Modul t4

QUADRATIC EQUATION
Past Year SPM Questions
1. Solve the equation 3x2 = 2(x – 1) + 7.
2. Solve the quadratic equation .2
24
3 2 p
p
3. Solve the equation .2
1
52 2
m
mm
4. Solve the quadratic equation mm
2
7
12 2
.
5. Solve the quadratic equation k
k
3
3
52 2
.

6. Solve the quadratic equation .21532 2
xxx
7. Solve the quadratic equation 6
2
)1(3
x
xx
8. Using factorization, solve the following quadratic equation 2
5 8 6 5x x x
9. Using factorization, solve the following quadratic equation 2
4 15 17x x
10. Solve the quadratic equation 2
(2 1) 2x x x
11. Solve the quadratic equation
6 3
1
2
x
x
x

SETS
Questions based on the Examination Format (Paper 2)
1. The Venn diagram in the answer space shows sets A, B and C. It is given that
ξ = A ∪ B ∪ C.
On the diagram provided in the answer spaces, shade the area of ,
(a) C′ ∩ (A ∪ B)
(b) (A′ ∩ C) ∪ (B ∩ C)
Answer :
2.
9
8-x
7-x
89-x7
Given that CBA and that 34)(n , find
(a) the value of x
(b) )'( CBAn
A B
x
C

3. Given that the universal set, RQP .
The Venn diagrams below show sets P, Q and R. On the diagrams, shade
(a) QP (b) 'RP
P
Q
R
P
Q
R
(c) '' RQP
P
Q
R
4. Given the universal set, xxx ,2510:{ is an integer}
Set P = {x : x is a prime number}
Set Q = {x : x is a multiple of 4} and
Set R = {x: x is a number where one of its digits is more than 7}
(a) Find the elements of set P
(b) Find the elements of set P R.
(c) Find n( RQP )’.
5. The Venn diagram shows the set E, F and G. If the universal set is GFE , in
the answer part, shade the regions for
(a) FE
(b) GFE
Answer :
(a) E F G (b) E F G

6. Given P = {3, 5}
Q = {2, 4, 7, 8}
R = {1, 3, 5, 7, 9}
and universal set RQP
Draw a Venn diagram to show the relationship between sets P, Q and R.
List all the elements in set .RP
7. The Venn diagram in the answer space shows sets P, Q and R. On the diagram
provided in the answer spaces, shade
a) the set Q’ R,
b) the set P Q R
Answer:
a) b)
8. The Venn diagram in the answer space shows sets P, Q and R. Given that
universal set
ξ = P Q R. In the answer spaces, shade
a) Q (P R),
b) (Q R)’ P.
Answer:
a) b)
P
Q
R
Q
R
P
P R
P
Q R
Q

9. Given that universal set ξ = { x : 31 ≤x ≤ 45, x are integers}.
Set P = { x : x are prime numbers }
Set Q = { x : x are odd numbers }
Set R = { x : x is a number where the sum of its digits is more than 8 }
List out all the elements in set R,
Find n(P Q R)’.
10. The Venn diagram in the answer space shows sets P, Q and R. Given that the
universal set
ξ = P Q R. On the answer spaces, shade
a) (P ∩ Q) U R b) (P U R)’ ∩ Q
Past Year SPM Questions
Paper 2
1. June 2004
The Venn diagram in the answer space shows sets P, Q and R such that the universal set
ξ = P Q R.
On the diagram in the answer space, shade
the set Q’.
the set P (Q R).
Answer:
P
Q
R
P
Q
R
Q
P
R
Q
P
R

2. November 2004
The Venn diagram in the answer space shows sets A, B and C. On the diagram provide in
the answer spaces, shade
a) the set A B’
b) the set A B C’.
Answer:
a) b)
3. June 2005
ξ = P Q R.
the set P R’.
the set (P’ Q’) R.
Answer:
a) b)
4. June 2006
ξ = P Q R.
A B
C
A B
C
Q
P
R
Q
P
R

the set P R.
the set (P R) Q’.
Answer:
5. November 2006
The Venn diagram in the answer space shows sets P, Q and R such that the universal set,
P Q R .
On the diagrams in the answer space, shade
the set '
P Q
the set 'P Q R
Answer:
6. June 2007
The Venn diagrams in the answer space shows sets K, L and M such that the universal
set, MLK .
On the diagrams in the answer space, shade the set
a) MK' ,
b) )M'(LK . [3 marks]
Answer :
a)
Q
P
R
Q
P
R
P
Q
R
P
Q
R
L
K
M

b)
7. Nov 2008
The Venn diagrams in the answer space shows sets P, Q and R such that the universal
set, RQP .
On the diagrams in the answer space, shade [3 marks]
a) QP ,
b) R)(P'
Q .
Answer
a)
b)
L
K
M
R
P Q
R
P Q

MATHEMATICAL REASONING
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
Answer : a) ………………………….
Implication 1 :
Implication 2 :
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:
All multiples of 2 are divisible by 4.
y < x if and only if –y > -x

Answer : a) ……………………
Implication 1 :
Implication 2 :
Year 2004 (Nov)
a) State whether the following statement is true or false.
b) Write down two implications based on the following sentence
Implication 1 :
Implication 2 :
c) Complete the premise in the following argument :
Premise 1 : All hexagons have six sides.
Premise 2 : …………………………………………………………………………….……
Conclusion : PQRSTU has six sides.
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.
Make a general conclusion by induction for a list of number 3, 17, 55, 129, … which
follows the following pattern:
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 : …………………………………………………………………
Year 2006 (July)
State whether each of the following statements is true or false.
(i) 4643
(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
60 0
.

Year 2006 (Nov)
(a) Complete each of the following statements with the quantifier “all” or “some” so
that it will become a true statement
………………………………… of the prime numbers are odd numbers.
………………………………... pentagons have five sides.
(b) State the converse of the following statement and hence determine whether its
converse is true or false.
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)
State whether the following statement is true or false.
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.
Year 2007 (Nov)
Premise 1 : If M is a multiple of 6, then M is a multiple of 3.
Premise 2 : ……………………………………………………..
Some even numbers are multiples of 3
If x > 9 , then x > 5

Conclusion : 23 is not a multiple of 6.
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
… = …………
Write down two implications based on the following statement:
“ p – q > 0 if and only if p > q”
Implication 1 :……………………………………………………………………
Implication 2 : …………………………………………………………………...
Year 2008 (June)
State whether the following compound statement is true or false.
7 x 7 = 49 and (-7)2 = 49
Write down two implications based on the following compound statement:
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.
Year 2008 (Nov)
KLM is an isosceles triangle if and only if two angles in KLM are equal.

State whether the following compound statement is true or false:
Write down two implications based on the following compound statement:
53
= 125 and -6 < -7
x3
= -64 if and only if x = -4.

SIMULTANEOUS LINEAR EQUATIOAN
Questions based on Examination Format
1) 443 nm
82nm
2) 1232 kh
104 kh
3) 832 vu
95vu
4) 153 nm
2
3
2
nm
5) 523 wv
46 wv
6) 232 qp
3
2
1
qp

7) 116
4
1
xx
1283 yx
8) 722 mh
164 mh
9) 932 sr
36sr
10) 643 wv
82wv

LINES AND PLANES IN 3 DIMENSIONS
9. 1 Angle Between Lines And Planes
9.1.1 a)Based on the diagram, calculate the angle between the line and the plane given
Example 1: Plane :EFGH
Line :GC
Angle : CGH
tan CGH =
GH
CH
=
8
4
CGH = 26.57o / 26o
34’
1. a) Plane : ABCD
Line : DV
Angle :
b) Plane : SRLK
Line : QL
Angle :
Example 2 : Plane : PSK
Line : KR
Angle : RKS
tan RKS =
KS
SR
2. a) Plane : CDEH
Line : FD
Angle :
b) Plane : URST
Line : RX
Angle :
P Q
RS
LK
G H
EF
DA
B C
6 cm
8 cm
4 cm
P Q
RS
LK
12 cm
7 cm
5 cm
A B
CD
V
10 cm
8 cm
3 cm
12 cm
5 cm
G H
EF
DA
B C
15 cm
6 cm
8 cm
R
S
U T
YX
24 cm
7 cm
4 cm

=
5
12
RKS = 67.38o / 67o
23’
Example 3 : Plane : JKLM
Line : NK
NM = 11 cm
Angle : NKM
KM = 22
912 = 15 cm
tan NKM =
KM
NM
=
15
11
NKM = 36.25o / 36o
15’
3. a) Plane : ABCD
Line : AV
b) Plane : ABCD
Line : DG
9. 2 Angle Between Two Planes
a) Calculate the angle between the two planes.
Example 1: Plane EFGH
and plane GHDA 1. a) Plane KLSP and
plane JKLM
b) Plane PSWV and plane
VUXW
J K
LM
N
12 cm
9 cm
A B
CD
V
8 cm
6 cm
4 cm
G H
EF
DA
B C
6 cm
5 cm
12 cm
G H
EF
DA
B C
9 cm
RS
X
W
V
U
4 cm
6 cm
Q
M
RS
P
L
12 cm
15 cm

Angle :
DHE = AGF
tan DHE =
GF
AF
=
6
9
DHE = 56.31o /
56o19’
Example 2 : Plane PQLK
and plane SRLK
Angle :
QLR = PKS
tan QLR =
LR
QR
=
7
10
QLR = 55o
2. a) Plane ABCD and
plane ADEF
b) Plane URST and plane
XRSY
Example 3 : Plane TRQ
and plane SRQP
3. a) Plane ABCD and
plane ABV
b) Plane PQSR and plane
PQKL
A B
CD
V
P Q
RS
LK
T
RS
QP
8 cm
6 cm
P Q
RS
LK
12 cm
10 cm
7 cm
BA
F
E
D C
20 cm
10 cm
13 cm
5 cm
5 cm
11 cm
4 cm
8 cm
5 cm 4 cm
3 cm
R
S
U T
YX
12
cm
9 cm
5 cm

Angle : TRS
tan TRS =
RS
TS
=
11
4
QLR = 19.98o / 19o59’
PROBABILITY
July 2006
6. Table below shows the number of teachers in a two-session school.
Session
Number of teachers
Men Women
Morning 6 10
Afternoon 4 8
Two teachers from the school are chosen at random to attend an assembly of Teacher’s
Day at the state level.
Calculate the probability that both teachers chosen
are men,
are from the same session
8. June 2007
Table 1 shows the number of a group of students in Form 1 Alpha and Form 1 Beta
who
are entitled to receive school bags.
Form
Gender
1 Alpha 1 Beta
Boys 3 6
Girls 5 2
Table 1
Two students from the group are chosen at random to receive a school bag each.
Find the probability that both students chosen
A are boys
B are girls from the same class
[5 marks]
[5 marks]

9. November 2007
Diagram 4 shows ten labeled cards in two boxes.
Box P Box Q
Diagram 4
A card is picked at random from each of the boxes.
By listing the outcomes, find the probability that
A both cards are labeled with a number
B one card is labeled with a number and the other card is labeled with a letter.
[5 marks]
11. November 2008
Diagram 10 shows three numbered cards in box P and two cards labeled with letters in
box Q.
P Q
Diagram 10
A card is picked at random from box P and then a card is picked at random from box Q.
By listing the sample of all possible outcomes of the event, find the probability that
a card with an even number and the card labeled Y are picked,
a card with a number which is multiple of 3 or the card labeled R are picked.
[5 marks]
A B2 C D E3 4 F G
2 63 Y R

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