5
55
8. J00 Q19 How many integer solutions does the following equation have?
x (A)
1
(B)
3
(C)
2
3
x 1
x 2000
(D)
1.
4
(E)
5
3.8 Standard Form Use the standard form A 10 n where n is a positive of negative integer, and 1 A 10 Deduce the number of digits of a number from its standard form (or its variations e.g. standard form minus one (J04 Q29)) (Questions like this typically require students to extract 2a and 5a from a given number, this allows the number to be expressed in the standard form (J02 Q7, J98 Q2) Write algebraic expressions (e.g. abcd) as as linear combinations of 10 0 ,101 ,10 2 , 1. J95 Q2 If 8.047 3 521.077119823 , what is 0.80473 equal to? (A) (D)
0.521077119823 0.00521077119823
(B) (E)
52.1077119823 0.052107119823
(C)
521077.119823
2. J96 Q7 The square of the number 12345678 is an m-digit number. What is the value of m? (A)
13
(B)
14
(C)
15
(D)
16
(E)
17
15 3. J98 Q2 What is the number of digits in the number 46 58 ?
3.9 Applications of algebra in arithmetic computation Simplify arithmetic computation by st pairing terms in numerical expressions (e.g. pairing 1 and last term, pairing adjacent terms) using the method of difference (such a method typically involves the knowledge and use of partial fractions) using a suitable algebraic form to model the question factorizing (repeatedly e.g. J99 Q15, J98 Q3) complex numerical expressions using a suitable substitution using the technique – cancellation of numerators and denominators (by first writing numerical expressions into suitable fractional forms) using estimation and approximation 0 1 2 expressing numbers as linear combinations of 10 ,10 ,10 , expressing each numbers as a difference of two numbers Classic case: (J02 20 – 9 + 99 + 999 + … = 10 – 1 + 100 – 1 + 1000 – 1 +…) extrapolating the result of simple arithmetical calculation to cases involving large numbers (e.g. J04 Q8 – What is the sum of all the digits in the number 10 2004 2004 ?) 1.
J95 Q22
Evaluate
246912 . 123457 123456 123458 2
2.
J96 Q14 (1987654)(1987654) (1897645)(1897645) Evaluate . 180018
3. J96 Q25 1 ). Evaluate (1 12 )(1 13 )(1 14 ) (1 1n ) (1 101
4. J96 Q28 What is the unit digit for the sum 13 23 33 43 53 63 193 ?
5. J97 Q4 Which of the following is the closest value to (487,000)(12,027,300) (9,621,001)(487,000) ? (19,367)(0.05) (A)
10,000,000
(B)
100,000,000
(C)
1,000,000,000
16 (D)
10,000,000,000
(E)
100,000,000,000
6. J97 Q12 The difference between the sum of the last 1997 even natural numbers less than 4000 and the sum of the last 1997 odd numbers less than 4000 is (A)
1996
(B)
1997
(C)
1998
(D)
3994
(E)
3996
7. J98 Q3 Find the value of
10002 (2522 2482 ) (2524 2484 ) . 2528 2488
8. J98 Q8 Find the value of 12 22 32 42 19972 19982 .
9. J98 Q10 Find the value of
1998 19971997 1997 19981997 .
10. J99 Q21 What is the product of
1 1 1 1001 1 (1 ) 1 2000 ? 2 2 2 1002 1001 2000
11. J00 Q1 Let x be the sum of the following 2000 numbers: 4, 44, , 44444 . 2000 digits
Then the last four digits (thousands, hundreds, tens, units) of x are (A)
0220
12.
J00 Q4
(B)
0716
(C)
Find the value of the product (1
(A)
2001 (B) 4000
1001 (C) 2000
1884
(D)
2880
(E)
5160
1 1 1 1 )(1 2 )(1 )(1 ). 2 2 2 3 1999 20002 101 201
(D)
21 40
(E)
11 20
17
13. J00 Q7 Let S n 1 2 3 4 (1) n1 n . Find the value of S2000 S2001. (A)
–1
(B)
0
(C)
1
(D)
2
(E)
3
14. J00 Q14 Evaluate 2 3 99 1 2 3 98 1 2 3 99 2 3 98 ( )( ) ( )( ) . 3 4 100 2 3 4 99 2 3 4 100 3 4 99
15. J00 Q28 Evaluate 11111 22222 . 2000 digits 1000 digits (Hint: Your answer should be an integer.)
3.10 Sequences and Series Continue/complete given number/alphabetical sequences Apply the first principles of arithmetic progressions (pairing of terms i.e. first and last etc) Recognise arithmetic progressions Use the formula for the nth term to solve problems involving arithmetic progressions Use the formula for the sum of the first n terms to solve problems involving arithmetic progressions Use the result Tn Sn Sn 1 Recognise geometric progressions Use the formulae for the sum of the first n terms to solve problems involving geometric progressions Determine the largest (smallest) term of a sequence by comparing the nth and (n+1)th term Use the method of differences to obtain the sum of a finite series e.g. by expressing the term in partial fractions Arithmetic Progression (AP) The nth term of an AP (with common difference d) is given by an a1 (n 1) d The sum of the first n terms of an AP (with common difference d) is given by n n n a1 a1 a 2 a n (a1 a n ) (2a1 (n 1)d ) 2 2 i 1
18
If x, y and z are three consecutive terms of an AP, then xz (arithmetic mean) z y y z or 2 y x z or y 2 Geometric Progression (GP) The nth term of a GP (with common ratio r) is given by an a1r n1 The sum of the first n terms of a GP (with common difference r) is given by n a1 (r n 1) a1 (1 r n ) a a a a 1 1 2 n (r 1) 1 r i 1 If x, y and z are three consecutive terms of a GP, then z y or y 2 xz or y xz (geometric mean) y x Some important formulae
n(n 1) 2 n(1 2n 1) The sum of first n odd numbers: 1 3 5 7 (2n 1) n2 2 n n n 1 n2 2 The factorization of x 1 : x 1 ( x 1)( x x x x 1) (e.g. x 2 1 ( x 1)( x 1) )
The sum of first n natural numbers: 1 2 3 4 n
1. J96 Q17 The value of 11 22 33 1100 is
2.
.
J97 Q17 1 1 1 1 1
1 2
3 4
1 3
6 etc.
1 4
1
Pascal’s triangle is an array of positive integers (see above), in which the first row is 1, the second row is two 1’s, each row begins and ends with 1, and the kth integer in any row when it is not 1, is the sum of the kth and (k–1)th numbers in the immediately proceeding row. Find the ratio of the number of integers in the first n rows which are not 1’s and the number of 1’s.
(E)
n2 n (B) 2n 1 None of the above.
3.
J97 Q18
(A)
n2 n 4n 2
(C)
n 2 2n 2n 1
(D)
n 2 3n 2 4n 2
19 The integers greater than one are arranged in five columns as follows: A
B 2 8 10 16
9 17
C 3 7 11 15
D 4 6 12 14
E 5 13
In which columns will the number 1000 fall? (A)
A
(B)
B
(C)
C
(D)
D
(E)
E
4. J97 Q21 In a game, a basket and 16 potatoes are placed in line at equal interval of 3 m. (Note that the basket is placed at one end of the line). How long will a competitor take to bring the potatoes one by one into the basket, if he starts from the basket and run at an average speed of 6 m a second?
5. J98 Q24 The sequence {a1 , a2 , a3 ,} is defined by a1 2, and an1 an 2n for n 1,2,3, . Find the value a100 . 6. J99 Q1 The next letter in the following sequence B, C, D, G, J, O, ____ is (A)
P
(B)
Q
(C)
R
(D)
S
(E)
T
7. J99 Q11 The number 1001997 is expressed as a sum of 999 consecutive odd positive integers. The largest possible such odd integer is (A)
1997
(B)
1999
(C)
2001
(D)
2003
(E)
2005
8. J99 Q25 Let n! denote the product n (n 1) (n 2) 2 1 . For what value of the positive integer n
100 n is / n! largest? 3
20 9. J00 Q5 Consider the following array of numbers: A
B 2 20 26 44
23 47
C 5 17 29 41
D 8 14 32 38
E 11 35
In which column does the number 2000 appear? (A)
A
(B)
B
(C)
C
(D)
D
(E)
E.
10. J00 Q6 Find the sum of all positive integers which are less than or equal to 200 and not divisible by 3 or 5. (A)
9367
(B)
11.
J00 Q10
9637
(C)
10732 (D)
For which positive integer k does the expression (A)
1998
(B)
1999
(C)
2000
(D)
12307 (E)
17302.
k 2 have the largest value? 1.001k
2001
(E)
2002
21
3.11 Inequalities Know and use the properties of inequalities (students also need to be aware of self-evident properties of real numbers; refer to classic: J99 Q10) e.g. If a 1, a n 1 for n (J04 Q9) a e.g. 0 1 for a, b 0 (J02 Q15) ab Manipulate inequalities Substitute equations into inequalities Construct inequalities from given situation Solve linear inequalities Solve quadratic inequalities (by factorization, etc) Solve quadratic inequalities through non-routine techniques (using the property of integers) Solve cubic inequalities Students should be able to solve these inequalities using non-routine methods e.g. by approximation which in terms requires familiarity with the values of manageable” numbers raised to the power of n, where n is a small integer J98 Q5 Solve complex inequalities (involving combination of different functions) e.g. (exponential and linear), involving surds J98 Q22 by non-routine methods (e.g. trial and error) (e.g. manipulating inequalities J98 Q22), (involving greatest integer functions by trial and error J96 Q22) Solve simultaneous inequalities/equations/inequations (e.g. S03 Q25) Use “Squeeze” theorem i.e. find suitable lower and upper bounds for algebraic/numerical expressions Compare the magnitude of numbers using a variety of techniques e.g. J03 by evaluating their difference, J02 rewriting numbers to that they have the same exponent, J01 rewriting numbers as fractions with the same numerator but different x y denominator using the identity x y , J96 by evaluating their x y differences/ considering specific cases Determine the intersection of solution sets of at least 2 inequalities
1. J95 Q3 If x is a positive number, which of the following expressions must be less than 1? (A)
1 x
(B)
1 x x
(C)
x2
(D)
1 x x
(E)
x x 1
22 2. J95 Q15 If 0 x 1 , y x x and z x y , what are the three numbers arranged in order of increasing magnitude? (A)
x, y, z (B)
3.
J96 Q26
x, z, y (C)
y, z, x (D)
If a, b, c and d are positive integers such that 1 ascending order.
z, y, x (E)
z, x, y
a c , arrange the following quantities in b d
b d bd b d , , , ,1. a c ac a c
4. J96 Q27 Find all possible real values y such that 7 x 4 y 168 and 5x 3 y 121.
5. J97 Q28 If the solution of the inequality x 2 ax 6 0 is c x 6 , find c.
6. J98 Q5 Find the positive integer n such that 60000 n 3 180000 and the unit digit of n 3 is 3.
7. J98 Q22 What is the smallest positive integer n such that 4n 1 4n 3 0.02 ?
8. J99 Q3 Suppose a 6 2 and b 2 2 6 . Then (A)
a b (B)
a b (C)
a b (D)
b 2a
(E)
a 2b
bca
(D)
bac
9. J99 Q7 Let a 2 48 , b 336 , and c 524 . Then (A) (E)
abc a c b.
(B)
cba
(C)
23 10. J99 Q8 (properties of fraction) The integer part of the fraction 1 1984
1 1 1999
1 1985
is (A)
121
(B)
122
(C)
123
(D)
124
(E)
125
11. J99 Q10 Let a and b be two numbers such that a > b. Consider the following inequalities: (i) a 2 b 2 (ii) 1a b1 (iii) ba 1 (iv) ab 0 (A) (D)
All are true (B) (i), (ii), (iii) are true
only (i) is true (C (E) None is true
only (ii) is true
3.12 Surds Perform operations with surds, including rationalizing the denominator 3.13 Logarithm Use the laws of logarithm 3.14 Identities Substitute appropriate values for x into identities by observation (usually to find solutions of (linear combinations of) coefficients) Identities P( x) Q( x)
P( x) Q( x) for all values of x
To find unknowns in an identity, (a) substitute values of x, or (b) equate coefficients of like powers of x.
1. J00 Q17 2 If (3x 3x 7)100 2a0 a1 x a2 x 2 a200 x 200 , find the value of
2a0 2a2 2a4 2a6 2a8 2a198 2a200. (A)
0
(B)
1
(C)
200
(D)
2000
(E)
7100
24 3.15 Binomial theorem Use the Binomial Theorem for expansion of (a b) n for positive integral n The Binomial Theorem for positive integer, n
(a b) n = a n + n C1a n1b + n C2 a n2 b 2 + n C3 a n3b 3 + … + b n There are n + 1 terms. The powers of a are in descending order while the powers of b are in ascending order. The sum of the powers of a and b in each term of the expansion is always equal to n. Tr 1 = n Cr a nr b r If a = 1, (1 b) n = 1 + n C1b + n C 2 b 2 + n C3b 3 + … + b n
Tr 1 = n C r b r
Unit 4:
Geometry
4.1 Mensuration: Perimeter, Area and Volume Calculate area and perimeter of geometrical figures (including triangles, circles, sectors, squares, rectangles etc) Calculate area of “irregular”/geometrical figures indirectly Know and use the formulae for surface area and volume of spheres, cubes, cones
1. J95 Q20 An equilateral triangle ABC has an area of 3 and side of length 2. Point P is an arbitrary point in the interior of the triangle. What is the sum of the distances from P to AB, AC and BC?
2. J95 Q26 In the diagram, congruent radii PS and QR intersect tangent SR. If the two disjoint shaded regions have equal areas and if PS = 10 cm, what is the area of quadrilateral PQRS? R S
P
Q
25 4.2 Radian measure Solve problems including arc length and sector of a circle, including knowledge and use of radian measure
4.3 Angles Use the following geometrical properties alternate angles sum of angles at a point exterior angle = sum of interior opp angles angle sum of triangle
1. J96 Q8 In the diagram, ABCD is a rectangle with AD = 2AB. M and N are midpoints of AD and BC respectively. Triangle ABE is an equilateral triangle. Calculate MEN. N B C
E
A
D
M
2. J96 Q29 In the following figure, AB = AC = BD. Find y in terms of x. A
x
B
y
D
C
26
4.4 Properties of Geometrical Figures Know the properties of (equilateral and isosceles) triangles Know the properties of quadrilaterals (square, rhombus, parallelogram, rectangle, kite) 1. J95 Q9 A four-sided closed figure has opposite sides equal in length. Which of the following statements about this figure must be true? (A) (B) (C) (D) (E)
If all its sides are equal in length, then the diagonals are equal in length. If the adjacent sides are perpendicular, then all its sides are equal in length. If its diagonals are equal in length, then the adjacent sides are perpendicular. If its diagonals are perpendicular, then the adjacent sides are perpendicular. If its diagonals are equal in length, then all its sides are equal in length.
2. J96 Q20 ABCD is a trapezium with AB parallel to DC. The point E on CD is such that DAE = BAE and CBE = ABE. Given that AD = 13 cm and BC = 12 cm, calculate the length of CD.
4.5 Polygons Calculate interior and exterior angles of polygons 1. J99 Q5 The sum of the angles A + B + C + D + E + F + G in the diagram is A B
G
C F D
(A)
240
(B)
280
(C)
350
E
(D)
360
(E)
420
27
4.6 Three Dimensional Figures Draw the nets for a given solid (cube etc) Know the relationship between a cone and a sector Cones A cone is a solid defined by a closed plane curve (forming the base) and a point (not on the same plane) called the vertex. When the base of a cone is a circle, it is called a circular cone. A right circular cone can be generated by the rotation of the right-angled triangle VOC about VO, which represents the height of the cone. Every point on the circumference of the base is the same distance l from the vertex V. The length l is called the slant height.
V
V Cut along VC
V l
h
l
C1
l
O
C2
r C
C1
C2
Answer the following questions before you proceed to deduce the formula for the curved surface of a cone: 1.
If a cut is made along VC and the cone is opened up and laid flat, what does it form? __________________________________
2.
What length of the sector corresponds to the slant height l? __________________________
3.
What length in the cone corresponds to the arc C1C2 in the sector? ____________________
28 Now, fill in the blanks:
Area of sector Arc length Area of circle Circumference 360
Arc length Given that , write down the ratio of in terms of r and l. 360 Circumfere down an expression for nce 360
Area of sector
Arc length Area of circle Circumfere nce
Curved surface area of a cone = ____________________ Total surface area of a closed cone = _________ + _________
1. J99 Q27 The following diagram shows a solid cube of volume 1 cm3. Let M be the midpoint of the edge BC. What is the shortest distance in cm for an ant crawl from the vertex A to M? H G E
F
D
A
M
C
B
29 4.7 Circle Properties Solve problems using the geometrical properties:
a straight line drawn from the centre of a circle to bisect a chord (not a diameter), is perpendicular to the chord and vice versa rt. angle in a semi-circle angle at centre is twice angle at circumference angles in the same segment angles subtended by arcs of equal length tangent perpendicular to radius tangents from exterior point are equal
Symmetrical/Angle Properties of Circles 1. a straight line drawn from the centre of a circle bisect a chord is perpendicular to the chord 2. equal chords are equidistant from the centre of a circle
1. Tangent perpendicular to radius 2. If TA and TB are tangents to a circle with centre O, then - TA = TB - ATO = BTO - AOT = BOT
Angle at centre is twice angle at circumference
Angles in the same segment are equal
Angles at the circumference subtended by equal arcs are equal
Right angle in a semicircle
1. opposite angles of cyclic quadrilateral 2. exterior angle of cyclic quadrilateral
30
Alternate segment theorem The angle between a tangent and a chord is equal to the angle made by that chord in the alternate segment.
1. J97 Q13 In the diagram, AM = MB = MC = 5 and BC = 6. Find the area of triangle ABC. C
A
M
B
2. J98 Q14 In the figure below, A, B, C, D are four points on a circle, and the line segments BA and CD are extended to meet at the point E. Suppose E = 42, and the arcs AB, BC and CD all have equal lengths. Find the measure of BAC + ACD in degrees.
B A E C
D
31 3. J00 Q26 In the diagram below, A, B, C, D lie on the line segment OE, and AC and CE are diameters of the circles centred at B and D respectively. The line OF is tangent to the circle centred at D with the point of contact F. If OA = 10, AC = 26 and CE = 20, find the length of the chord GH. H
F
G O
A
B
C
E D
4.8 Loci Use the following loci and method of intersecting loci sets of points in two dimensions which are equidistant from two given intersecting straight lines 1. J97 Q16 Line l2 intersects l1 and line l3 is parallel to l1. The three lines are distinct and lie in a plane. Determine the number of points that are equidistant from all three lines.
4.9 Triangles
Use properties of congruency Know and use appropriate tests to verify if 2 triangles (figures) are congruent Use properties of similar figures (including non-triangles) Know and use appropriate tests to verify if 2 triangles (figures) are similar Use the relationship between volumes of similar solids Use the theorem – ratio of area of triangles with common height = ratio of bases
32
Congruent Triangles Two triangles are congruent if they are identical, i.e. they are of the same shape and size. ABC is congruent to XYZ (written as ABC XYZ) if AB = XY, BC = YZ, CA = ZX and A = X, B = Y, C = Z A
C
B Test for congruency SSS: SAS: AAS (or ASA): RHS:
X
Z
Y
3 sides on one triangle are equal to 3 sides on the other triangle 2 pairs of sides and the included angles are equal 2 pairs of angles and a pair of corresponding sides are equal Right-angled triangle with the hypotenuse equal and one other pair of sides equal
Similar Triangles Two triangles are similar if they have the same shape, i.e. the corresponding angles are equal and the corresponding sides are proportional. ABC is similar to XYZ (written as ABC XYZ) if X, B = Y, C = Z
AB BC CA and XY YZ ZX
A X
B
C
Y
Tests for similarity The corresponding angles are equal (AA) The corresponding sides are proportional Two corresponding sides are proportional with included angles equal
Z
A =
33 Properties of Similar Figures If X and Y are two similar solids/figures, then l x hx l y hy 2
h Ax l x x h Ay l y y V x l x V y l y
3
hx h y 3
2
3
3
m x l x hx (if they have the same density) m y l y h y Triangles sharing the same height Consider two triangles, with areas A1 and A2 sharing the same height, h. 1 b1 h A1 b 2 1 A2 1 b2 b2 h 2 h
A2
A1 b1
b2
1. J95 Q14 In the diagram, ABC and CDE are right angles. Given that CD = 6 cm, AD = 7 cm and AB = 5 cm, what is the area of quadrilateral ABED? A
D
B
E
C
2. J96 Q2 In the following triangle ABC, M and N are points on AB and AC respectively such that AM : MB = 1 : 3 and AN : NC = 3 : 5.
34 A 1
3
M
N 5
3
B C Find the ratio of the area of triangle MNC : area of triangle ABC.
3. J96 Q13 A quadrilateral has sides of length 4 cm, 6 cm, 8 cm and 9 cm respectively. Another similar quadrilateral has a side of length 12 cm. What is the largest possible perimeter of this similar quadrilateral?
4. J97 Q6 In the diagram, the radii of the sectors OPQ and ORS are 5 cm and 2 cm respectively. Find the ratio of the area of the shaded region to the area of the sector OPQ.
P
Q R
S O 5.
J98 Q17
In the figure below BAC = 90 and DEFG is a square. If the length of BC is
185 6
and the
area of ABC is 1369, find the area of the square DEFG. A D
B
E
G
F
C
6. J98 Q23 In the figure below, AP is the bisector of BAC, and BP is perpendicular to AP. Also, K is the midpoint of BC. Suppose that AB = 8 cm and AC = 26 cm. Find the length of PK in cm.
35
A
P B
K
C
7. J98 Q25 In the diagram below, ABC is a right-angled triangle with B = 90. Suppose that BP AQ 2 and AC is parallel to RP. If the area of triangle BSP is 4 square units, find the CP CQ area of triangle ABC in square units. A
Q
R S
B
P
C
8 J99 Q12 In the diagram below, ABCD is a square and AE BF CG DH m . EB FC GD HA n A
H
9. J99 Q17 In triangle ABC, D, E and F are points on the sides BC, AC an AB respectively such that BC = 4CD, AC = 5AE and AB = 6BF. Suppose the area of ABC is 120 cm2, what is the area of DEF in cm2?
36 4.10 Coordinate Geometry Calculate the gradient of a straight line from the coordinates of two points on it
37 Unit 5: Trigonometry 5.1 Triangles Use triangle inequality Use Pythagoras’ theorem Apply the sine, cosine and tangent ratios to the calculation of a side or of an angle of a right-angled triangle Recall and use the exact values of trigonometrical functions of special angles 1 Solve problems using the sine and cosine rules and the formula ab sin c for the area 2 of a triangle Know the range of values of for which cos is positive or negative
Triangle inequality In any triangles ABC, the sum of the lengths of two sides is greater than the length of the third side. This is known as the triangle inequality i.e. A AB < BC + AC BC < AB + AC AC < AB + BC
c
B
opposite a hypothenus c adjacent b cos hypothenus c opposite a tan adjacent b
C
a
Simple trigonometrical ratios of an acute angle sin
b
C
c
a
A
The signs of the trigonometrical ratios y
b
2nd Quadrant
1st Quadrant
sin positive
ALL positive
B
x 3rd Quadrant
4th Quadrant
tan positive
cos positive
The trigonometric ratio of special angles
38
sin cos tan
0 0 0 2 4 1 2 0
30
45
60
90
1 1 2 2 3 2 1
2 2 2 2
3 2 1 1 2 2
4 1 2 0 0 2
1
3
3
C
Pythagoras’ theorem c
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides i.e. AC 2 AB 2 BC 2 c2 b2 a2
A
a
b
B
A
Sine rule c a b c In any triangle ABC, 2R , sin A sin B sin C where R is the circumradius of the triangle. B
Cosine rule In any triangle ABC, a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2ac cos B c 2 a 2 b 2 2ab cos C
Area of a triangle 1 base height 2 1 1 1 Area = ab sin C = bc sin A = ac sin B 2 2 2 abc Area = s(s a)(s b)(s c) , s (Heron’s formula) 2
Area =
b
a
C
39 Triangle Inequality 1. J97 Q23 The lengths of the sides of a quadrilateral are given by 1996 cm, 1997 cm, 1998 cm and cm. If z is an integer, what is the largest possible value of z?
z
2. J98 Q11 Find the total number of triangles such that the lengths (in cm) of all three sides of each triangle are positive integers and the length of the longest side of each triangle is 15 cm.
3. J00 Q29 Determine the number of acute-angled triangles (i.e. all angles are less than 90) having consecutive integer sides (say n – 1, n, n + 1) and perimeter not more than 2000.
Miscellaneous 1. J95 Q10 In the diagram, M is the midpoint of the semi-circular arc drawn on one side of a 6 cm by 7 cm rectangle. What is the perimeter of isosceles triangle MBC? C B
7 cm
M
A
6 cm
D
2. J95 Q18 Four congruent circles, each of which is tangent externally to two of the other three circles, are circumcised by a square of area 144 cm2. If a small circle is then placed in the center so that it is tangent to each of the circles, what is the diameter of this small circle?
40
3. J95 Q19 ABCD is an isosceles trapezium with AB parallel to DC, AC = DC and AD = BC. If the height of the trapezium is equal to AB, find the ratio of AB: DC. A B
D
C
4. J95 Q27 In a right angled triangle, the lengths of the adjacent sides are 550 and 1320. What is the length of the hypotenuse (correct to the nearest whole number)?
5. J95 Q29 An isosceles right-angled triangle is removed from each corner of a square piece of paper so that a rectangle remains. What is the length of a diagonal of the rectangle if the sum of the areas of the cut-off pieces is 200 cm2?
6. J96 Q3 A rectangle whose length is twice that of its breadth has a diagonal equal to that of a given square. What is the ratio of the area of the rectangle to the area of the square?
41 7. J96 Q5 In the following figure, 6 right-angled triangles are assembled together. Given that a and QR = 8a, find the expression (b – a)(b + a), in terms of a. a
a
PQ =
a
a R
b
Q
P
8. J96 Q12 In the diagram, CD = 10 cm, CE = 6 cm and DE = 8 cm. Find the area of the rectangle ABCD. C D
A
E
B
9. J96 Q23 In the diagram, AB = BD = 5 cm, ABD = 90 and DC = 2AD. Calculate the length of BC. B
A
D
C
10. J97 Q5 A girls’ camp is located 300 m from a straight road. A boys’ camp is located on this road and its distance from the girls’ camp is 500 m. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. What is the distance of the canteen from each of the camps?
42 11. J97 Q20 Triangle ABC has sides AB = AC = 13 cm and BC = 10 cm. Another triangle, PQR, has the same area as ABC with PQ = PR = 13 cm. Given that the two triangles are not congruent, calculate the length of QR.
12. J98 Q4 In the figure below, the ratio of the area of the quarter circle to that of the inscribed rectangle is 50 : 21 . If the radius of the quarter circle is 5 cm, find the perimeter of the rectangle in cm.
43 Unit 6 Combinatorics 6.1 Counting
Use the strategy of systematic listing/counting Use the addition principle Use the multiplication principle Recognize and distinguish between a permutation case and a combination case Know and use the notation n! and the expressions for permutations and combinations of n items taken r at a time Answer problems on arrangement and selection (can include cases with repetition of objects, or with objects arranged in a circle or involving both permutations and combinations) 1. J95 Q11 Each time the two hands of a certain standard 12-hour clock form a 180 angle, a bell chimes once. From noon today till noon tomorrow, how many chimes will be heard? (A)
20
(B)
21
(C)
22
(D)
23
(E)
24
2. J97 Q29 How many numbers greater than ten thousand can be formed with the digits 0, 1, 2, 2, 3 without repetition? (Note that the digit 2 appears exactly twice in each number formed.)
3. J98 Q19 Seven identical dominoes of size 1 cm 2 cm and with identical faces on both sides are arranged to cover a rectangle of size 2 cm 7 cm. One possible arrangement is shown in the diagram below. Find the total number of ways in which the rectangle can be covered by the seven dominoes.
4. J99 Q13 Two different numbers are to be chosen from the set {11, 12, 13, …, 33} so that the sum of these two numbers is an even number. Find the number of ways to choose the two numbers.
44 5. J99 Q19 In a quiz containing 10 questions, 4 points are awarded for each correct answer, 1 point is deducted for each incorrect answer and no point is given for each blank answer. The number of possible scores is (A)
10
(B)
40
(C)
44
(D)
45
(E)
50
6. J99 Q 20 The number of positive integers from 1 to 500 that can be expressed in the form a b with a and b being integers greater than 1 is (A)
25
(B)
27
(C)
29
(D)
33
(E)
35
7. J99 Q23 How many ways are there to form a three-digit even integer using the digits 0, 1, 2, 3, 4, 5 without repetition?
8. J00 Q12 How many numbers greater than 2000 can be formed by using some or all of the digits 1, 2, 3, 4, 5 without repetition?
9. J00 Q25 How many of the integers between 20000 and 29999 have exactly one pair of identical digits? (Note: The two identical digits need not be next to each other. For example, 20130 is one of the numbers we are looking for as it contains exactly one identical pair of digits, namely, 0; whereas 20230 and 20030 are not.)
6.2 Graph Theory Use graphs to model and solve problems 1. J95 Q28 Ten players took part in a round-robin tournament (i.e. every player must play against every other player exactly once). There were no draws in this tournament. Suppose that the first player won x1 games, the second player won x 2 games, the third player x3 games and so on. Find the value of x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 .
45 2. J98 Q7 In a league competition which consists of 11 team, each team plays against every other twice. Each match between two teams always results in a winner, and the winning team in each match will be given an amount of $200 as prize-money. What is the total amount of prizemoney, in dollars, given out in the whole competition?
6.3 Pigeonhole Principle Know and use the pigeonhole principle
46
Unit 7: Elementary Number Theory 7.1 Properties of Numbers Know the (self-evident) properties of rational, irrational and real numbers n e.g. If a 1, a 1 for n (J04 Q9)
1.
e.g. a 0 and a n 0 n is even (J00 Q19) e.g. the cube of a fraction cannot be an integer unless the fraction is an integer (J98 Q30)
J95 Q5
Consider the following statements: is a non-recurring decimal. (i) is an irrational number. (ii) (iii) 227 . (iv) is approximately 3.142. is a real number. (v) (A) (C) (D) (E)
Only (iii) is true. (B) Only (ii) and (iii) are true. Only (i), (ii), (iv) and (v) are true. Only (iii) and (iv) are true. Only (iii) and (v) are true.
7.2 Functions (Greatest Integer Function) Know and use the greatest integer function Know the properties of the greatest integer function Use the result [a] {a} a 1. J96 Q22 The symbol [x] is defined as the greatest integer less than or equal to the number x. If a [a] = 68 and b [b] = 109, what is the value of [a] [b] – [a + b]?
2. J00 Q30 Find the total number of integers n between 1 and 10000 (both inclusive) such that n is divisible by [ n ] . Here [ n ] denotes the largest integer less than or equal to n .
47
7.3 Prime factorization Determine the HCF and LCM of two or more numbers Use prime factorization to determine the factors of a number Note: Students should develop sensitivity towards numbers e.g. J02 requires recognizing the common factors of seemingly unrelated numbers – 396 = 4(99), 297 = 3(99), 198 = 2(99) Use prime factorization to determine the number of factors of a number (need knowledge of combinatorics) 1. J96 Q19 Let a, b, c, d be integers, and (a 2 b2 )(c 2 d 2 ) 29 . Find the value of a 2 b 2 c 2 d 2 .
2. J96 Q30 The symbol n! is defined as 1 2 3 n . For example, 5! = 1 2 3 4 5 120 . Given that n! 223 313 56 73 112 132 17 19 23 . What is the value of n?
3. J97 Q10 If x and 221 are both integers, what is the total number of possible values of x? x
4. J98 Q6 A card is chosen at random from a pack of 8 cards which are numbered 2, 3, 5, 7, 11, 13, 17, 19 respectively. The number of the card is recorded, and then the card is placed back with the other cards. The cards are then shuffled, and the above process is repeated until a total of four cards are chosen. Suppose the product of the four numbers thus obtained is P. How many of the numbers 136, 198, 455, 1925, 3553 cannot be equal to P?
5. J98 Q9 Find the total number of positive integers x such that 324000 is divisible by x and x is divisible by 20.
6. J98 Q21 372 identical cubes are placed together to form a rectangular solid. Find the total number of different rectangular solids which can be formed in this way.
7. J99 Q15 The number of positive integers that are factors of 62 (633 632 63 1) 1 is (A)
4
(B)
16
(C)
25
(D)
32
(E)
45
48
8. J00 Q8 How many (positive integer) factors does the number 1017 have? (A)
289
(B)
290
(C)
323
(D)
324
(E)
none of the above
9. J00 Q22 Let a, b, c, d be four distinct positive integers whose product abcd is equal to 2000. What is the largest possible value of the sum a + b + c + d?
7.4 Modular Arithmetic Know and use the Quotient – Remainder Theorem Know and use the properties of modular arithmetic (e.g. J01 Q20 x sum of digits of x (mod 9)) Know and use the periodic property of modular arithmetic to determine the last (few) digit(s) of a number (Classic: J02 Determine the last digit of 20022002) Deduce the units digit of a number, a given the units digit of a n for n = 1 or 2 or 3 etc 1. J95 Q4 What is the unit digit in (24310 )(1639 )(6338 ) ?
2. J95 Q12 Twenty soldiers, numbered 1 through 20, stood in a circle clockwise numerical order, all facing the center. They began to count out loud in clockwise order: the first soldier called out the number 1, the second called out 2; and each soldier then called out the numbers 1 more than the number called to his right. What was the number of the soldier who called out the number 1995?
3. J95 Q23 A natural number gives the same remainder (not zero) when divided by 3, 5, 7 or 11. Find the smallest possible value of this natural number.
4. J97 Q19 A number x is divisible by the numbers 2, 3, 4, 5, 8 and 9, but leaves a remainder of 5 when divided by 7. Find the smallest possible value of x.
5. J97 Q30 Find the smallest positive integer n such that 1000 n 1100 and 1111n 1222 n 1333n 1444 n is divisible by 10.
49 6. J98 Q16 What is the unit digit of 19971998 19981999 19991997?
7. J00 Q11 What is the units digit of (31999)(7 2000)(17 2001) ?
7.5 Divisibility Know and use the divisibility tests for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13 and 25 Calculate the number of integers (within a given range) divisible by a certain number (integer) (use of the greatest integer function is optional) Use the divisibility property – the remainder of A when divided by n is the same as the remainder of the sum of the digits of A when divided by n Use the divisibility property – if a n and b a , then b n Use the divisibility property – if a n and b a , then ab n Use the divisibility property – if a b c and a b , then a c Use the divisibility property – if ab n , then a n and b n Use the divisibility property – if a m and b m leave the same remainder, then a – b is divisible by m Deduce the (unknown) numerator of a fraction (numerically equal to an integer) by observing its denominator … etc 1. J96 Q24 Mr A, Mr B, Mr C and Mr D are car salesmen. In the period from 1985 to 1995, Mr A sold 8 times as many cars sold by Mr B, times as many sold by Mr C and 12 times as many sold by Mr D. During this period, the total number of cars sold by the four salesmen was less than 600. What is the greatest possible number of cars which Mr A could have sold from 1985 to 1995?
2. J98 Q12 When the three numbers 1238, 1596, 2491 are divided by a positive integer m, the remainders are all equal to a positive integer n. Find m + n.
3. J99 Q28 What is the smallest positive integer n such that the digits of n are either 0 or 1 and n is divisible by 225?
50 7.6 Implicit Properties of Digits (of a number) Recognize that each digit of number lies between 0 and 9 (inclusive)
7.7 Legendre’s Formula
n n n Use the formula – 2 r such that p r 1 n p r to determine the p p p exponent of the greatest power of a prime p dividing n! (Classic example: Determine the number of zeros at the end of n!) 1. J00 Q18 How many (consecutive) zeros are there at the end of the number 100! 1 2 3 99 100 ? (For example, there are 2 (consecutive zeros) at the end of the number 30100.)
Practice 1. J95 Q23 A natural number (> 2) gives the same remainder (not zero) when divided by 3, 5, 7 or 11. Find the smallest possible value of this natural number.
7.8 Diophantine equations Solve diophantine equations 1. J97 Q22 A 2-digit number represented by BC is such that the product of BC and C is a 3-digit number represented by ABC . Find all the possible2-digit numbers represented by BC.
2. J97 Q24 A solution of the equation ( x a)( x b)( x c) 5 0 is x = 1, where a, b, and c are different integers. Find the value of a b c .
51 3. J97 Q26 A rectangle has length p cm and breadth q cm, where p and q are integers. If p and q satisfy the equation pq q 13 q 2 , calculate the maximum possible area of the rectangle.
4. J97 Q27 Suppose x, y, and z are positive integers such that x > y > z > 663 and x, y and z satisfy the following: x + y + z = 1998 2x + 3y + 4z = 1998
5. J98 Q13 Suppose a, b are two numbers such that a 2 b 2 8a 14b 65 0 . Find the value of a 2 ab b2 .
6. J98 Q18 The age of a man in the year 1957 was the same as the sum of the digits of the year in which he was born. Find his age in the year 1998.
7. J98 Q20 Let x and y be two positive integers such that x – y = 75 and the least common multiple of x and y is 360. Find the value of x y .
8. J98 Q28 Find the total number of positive four-digit integers x between 1000 and 9999 such that x is increased by 2088 when the digits of x are reversed. [As an example, the integer 1234 is changed to 4321 after reversing the digits.]
9. J98 Q29 Let a, b, c be positive integers such that ab + bc = 518 and ab – ac = 360. Find the largest possible value of the product abc .
10. J98 Q30 Suppose a, b, c, d are four positive integers such that a 3 b 2 , c 3 d 2 , and c a 73 . Find the value of a c .
11. J99 Q18 If a and b are positive integers such that a 2 b 2 15 and a3 b3 28 , then the number of possible pairs of (a, b) is (A)
0
(B)
1
(C)
2
(D)
3
(E)
None of the above
52 12. J99 Q22 Suppose that p, q, (2p – 1)/q, (2q – 1)/p are positive integers and p, q > 1. What is the value of p q ?
13. J00 Q3 A 4-digit number abcd consisting of 4 distinct digits satisfies 9 abcd dcba . Then the second digit b is (A)
0
(B)
1
(C)
2
(D)
3
(E)
4
14. J00 Q21 Let x be a 3-digit number such that the sum of the digits equals 21. If the digits of x are reversed, the number thus formed exceeds x by 495. What is x?
15. J00 Q23 One of the integers among 1, 2, 3, …, n is deleted. The average of the remaining n – 1 numbers is 602 . Which number was deleted? 17
16. J00 Q27 Let n be a positive integer such that n + 88 and n 28 are both perfect squares. Find all the possible values of n.
