MATHEMATICAL REASONING

6.1: STATEMENT - is a sentence which states a definite true or false but not both

- it cannot be a question (?)

- it cannot be an exclaimation (!)

- it cannot be an instruction.

6.1.1: Determine whether each of the following is a statement ( ) or not a statement ( ).

Example Answer Exercise Answer

1 18 is an odd number 248124 xxx =

2 X + 4 642000 = 6.42 10 3 3 21 + 4 = 25 (2.56 10 4) 2 4 23 > 34 I good in mathematics

5 43 + 25 68 3.46 is an integer

6 All octagons have 3 edges 7 + 91

7 What is the price of the dictionary? Please try again

8 89 is a perfect square A parallelogram is a circle

9 Some even numbers can be divided by 5 { },5,4,3,2,1,04,3,1 10 Finish your mathematics` exercise 435 2 + xx

6.1.2: Determine whether each of the following is true or false.

Example Answer Exercise Answer

1 1 > 31

False The root of x2 3

2 is x = 3

2 81 is a perfect square True 13 + 6 > 10 3

3 0.0002450 = 2.45 103 False )()22(22 2 xxxx +=+

4 4325 =+ False Zero is smaller than 1

5 41 is a prime number True { }8,6,4,2 6 All hexagons have 6 sides True 11255048 +== 7 13 is a factor of 69 False Ice melts at 10

oC

8 12 is multiple of 4 True { },10,9,8,7,611,10 9 All sets have as its subset True (5 3) 2 = 2 6 10 { } =0 False x = 4 is a root of x2 5x + 4 = 0 11 What is the square root of 9? Draw a graph of y = 3x

3 2.

12 A parallelogram is a quadrilateral. A heptagon has nine sides.

6.2 QUANTIFIER ALL AND SOME

6.2.1 Based on the information given, construct a true statement using the quantifier

all or some

6.2.1 a) Exercise:

Object and property Answer (true statement)

1 Acute angle ; less than 90o All acute angles are less than 90

o

2 Negative number; smaller than zero All negative numbers are smaller than zero

3 Triangle ; right-angled triangles Some triangles are right-angled triangles

4 Sets; as its subset All sets have as its subset

6.2.1 b) Exercise:

6.2.2 Based on the information given, construct a false statement using the quantifier

all or some

6.2.2 a) Example:

Object and property Answer (false statement)

1 Null sets ; no elements Some null sets have no elements

2 Parallel line ; the same length All parallel lines have the same length

3 Quadrilaterals ; two parallel sides Some quadrilaterals have two parallel sides

4 Odd number ; perfect square All odd number are perfect squares

6.2.2 b) Exercise:

Object and property Answer (true statement)

1 Rhombuses ; four equal sides

2 Odd number ; prime number

3 Factor of 6 ; factor of 3

4 Isosceles triangle ; two equal sides

5 Even number ; divisible by 10

Object and property Answer (false statement)

1 Multiple of 2 ; multiple of 4

2 Orchid flower ; yellow in colour

3 Animal ; can swim

4 Human being ; heart

5 Multiples of 8 ; can be exactly divided

by 2

6.3: Operations on statement

6.3.1: Change the truth of each of the following statements by using the word not or no

6.3.1 a) Example:

Statement Truth

1 4 is a factor of 32 True

4 is not a factor of 32 False

2 Human being have legs True

Human being have no legs False

3 All triangles have a sum of interior angles of 180o True

Not all triangles have a sum of interior angles of 180o False

4 Rambutan have thorns False

Rambutan have no thorns True

5 12 + 3

2 is more than 3

2 True

12 + 3

2 is not more than 3

2 False

6 Fish has fins True

Fish has no fins False

7 Mammal is warm blooded True

Mammal is not warm blooded False

8 All perfect squares are integers True

Not all perfect squares are integers False

9 56 can be exactly divided by 6 False

56 can not be exactly divided by 6 True

10 122 is equal to 144 True

122 is not equal to 144 False

6.3.1 b) Exercise:

Change the truth of each of the following statements by using the word not or no

Statement Truth

1 Some even numbers are divisible by 10

2 All factors of 7 are factors of 14

3 All trapeziums have a pair of parallel lines

4 44 is a multiple of 11

5 2

3

100 is equal to 102

6 Nucleus is an organelle

7 Plants have hair roots to absorb water and minerals

8 10 and 120 are multiples of 10

9 20 is equal to 2

10 All prime numbers are not divisible by 2

6.3.2 : Forming a compound statement by combining two given statements using the word

and or or

Concept: The truth table for p and q

p Q p and q

true true true

true false false

false true false

false false false

Concept: The truth table for p or q

6.3.2 a) Example:

Form a true statement for each of the two given statements.

(The first statement is p and the second statement is q)

Statements p q Compound statement (true statement)

1 5125;525 3 == 5125525 3 == and @ 5125525 3 == or 2 aaaa =+= 1;1 aaoraa =+= 11 3 100 is an even number ;

2 is a prime number

100 is an even number and 2 is a prime number @

100 is an even number or 2 is a prime number

4 { } { } { }baabaa ,;, { } { } { }baaorbaa ,,

p Q p or q

true true true

true false true

false true true

false false false

5 6 is a factor of 12 ;

6 is a factor of 18

6 is a factor of 12 and 6 is a factor of 18 @

6 is a factor of 12 or 6 is a factor of 18

6 53 > 12 ; 24 2 = 8 53 > 12 or 24 2 = 8

7 2 m = 200 cm ; 1 m = 100 cm

2 m = 200 cm and 1 m = 100 cm

8 A triangle has 3 sides

A hexagon has 5 sides

A triangle has 3 sides or a hexagon has 5 sides

9 4 < 2 ; 8 0 = 1 4 < 2 and 8

0 = 1

10 4 + 9 = 5 ; 2 > 32 4 + 9 = 5 or 2 > 32

6.3.2 b) Exercise:

Determine the truth of each of the following compound statement.

Statements p q Compound statement

1 55 < 188115 = and

False

2 35 or 45 is a multiple of 10

3 4 is a factor of 24 or 30

4 A rectangle has 4 sides and a pentagon has 6 sides

5 7 is a factor of 49 and a prime number

6 12 + 2

2 = 3

2 and 3

2 + 4

2 = 5

2

7 2 is equal to 20 or (2

1)

1

8 Some even numbers are divisible by 2 or all odd numbers are

divisible by 3

9 36 is a perfect square and a multiple of 4

10 80 is a perfect square or an even number

11 17 is a prime number and a factor of 34

12 1 m2 = 10 000 cm

2 or 1 cm

3 = 1000 mm

2

13 Ant is an insect and has 4 legs

14 The symbols and { } denote a null set

15 5 % =

20

1 and

200

1%

5

1=

6.4: Implication

Implication If p, then q where p is the antecedent and q is the consequent.

If a compound statement consisting of if and only if ,

we can write its two implications as If p, then q and If q, then p (known as converse of an implication)

6.4.a) Example:

Write two implications from each of the following compound statements.

6.4. b) Exercise

Write two implications from each of the following compound statements.

Compound Statement Implications

a) 5 + x = 5 if and only if x = 0

Implication 1 : If 5 + x = 5, then x = 0

Implication 2 : If x = 0, then 5 + x = 5

b) PQP = if and only if PQ

Implication 1 : If PQP = , then PQ Implication 2 : If PQ , then PQP =

c) x is a multiple of 4 if and

only if x is divisible by 4

Implication 1 : If x is a multiple of 4, then x is divisible by 4

Implication 2 : If x is divisible by 4, then x is a multiple of 4

d) 331

=y if and only if y = 27 Implication 1 : If 331

=y , then y = 27

Implication 2 :If y = 27, then 331

=y

e) x2 = 9 if and only if x = 3 Implication 1 : If x2 = 9, then x = 3

Implication 2 : If x = 3, then x2 = 9

Compound Statement Answer

a) 10

a = 1

if and only if a = 0

Implication 1 :

Implication 2 :

b) x3 = 64 if and only if x = 4 Implication 1 :

Implication 2 :

c) Abu will be punished if and

only if he is late to school

Implication 1 :

Implication 2 :

d) x + 3 = 7 if and only if

x 8 = 18

Implication 1 :

Implication 2 :

e) BA if and only if ABA =

Implication 1 :

Implication 2 :

f) y2 4y = 4 if and only if

y = 2

Implication 1 :

Implication 2 :

g) k is a perfect square if and

only if k is an integer

Implication 1 :

Implication 2 :

h) m is a negative number if and

only if m3 is a negative number

Implication 1 :

Implication 2 :

i) 10 1

=z

1 if and only if z =10 Implication 1 : If 10

1 =

z

1, then z =10

Implication 2 :

j) 5=m if and only if 52 = m Implication 1 :

Implication 2 :

6.5: Argument

Argument is the process in making a conclusion based on two given premises.

A premise is a given statement.

There are three simple types of arguments which can be used to make a conclusion.

Argument Type I

Premise 1: All A are B

Premise 2: C is A

Conclusion: C is B

Argument Type II

Premise 1: If p then q

Premise 2: p is true

Conclusion: q is true

Argument Type III

Premise 1: If p then q

Premise 2: Not q is true

Conclusion: Not p is true

6.5 a) Complete each of the following arguments.

Example Exercise

1 Premise 1: If m < n, then m n < 0.

Premise 2 : m < n

Conclusion: m n < 0.

Premise 1: If m > n then m n > 0.

Premise 2: m > n

Conclusion: .

2 Premise 1 : All rectangles have four right angles

Premise 2 : ABCD is a rectangle

Conclusion : ABCD has four right angles

i.) Premise 1:..

Premise 2 : 20 is a negative number

Conclusion : 20 is smaller than zero

ii.) Premise 1: All numbers with a last digit 0 is a

multiple of 10.

Premise 2: 2340 is a number with a last digit 0

Conclusion:

3 Premise 1 : All odd numbers are not divisible by 2

Premise 2 : 23 is an odd number

Conclusion : 23 is not divisible by 2

Premise 1: All pentagons have a sum of their

interior angles equal to 540o

Premise 2 :..

.

Conclusion: MNOPQ has a sum of the interior

angles equal to 540o

4 Premise 1 : If set B = , then n(B) = 0

Premise 2 : n(B) 0

Conclusion : set B

Premise 1 : If x + 7 = 10, then x = 3

Premise 2 : x 3

Conclusion:.

5 Premise 1 : All factors of 12 are factors of 24

Premise 2 : 4 is a factor of 12

Conclusion : 4 is a factor of 24

Premise 1: If 90o < < 180o, then is an obtuse

angle

Premise 2:

Conclusion : 100o is an obtuse angle

6 Premise 1 : If x = - 3, then x3 = - 27

Premise 2 : x3 -27

Conclusion: x 3

Premise 1 :If { }8,6,4,2x , then x is an even number

Premise 2:

Conclusion: x is not an even number

7 Premise 1: If x and y are odd numbers,

then the product of x and y is an odd

number

Premise 2 : 3 and 5 are odd numbers

Conclusion : The product of 3 and 5 is an odd

number

Premise 1 : If KLM is an equilateral triangle,

then KL = LM = KM

Premise 2: KLM is an equilateral triangle

Conclusion: ..

8 Premise 1 : If p > 3 , then 6p > 18

Premise 2 : 6p < 18

Conclusion : p < 3

Premise 1 : If C is a subset of D, then n(C) n(D) Premise 2 : n(C) > n(D)

Conclusion:

6.6: Deduction and Induction

6.6.1 Deduction is making a conclusion for a specific case based on given general

statements.

6.6.1 a) Example :

Make a conclusion by deduction for each of the following cases.

1 All perfect squares can be written in the form of x2.

36 is a perfect square

Conclusion: 36 = 62.

2 The sum of the interior angles of a polygon is (n 2) 180o. Hexagon is a polygon

Conclusion : The sum of the interior angles of a hexagon is (6 2) 180o = 720o

3 All sets have an empty set, as subset Set N = { }6,5

Conclusion : Set N has an empty set, as subset

4 The radius of a circle is 3 cm

The circumference of the circle with a radius of r cm is 2 rh

Conclusion : The circumference of the circle with a radius of 3 cm is 2 )3(h = h6 cm

5 All parallelograms have two pairs of parallel lines.

ABCD is a parallelogram

Conclusion : ABCD has two pairs of parallel lines

6.6 b) Exercise

Make a conclusion by deduction for each of the following cases.

1 It is compulsory for all form 5 students to sit for the SPM examination.

Ali sat for the SPM examination.

Conclusion: .

2 All herbivores eat grass

Goats are herbivores

Conclusion: .

3 All cuboids have 12 edges.

Object D is a cuboid

Conclusion: ..

4 All quadratic equations have 2 as the highest power of the unknown.

x2 + 2x 14 = 0 is a quadratic equation.

Conclusion:

5 All those who are wearing school uniforms are students.

Abu was not wearing the school uniform.

Conclusion: .

6.6.2 Induction is making generalization based on the pattern of a numerical

sequence, or specific cases.

6.6.2 a) Example:

Make a conclusion by induction for each of the following cases.

1 Given 1, 7, 17, 31

and 1 = 2(12) 1

7 = 2(22) 1

17 = 2(32) 1

31 = 2(42) 1,

...

General conclusion: 2n2 1 where n = 1, 2, 3, 4

2 Given 5, 11, 17, 23

and 5 = 6(0) + 5

11 = 6(1) + 5

17 = 6(2) + 5

23 = 6(3) + 5

..

General conclusion: 6(n) + 5 , where n = 0, 1, 2, 3, .

3 Given 5, 11, 21, 35

and 5 = 2(1)2 + 3

11 = 2(2)2 + 3

21 = 2(3)2 + 3

35 = 2(4)2 + 3 , .. make a general conclusion and find the 9th number

General conclusion: 2(n)2 + 3 where n = 1, 2, 3, 4,.

Hence, the 9th number is 2(9)2 + 3 = 165

6.6.2 b.) Make a conclusion by induction for each of the following cases.

1 Given 5, 14, 29, 50

and 5 = 2 + 3(1)2

14 = 2 + 3(2)2

29 = 2 + 3(3)2

50 = 2 + 3(4)2

General conclusion

2. The numerical sequence 88, 82, 72, 58, .

can be written as

88 = 90 2 1 82 = 90 2 4 72 = 90 2 9 58 = 90 2 16 ..

General conclusion

3 Given 2, 9, 16, 23,

And 2 = 2 + 7(0)

9 = 2 + 7(1)

16 = 2 + 7(2)

23 = 2 + 7(3)

..

General conclusion :..

4. Given 3, 24, 81, 192,

and 3 = 3(1)3

24 = 3(2)3

81 = 3(3)3

192 = 3(4)3

General conclusion :..

5 Given 1, 4, 7, 10, 13,

and 1 = 3 1 2 4 = 3 2 2 7 = 3 3 2 10 = 3 4 2 13 = 3 5 2

General conclusion :

1) i: State whether the following statement is true or false.

ii : Complete the premise in the following argument.

Premise 1: If JKL is an equilateral triangle, then the value of its interior angle is 60o

Premise 2: ______________________________________________________

Conclusion: The value of the interior angle of JKL is 60o.

iii : Write down two implications based on the following sentence.

Answer:

i. .

ii. Premise 2:

.

iii. Implication 1 : .

Implication II : .

2) i : Is the sentence below a statement or a non-statement ?

ii : Write down two implications based on the following sentence.

iii : Based on the information below, make a general conclusion by induction regarding the sum of

the interior angles of a triangle.

Answer:

9 > 6 and 42 = 8

x > y if and only if x y > 0

5 is an even number

The sum of the interior angles of triangle ABC = 180o

The sum of the interior angles of triangle JKL = 180o

The sum of the interior angles of triangle PQR = 180o

PQR is a right-angled triangle if and only if PR2 = PQ

2 + QR

2

i. .

ii. Implication 1 : .

Implication II :

iii. General conclusion :

3. a) Determine whether the following statement is true or false.

b) Write two implications from the statement given below.

c) Complete the premise in the following argument.

Premise 1 : If 2y = 10, then y = 5.

Premise 2 : ..

Conclusion : 2y 10.

Answer:

a)

b) Implication I:

Implication II:

c) Premise 2: ..

4. a) Complete the conclusion in the following argument.

Premise 1 : All regular hexagons have 6 equal sides.

Premise 2 : ABCDEF is a regular hexagon.

Conclusion : .

b) Make a conclusion by induction for a list of numbers 9,29, 57, 93,that follow the patterns

below :

9 = 4(2)2 7

29 = 4(3)2 7

57 = 4(4)2 7

93 = 4(5)2 7

c) Combine the two statements given below to form a true statement.

34 = 12 or

4

5 = 1.25

x = 4 if and only if x3 = 64

i) 15 ( 5) = 5 ii) 32 is a multiple of 8.

Answer:

a.) Conclusion:

b.) .

c.) ..

5. a) Below are three statements : 42 = 8

: 75.04

3=

: 5 < 2

b) Complete the following argument.

Premise 1 : If a = 6, then 5a = 30 .

Premise 2 : 5a 30 Conclusion : .. .

c) Write down two implications based on the following;

Answer:

a.)

b.) Conclusion:

c.). Implication 1 : .

Implication 2 : ....

Combine any of the two statements

to form a false statement.

3r > 6 if and only if r > 2

SPM PAST YEAR QUESTIONS

Year 2003 (Nov)

a) Is the sentence below a statement or non-statement?

4 is a prime number

b) Write down two implications based on the following sentence;

'' PRifonlyandifRP

c) Based on the information above, make a general conclusion by induction regarding the number of

subsets in a set with k elements. (5 marks)

Answer : a) Statement

b) Implication 1 : If RP , then '' PR Implication 2 : If '' PR , then RP

c) The number of subsets in a set with k elements is 2k

Year 2004 (July)

a) State whether the following sentence is a statement or a non-statement.

b.) Write down a true statement using both of the following statements:

Statement 1: 1052 =

Statement 2: 1001010 =

c.) Write down two implications based on the following sentence:

(4 marks)

Answer : a) Statement

b) 52 = 10 or 10 x 10 = 100

c) Implication 1 : If y < x then -y > -x

Implication 2 : If -y > -x then y < x

The number of subsets in a set with 2 elements is 22.

The number of subsets in a set with 3 elements is 23.

The number of subsets in a set with 4 elements is 24.

All multiples of 2 are divisible by 4.

y < x if and only if y > -x

Year 2004 (Nov)

a) State whether the following statement is true or false.

b) Write down two implications based on the following sentence

c) Complete the premise in the following argument :

Premise 1 : All hexagons have six sides.

Premise 2 : .

Conclusion : PQRSTU has six sides. (5 marks)

Answer : a) True

b) Implication 1 : If m3 = 1000 then m = 10

Implication 2 : If m = 10 then m3 = 1000

c) PQRSTU is a hexagon

Year 2005 (July)

a) Determine whether the following sentence is a statement or non-statement.

b) Write down the converse of the following implication, hence state whether the converse is true or false.

a) Make a general conclusion by induction for a list of number 3, 17, 55, 129, which follows the following

pattern:

(5 marks)

8 > 7 or 32

= 6

m3 = 1000 if and only if m = 10

0352 2 =+ mm

If x is an odd number then 2x is an even number.

1)4(2129

1)3(255

1)2(217

1)1(23

3

3

3

3

+=

+=

+=

+=

Year 2005 (Nov)

a) State whether each of the following statement is true or false.

i) 8 2 = 4 and 82 = 16. ii) The elements of set A = { }18,15,12 are divisible by 3 or the elements of set B = { }8,6,4 are multiples of 4.

b) Write down premise 2 to complete the following argument .

Premise 1 :If x is greater than zero, then x is a positive number.

Premise 2 : .

Conclusion : 6 is a positive number.

c) Write down 2 implications based on the following sentence.

3m > 15 if and only if m > 5

Implication 1 :

Implication 2 : (5 marks)

Year 2006 (July)

a.) State whether each of the following statements is true or false.

(i) 4643 =

(ii.) -5 > - 8 and 0.03 = 3 110

b) Write down two implications based on the following sentence.

ABC is an equilateral triangle if and only if each of the interior angle of ABC is 600.

c.) Complete the premise in the following argument:

Premise 1 : .

Premise 2 : .18090 00 x

Conclusion : sin 0x is positive. (5 marks)

Year 2006 (Nov) (a) Complete each of the following statements with the quantifier all or some so that it will become

a true statement

(i) of the prime numbers are odd numbers.

(ii) ... pentagons have five sides.

(b) State the converse of the following statement and hence determine whether its converse is true or false.

(c) Complete the premise in the following argument:

Premise 1 : If set K is a subset of set L, then L L K =

Premise 2 :

Conclusion: Set K is not a subset of set L

Year 2007 (June)

a) State whether the following statement is true or false.

b) Write down Premise 2 to complete the following argument:

Premise 1 : If a quadrilateral is a trapezium, then it has two parallel sides.

Premise 2 : ..

Conclusion: ABCD is not a trapezium.

c) Based on the information below, make a general conclusion by induction regarding the

sum of interior angles of a polygon with n sides.

c) Write down two implications based on the following statement:

Matrix

dc

ba has an inverse if and only if ad bc 0

[6 marks]

Some even numbers are multiples of 3

Sum of interior angles of a polygon with 3 sides is ( 3 2 ) x 1800

Sum of interior angles of a polygon with 4 sides is (4 2 ) x 1800

Sum of interior angles of a polygon with 5 sides is (5 2 ) x 1800

If x > 9 , then x > 5

Year 2007 (Nov)

a) Complete the following statement using quantifier all or some, to make it a true statement.

b) Write down Premise 2 to complete the following argument:

Premise 1 : If M is a multiple of 6, then M is a multiple of 3.

Premise 2 : ..

Conclusion : 23 is not a multiple of 6.

c) Make a general conclusion by induction for the sequence of numbers 7, 14, 27,

which follows the following pattern.

7 = 3(2)1 + 1

14 = 3(2)2 + 2

27 = 3(2)3 + 3

=

d) Write down two implications based on the following statement:

p q > 0 if and only if p > q

Implication 1 :

Implication 2 : ...

[6 marks]

Year 2008 (June)

a) State whether the following compound statement is true or false.

7 x 7 = 49 and (-7)2

= 49

................................quadratic equations have two equal roots.

b) Write down two implications based on the following compound statement:

c) Write down Premise 2 to complete the following argument:

Premise 1:

If PQRS is a cyclic quadrilateral, then the sum of the interior opposite angles of PQRS is

1800

.

Premise 2:

Conclusion: PQRS is not a cyclic quadrilateral.

[5 marks]

Year 2008 (Nov)

a) State whether the following compound statement is true or false:

b) Write down two implications based on the following compound statement:

c) It is given that the interior angle of a regular polygon of n sides is 2

1 180n

.

Make one conclusion by deduction on the size of the interior angle of a regular hexagon.

KLM is an isosceles triangle if and only if two angles in KLM are equal.

53

= 125 and -6 < -7

x3

= -64 if and only if x = -4.

[5 marks]

Answer

Chapter 6: Mathematical Reasoning

6.1.1

6.1.2

6.2. b 1. All rhombuses have four equal sides

2. Some odd numbers are prime number

3. All factors of 6 are factor of 3

4. All isosceles triangles have two equal sides

5. Some even numbers are divisible by 10

6.2.2 b 1. Some multiples of 2 are multiples of 4

2. All orchid flowers are yellow in colour

3. All animals can swim

4. Some human beings have hearts

5. Some multiples of 8 can be exactly divided by 2

6.3.1 b)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1.True 2. True 3. True 4. False 5. True 6. True 7. False 8. True 9. False 10. True

Statement Truth

1 Some even numbers are divisible by 10 True

Some even numbers are not divisible by 10 False

2 All factors of 7 are factors of 14 True

Not all factors of 7 are factors of 14 False

3 All trapeziums have a pair of parallel lines True

Not all trapeziums have a pair of parallel lines False

4 44 is a multiple of 11 True

44 is not a multiple of 11 False

5 2

3

100 is equal to 102 False

2

3

100 is not equal to 102

True

6 Nucleus is an organelle True

Nucleus is not an organelle False

7 Plants have hair roots to absorb water and minerals True

Plants have no hair roots to absorb water and minerals False

8 10 and 120 are multiples of 10 True

10 and 120 are not multiples of 10 False

9 20 is equal to 2 False

20 is not equal to 2 True

10 All prime numbers are not divisible by 2 False

Not all prime numbers are not divisible by 2 True

6.3.2 b

p q True / False

1 False

2 False

3 True

4 False

5 True

6 False

7 True

8 False

9 True

10 True

11 True

12 True

13 False

14 True

15 False

6.4. b

a Implication 1 : If 10 a = 1, then a = 0

Implication 2 : If a = 0, then 10 a = 1

b Implication 1 : If x3 = 64, then x = 4

Implication 2 : If x = 4, then x3 = 64

c Implication 1 : If Abu is punished, then he was late to school

Implication 2 : If Abu is late to school, then he will be punished

d Implication 1 : If x + 3 = 7, then x 8 = 18

Implication 2 : If x 8 = 18, then x + 3 = 7

e Implication 1 : If BA , then ABA = Implication 2 : If ABA = , then BA

f Implication 1 : If y2 4y = 4 then y = 2

Implication 2 : If y = 2, then y2 4y = 4

g Implication 1 : If k is a perfect square, then k is an integer

Implication 2 : If k is an integer, then k is a perfect square

h Implication 1 : If m is a negative number, then m3 is a negative number

Implication 2 : If m3 is a negative number, then m is a negative number

i Implication 1 : If 10 1 =

z

1, then z =10

Implication 2 : If z =10, then 10 1

=z

1

j Implication 1 : If 5=m , then 52 = m

Implication 2 : If 52 = m, then 5=m

6.5 b

6.6.1 b

6.6.2 b

Questions According to Examination Format

1. i: False

ii : JKL is an equilateral triangle.

iii : If x > y, then x y > 0 .

If x y > 0, then x > y.

2. i : Statement

ii : 1 : If PQR is a right-angled triangle, then PR2 = PQ

2 + QR

2

2: If PR2 = PQ

2 + QR

2, then PQR is a right-angled triangle

iii : The sum of the interior angles of all triangles = 180o

3. a) True

b) If x = 4, then x3 = 64

If x3 = 64, then x = 4

c) y 5

4. a) ABCDEF has 6 equal sides.

1 Premise 2 : 5 < 12

2 i.)Premise 1 : All negative numbers are smaller than zero

ii.)Premise 2 : 2340 is a multiple of 10

3 Premise 2 : MNOPQ is a pentagon

4 Conclusion : x + 5 10 5 Premise 2 : 90o <

0100 < 180o 6 Conclusion : { }8,6,4,2x 7 Premise 2 : KL = LM = KM

8 Conclusion : C is not a subset of D

1 Conclusion : Ali is a form 5 student.

2 Conclusion : Goats eat grass

3 Conclusion : Object D has 12 edges

4 Conclusion : x2 + 2x 14 = 0 has 2 as the highest power of its unknown

5 Conclusion : Abu is not a student.

1 General conclusion : 2 + 3(n)2 where n = 1,2,3,4,..

2 General conclusion : 90 2(n)2 where n = 1,2,3,4,..

3 General conclusion : 2 + 7(n), where n = 0,1,2,3,..

4 General conclusion : 3(n)3 where n = 1,2,3,4,..

5 General conclusion : 3 n 2, or 3(n) 2 , where n = 1,2,3,4,5,..

b) 4(n)2

7 where n = 2, 3, 4, 5,

c) 15 ( 5) = 5 or 32 is a multiple of 8.

5. a) 75.04

3= and 5 < 2 @ 42 = 8 or 5 < 2 @ 42 = 8 or 75.0

4

3=

b) a 6 c) If 3r > 6, then r > 2.

If r > 2 , then 3r > 6.

PAST YEARS SPM QUESTIONS

June 2004

1. a) Statement

b) 1052 = or 1001010 = c ) If y < x , then xy > If xy > , then xy <

Nov 2004

2. a) True

b) If m3 = 1000 , then m = 10

If m = 10, then m3 = 1000

c) PQRSTU is a hexagon.

June 2005

3. a) Statement

b) If 2x is an even number, the x is an odd number. (True)

c) ,12 3 +n where n = 1, 2, 3

Nov 2005

4. a) i: False

ii: True

b) 6 is greater than zero.

c) If 3m > 15, then m > 5.

If m > 5, then m > 5.

June 2006

5. a) (i) True

(ii) False

b) If ABC is an equilateral triangle, then each of the interior angle of ABC is 600.

If each of the interior angle of ABC is 600, then ABC is an equilateral triangle.

c) If 00 18090 x , then 0sin x is positive.

7. Nov 2006

a) (i) Some

(ii) All

b) If x > 5 , then x > 9 , False

c) L L K

8. June 2007

a) True

b) ABCD has no two parallel sides

c) (n 2 ) x 1800

d) Implication 1 : If matrix

dc

ba has an inverse then ad bc 0

Implication 2 : If ad bc 0 then

dc

ba has an inverse

9. Nov 2007

a) Some

b) 23 is not a multiple of 3

c) 3(2)n + n , n = 1, 2, 3,

d) Implication 1 : If p q > 0 then p > q

Implication 2 : If p > q then p q > 0

10. June 2008

a) True

b) Implication 1 : If KLM is an isosceles triangle, then two angles in

KLM are equals.

Implication 2 : If two angles in KLM are equals, then KLM is an

isosceles triangle.

c) The sum of the interior opposite angles of PQRS is not equal to 1800.

11. Nov 2008

a) False

b) Implication 1 : If x3 = -64 then x = -4

Implication 2 : If x = -4 then x3

= -64