Chapter 6 II Math Reasoning ENHANCE_edit

8/2/2019 Chapter 6 II Math Reasoning ENHANCE_edit

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Mathematical Reasoning 56

CHAPTER 4: MATHEMATICAL REASONING

4.1: 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.4.1.1: 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 = 6.42 103

3 21 + 4 = 25 (2.56 104

)2

4 23 > 34 I'm 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 Answer1 1 > 31 The root of x

232 is x = 3

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

3 0.0002450 = 2.45 103 )()22(22

2xxxx

4 4325 Zero is smaller than 1

5 41 is a prime number 8,6,4,2 6 All hexagons have 6 sides 11255048

7 13 is a factor of 69 Ice melts at 10oC

8 12 is multiple of 4 ,10,9,8,7,611,10 9

All sets have as its subset(5

3)

2= 2

6

10 0 x = 4 is a root of x25x + 4 = 011 What is the square root of 9? Draw a graph of y = 3x

32.

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


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6.2 QUANTIFIER ALLAND 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 90o

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


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6.3: Operations on statement

6.3.1: Change the truth of each of the following statements by using the word notor 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

+ 32

is more than 32

True

12 + 32 is not more than 32 False

6 Fish has fins TrueFish 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

52

3

100 is equal to 102


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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 wordand or or

Concept: The truth table for p andq

p Q pandq

true true true

true false false

false true false

false false false

Concept: The truthtable for p orq

6.3.2 a) Example:

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

(The first statement isp 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 por q

true true true

true false true

false true true

false false false


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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

2 35 or 45 is a multiple of 103 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 number6 1

2+ 2

2= 3

2 and 3

2+ 4

2= 5

2

7 2 is equal to 20or (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 cm2or 1 cm3 = 1000 mm2

13 Ant is an insect and has 4 legs

14 The symbols and denote a null set

155 % =

20

1and

200

1%

5

1


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6.4: Implication

ImplicationIf p, then qwherepistheantecedent and q is theconsequent.

If a compound statement consisting ofifand only if ,

we can write itstwo implications as If p, then q and If q, thenp(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 ifx = 0 Implication 1 : If 5 +x = 5, thenx = 0

Implication 2 : Ifx = 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) xis a multiple of 4 if and

only ifx is divisible by 4

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

Implication 2 : Ifx is divisible by 4, then xis a multiple of 4

d) 331

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

y , theny = 27

Implication 2 :Ify = 27, then 331

y

e) x2 = 9 if and only ifx = 3 Implication 1 : If x2 = 9, then x = 3Implication 2 : If x = 3, then x2 = 9

Compound Statement Answer

a) 10a

= 1if and only if a = 0 Implication 1 :

Implication 2 :

b)x

3

=64 if and only ifx =4 Implication 1 :

Implication 2 :

c) Abu will be punished if and

only ifhe is late to school

Implication 1 :

Implication 2 :


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d) x + 3 =7 if and only if

x8 =18

Implication 1 :

Implication 2 :

e) BA if and only ifABA Implication 1 :

Implication 2 :

f) y24y =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 andonly ifm3

is a negative number Implication 1 :

Implication 2 :

i) 101

=z

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

1=z

1, then z =10

Implication 2 :

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

Implication 2 :


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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 IPremise 1: All A are B

Premise 2: C is A

Conclusion: C is B

Argument Type IIPremise 1: Ifp then q

Premise 2: p is trueConclusion: q is true

Argument Type IIIPremise 1: Ifp then q

Premise 2: Not q is true

Conclusion: Notp is true

6.5 a) Complete each of the following arguments.

Example Exercise

1 Premise 1: Ifm < n, then mn < 0.

Premise 2 : m < n

Conclusion: mn < 0.

Premise 1: Ifm > n then mn > 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


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

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: Allpentagons 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 : Ifx + 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 obtuangle

Premise 2:

Conclusion : 100o is an obtuse angle

6 Premise 1 : Ifx = - 3, thenx3

= - 27

Premise 2 :x3 -27

Conclusion: x 3

Premise 1 :If 8,6,4,2x , thenx is an evennumber

Premise 2:

Conclusion: x is not an even number

7 Premise 1: Ifx andy are odd numbers,

then the product ofx and y is an odd

numberPremise 2 : 3 and 5 are odd numbers

Conclusion : Theproduct 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 : Ifp > 3 , then 6p > 18

Premise 2 : 6p < 18

Conclusion :p < 3

Premise 1 : IfC is a subset of D, then n(C) n(

Premise 2 : n(C) > n(D)

Conclusion:


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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 ofx2.

36 is a perfect square

Conclusion: 36 = 62.

2 The sum of the interior angles of a polygon is (n2) 180o.

Hexagon is a polygon

Conclusion : The sum of the interior angles of a hexagon is (62) 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 cmis 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: .


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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 + 2x14 = 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(1

2

)17 = 2(22)117 = 2(3

2)1

31 = 2(42)1,

...

General conclusion: 2n21 where n = 1, 2, 3, 4

2 Given 5, 11, 17, 23

and 5 = 6(0) + 5

11 = 6(1) + 517 = 6(2) + 5

23 = 6(3) + 5

..

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


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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

+ 3where 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 = 902 182 = 902 4

72 = 902 958 = 902 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 12

4 = 3 227 = 3 3210 = 3 4213 = 3 52

General conclusion:


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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 ifx 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

= PQ2

+ QR2


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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, theny = 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 patternsbelow :

9 = 4(2)27

29 = 4(3)2757 = 4(4)27

93 = 4(5)27

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

34

= 12 or4

5= 1.25

x = 4 if and only ifx3 = 64


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

Answer:

a.) Conclusion:

b.) .

c.) ..

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

: 75.04

3

: 5 6 if and only if r> 2


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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 = 100c) 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.



All multiples of 2 are divisible by 4.

y < x if and only ify > -x


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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) Trueb) 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 followingpattern:

(5 marks)

8 > 7 or 32 = 6

m3 = 1000 if and only if m = 10

03522 mm

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

1)4(21291)3(255

1)2(217

1)1(23

3

3

3

3


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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 :Ifx is greater than zero, thenx 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) 464

3

(ii.) -5 > - 8 and 0.03 = 3 1

10

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 : .1809000 x

Conclusion : sin0

x is positive. (5 marks)


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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 LLK

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

bahas an inverse if and only if adbc 0

[6 marks]

Some even numbers are multiples of 3

Sum of interior angles of a polygon with 3 sides is ( 32 ) x 1800

Sum of interior angles of a polygon with 4 sides is (42 ) x 1800

Sum of interior angles of a polygon with 5 sides is (52 ) x 1800

If x > 9 , then x > 5


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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.


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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 is1800 .

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 21 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.


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6.3.2 b

p q True / False

1 False

2 False3 True

4 False

5 True

6 False

7 True

8 False

9 True

10 True

11 True

12 True

13 False14 True

15 False

6.4. b

a Implication 1 : If 10a

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

a= 1

b Implication 1 : Ifx3

=64, thenx =4

Implication 2 : Ifx =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 x8 =18Implication 2 : If x8 =18, then x + 3 =7

e Implication 1 : If BA , then ABA

Implication 2 : If ABA , then BA

f Implication 1 : Ify24y =4 then y = 2

Implication 2 : Ify = 2, then y24y =4

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

Implication 2 : If k is an integer, thenk 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 =z1 , then z =10

Implication 2 : Ifz =10, then 101

=z

1

j Implication 1 : If 5m , then 52

= m

Implication 2 : If 52

= m, then 5m


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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 : Ifx >y, thenx y > 0 .Ifx y > 0, thenx >y.

2. i : Statement

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

= PQ2

+ QR2

2: If PR2

= PQ2

+ QR2, then PQR is a right-angled triangle

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

3. a) True

b) Ifx = 4, then x3

= 64If 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 zeroii.)Premise 2 : 2340 is a multiple of 10

3 Premise 2 : MNOPQ is a pentagon

4 Conclusion : x + 5 10

5 Premise 2 : 90o

2 , then 3r > 6.

PAST YEARS SPM QUESTIONS

June 2004

1. a) Statementb) 105

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

c) ,123 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.


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c) If00

18090 x , then 0sin x is positive.

7. Nov 2006a) (i) Some

(ii) All

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

c) LLK

8. June 2007

a)Trueb)ABCD has no two parallel sidesc)(n2 ) x 1800d) Implication 1 : If matrix

dc

bahas an inverse then adbc 0

Implication 2 : If adbc 0 then

dc

bahas an inverse

9. Nov 2007

a)Someb)23 is not a multiple of 3c)3(2)n + n , n = 1, 2, 3, d)Implication 1 : If pq > 0 then p > q

Implication 2 : If p > q then pq > 0

10. June 2008a) Trueb) 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) Falseb) Implication 1 : If x3 = -64 then x = -4

Implication 2 : If x = -4 then x3

= -64

Chapter 6 II Math Reasoning ENHANCE_edit

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