Chapter 6 II Math Reasoning ENHANCE

8/3/2019 Chapter 6 II Math Reasoning ENHANCE

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CHAPTER 6: MATHEMATICAL REASONING

6.1: STATEMENTS

- is a sentence which is either true or false but not both

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 4352

+xx

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

6.2 QUANTIFIER ALL AND SOME

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

Mathematical Reasoning

Example Answer Exercise Answer

1 1 > 31 False The root of x2 32 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 15 41 is a prime number True { }8,6,4,26 All hexagons have 6 sides True 11255048 +==

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

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 x25x + 4 = 0

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6.3.1 : Change the truth of each of the following statements by using the word not orno

6.3.1 a) Example:

Statement Truth

1 4 is a factor of 32 True4 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 has thorns False

Rambutan has no thorns True

5 12 + 32 is more than 32 True

12 + 32 is not more than 32 False

6 Fish has fins True

Fish has no fins False7 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 notbe 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

100is equal to 102

6 Nucleus is an organelle

Mathematical Reasoning 58


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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 pandq

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.

Statements p q Compound statement (true statement)

1 5125;525 3 == 5125525 3 == and @

5125525 3 == or2 aaaa =+= 1;1 aaoraa =+= 11

3 100 is an even number ;

2 is a prime number

100 is an even numberand 2 is a prime number @

100 is an even numberor 2 is a prime number


p Q por q

true True true

true false true

false true true

false false false

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

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 True / False1 55 < 188115 = and False

2 35 or 45 is a multiple of 10

3 4 is a factor of 2 4 or 30

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

5 is a factor of 49 and a prime number

6 12 + 22 = 32 and 32 + 42 = 52

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 numberand 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=

6.4 : Implication



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Implication Ifp, then qwherepisthe antecedentand 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, thenp (known as converse of an implication)

6.4.a) Example:

Write two implications from each of the following compound statements.

6.4. b) ExerciseWrite 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 ifPQ

Implication 1 : If PQP = , then PQ

Implication 2 : If PQ , then PQP =

c) xis a multiple of 4 if and

only ifxis divisible by 4

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

Implication 2 : Ifxis 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= 3

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

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Compound Statement Answer

a) 10 a = 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 :

d) x+ 3 = 7 if and only if

x 8 = 18

Implication 1 :

Implication 2 :

e) BA if and only ifABA =

Implication 1 :

Implication 2 :

f) y2 4y =4 if and only if

y = 2

Implication 1 :

Implication 2 :

g) kis a perfect square if and

only if k is an integer

Implication 1 :

Implication 2 :

h) m is a negative number if and

only ifm3 is a negative number

Implication 1 :

Implication 2 :

i) 10 1 =z

1if and only ifz =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 :

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6.5 : Argument

Argument : is the process in making a conclusion based on the premise. Premise is a given statement

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 : Ifm < n, then m n < 0.

Premise 2 :..

Conclusion : 5 12 < 0.

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

multiple of 10.

Premise 2 :..

Conclusion : 2340 is a number with a last digit 0

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 the sum of the

interior angles are 540o

Premise 2 :..

Conclusion: MNOPQ has the sum of the interio

angles is 540o

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

Premise 2 : n(B) 0

Conclusion : set B

Premise 1 : Ifx+ 5 = 10, then x = 5

Premise 2 : x 5

Conclusion:.

5 Premise 1 : All factors of 4 are factors of 12

Premise 2 : 4 is a factor of 12

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

obtuse angle

Premise 2:...



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Conclusion : 4 is a factor of 4Conclusion : 100o is an obtuse angle

6 Premise 1 : If x= 3, thenx2 = 9

Premise 2 :x 3

Conclusion : x2 9

Premise 1 :If { }8,6,4,2x , thenxis an evennumber

Premise 2 : x is not an even number

Conclusion:.

7 Premise 1: If xandy are odd numbers,

then the product ofxand y is an odd

number

Premise 2 : 3 and 5 are odd numbersConclusion : Theproduct of 3 and 5 is an odd

number

Premise 1 : If KLM is an equilateral triangle,

then KL = LM = KM

Premise 2:...

Conclusion: KLM is an equilateral triangle

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

Premise 2 : p < 3

Conclusion : 6p < 18

Premise 1 : IfC 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 conclusion for a specific case based on a given general statement.

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 (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



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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) 17 = 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) + 517 = 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 2 182 = 90 2 472 = 90 2 958 = 90 2 16..

General conclusion

3 Given 2, 9, 16, 23,

And 2 = 2 + 7(0)

4. Given 3, 24, 81, 192,

and 3 = 3(1)3



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9 = 2 + 7(1)

16 = 2 + 7(2)

23 = 2 + 7(3)

..

General conclusion :..

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 24 = 3 2 27 = 3 3 210 = 3 4 213 = 3 5 2

General conclusion :

Questions According to Examination Format

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 ?


9 > 6 and 42 = 8

x >y if and only if x y > 0

5 is an even number


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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:

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:


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

34 = 12 or4

5= 1.25

x = 4 if and only ifx3 = 64


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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.

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 < 2

b) Complete the following argument.

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

Premise 2 : 5a 30Conclusion : .. .

c) Write down two implications based on the following;

Answer:

a.)


Combine any of the two statements

to form a false statement.

3r> 6 if and only if r> 2


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b.) Conclusion:

c.). Implication 1 : .

Implication 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 2 k

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:


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.


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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

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.


8 > 7 or 32 = 6

m3 = 1000 if and only if m = 10

y < x if and only if y > -x

03522

=+ mm

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


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

following pattern:

(5 marks)

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) 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 60 0 .


1)4(2129

1)3(255

1)2(217

1)1(23

3

3

3

3

+=+=

+=

+=


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c.) Complete the premise in the following argument:

Premise 1 : .

Premise 2 : .1809000

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 becomea 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.


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


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

Matrix

dc

bahas an inverse if and only if ad bc 0

[6 marks]

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]


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


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Year 2008 (June)

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

7 x 7 = 49 and (-7)2 = 49

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:


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

53 = 125 and -6 < -7


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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

n

21 x 1800 .

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

[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


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

76

x3

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


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6.3.1 b)


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

52

3

100is equal to 102

False

2

3

100is 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 False8 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

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6.3.2 b

p q True / False1 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 : Ifx3 = 64, thenx= 4

Implication 2 : Ifx= 4, then x3 = 64

c Implication 1 : If Abu is punished, then he was late to schoolImplication 2 : If Abu is late to school, then he will be punished

d Implication 1 : If x+ 3 = 7, then x 8 = 18Implication 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 = 2Implication 2 : Ify = 2, then y2 4y =4

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

Implication 2 : If k is an integer, then kis 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 numberiImplication 1 : If 10 1 =

z

1, then z =10

Implication 2 : Ifz =10, then 10 1 =z

1

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

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

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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, then x y > 0 .

If x y > 0, thenx >y.

2. i : Statement


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+ 5 105 Premise 2 : 90o < 0100 < 180

o

6 Conclusion : { }8,6,4,2x7 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 grass3 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,..

79


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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.

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= and5 < 2 @ 42 = 8 or5 < 2 @ 42 = 8 or 75.0

4

3=

b) a 6c) 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 If xy > , then xy 15, then m > 5.

If m > 5, then m > 5.



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June 2006

5. a) (i) True

(ii) False

b) If ABC is an equilateral triangle, then each of the interior angle of ABC is 60 0 .

If each of the interior angle of ABC is 60 0 , then ABC is an equilateral triangle.

c) If 00 18090 x , then0sin x is positive.

7. Nov 2006

a) (i) Some

(ii) All

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

c) LLK 8. June 2007

a) True

b)ABCD has no two parallel sides

c) (n 2 ) x 1800

d) Implication 1 : If matrix

dc

bahas an inverse then ad bc 0

Implication 2 : If ad bc 0 then

dc

bahas an inverse

9. Nov 2007

a) Some

b)23 is not a multiple of 3c) 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



27/27

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

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Chapter 6 II Math Reasoning ENHANCE