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
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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
Mathematical Reasoning
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.
Mathematical Reasoning
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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Mathematical Reasoning
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 ?
Mathematical Reasoning 67
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:
Mathematical Reasoning 68
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.)
Mathematical Reasoning 69
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:
Mathematical Reasoning 70
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.
Mathematical Reasoning 71
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 .
Mathematical Reasoning 72
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.
Mathematical Reasoning 73
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]
Mathematical Reasoning 74
................................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:
Mathematical Reasoning 75
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
Mathematical Reasoning
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)
Mathematical Reasoning
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
Mathematical Reasoning
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
Mathematical Reasoning
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,..
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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
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Implication 2 : If x = -4 then x3 = -64