Modul t4

QUADRATIC EQUATIONPast Year SPM Questions

1. Solve the equation 3x2 = 2(x – 1) + 7.

2. Solve the quadratic equation

34

p2= p2

+2 .

3. Solve the equation

2m2 + 5 mm + 1

= 2 .

4. Solve the quadratic equation2 m2−1=−7

2m

.

5. Solve the quadratic equation

2 k2 − 53

= 3 k.

6. Solve the quadratic equation 2 x2+3 x=15+2 x .

7. Solve the quadratic equation

3 x ( x−1 )2

=x+6

8. Using factorization, solve the following quadratic equation

9. Using factorization, solve the following quadratic equation

10. Solve the quadratic equation

11. Solve the quadratic equation

SETSQuestions 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.

Given that ξ=A∪B∪C and that n( ξ )=34 , find

(a) the value of x(b) n( A∩B∩C ' )

3. Given that the universal set, ξ=P∪Q∪R . The Venn diagrams below show sets P, Q and R. On the diagrams, shade

A B

x

C 9

8-x7-x

89-x7

(a) P∩Q (b) P∪R '

(c) P '∩Q∩R '

4. Given the universal set, ξ=¿¿ 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(P∪Q∪R )’.

5. The Venn diagram shows the set E, F and G. If the universal set isξ=E∪F∪G , in the answer part, shade the regions for (a) E∪F

(b) ( E∩F )∪G 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

PQ

R

PQ

R

PQ

R

RQP

P R

P

Q RQ

Q

R

P

Q

R

Draw a Venn diagram to show the relationship between sets P, Q and R.List all the elements in set P∪R .

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 = P ξ Q R. In the answer spaces, shade a) Q (P R), b) (Q R)’ P.

Answer: a) b)

9. Given that universal set = { x : 31 ≤x ≤ 45, x are integers}.ξ 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,

P

P

Q

RP

Q

R

QP

R

QP

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 = P ξ 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 = P ξ ¿ Q ¿ R.

On the diagram in the answer space, shadethe set Q’.the set P¿ (Q ¿ R).

Answer:

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’

A BC B

C

QP

R

QP

R

QP

R

QP

R

b) the set A ¿ B ¿ C’.Answer: a) b)

3. June 2005

The Venn diagram in the answer space shows sets P, Q and R such that the universal set = P ξ Q R.

On the diagram in the answer space, shade the set P R’. the set (P’ Q’) R.

Answer:a) b)

4. June 2006

The Venn diagram in the answer space shows sets P, Q and R such that the universal set = ξP Q R.On the diagram in the answer space, shadethe 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,

.On the diagrams in the answer space, shade

the set

A

P

QR

P

QR

L

KM

L

KM

the set

Answer:

6. June 2007The Venn diagrams in the answer space shows sets K, L and M such that the universal set, ξ=K∪L∪M .On the diagrams in the answer space, shade the set

a) K'∩M ,

b) K ∪( L∩M' ) . [3 marks]

Answer :a)

b)

7. Nov 2008The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ=P∪Q∪R .On the diagrams in the answer space, shade [3 marks]

a) P∩Q ,

R

P Q

R

P Q

b) ( P'∩Q )∪R .

Answer

a)

b)

MATHEMATICAL REASONINGSPM 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;

P ⊂ R if and only if R ' ⊂ P '

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: 52=10

Statement 2: 10×10=100

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

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.

8 > 7 or 32 = 6

m3 = 1000 if and only if m = 10

All multiples of 2 are divisible by 4.

y < x if and only if –y > -x

2 m2+5 m−3=0

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:

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 = {12 , 15 , 18 } are divisible by 3 or the elements of set B = {4 , 6 , 8 } 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.

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

3=2(1)3+117=2(2)3+155=2(3)3+1129=2(4 )3+1

(i) 3√64=4

(ii.) -5 > - 8 and 0.03 = 3 ¿ 10−1

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

.

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 K ∩ L= L

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)

Write down Premise 2 to complete the following argument:

Some even numbers are multiples of 3

If x > 9 , then x > 5

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.

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:

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.Year 2008 (Nov)

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

Write down two implications based on the following compound statement:

SIMULTANEOUS LINEAR EQUATIOANQuestions based on Examination Format

1) 3 m + 4 n = 4

m − 2 n = 82) 2 h − 3 k = 12

4 h + k = 10

x3 = -64 if and only if x = -4.

3) 2 u − 3 v = 8

u + 5 v = −94) 3 m − n = 15

m − 2

3n = 2

5) 3 v − 2 w = 5

6 v − w = 46) 2 p + 3 q = −2

12

p − q = 3

7)

14

x − 6 x = 11

3 x + 8 y = 12

8) 2 h − 2 m = 7

4 h + 6 m = −1

G H

EF

DA

B C

6 cm

8 cm

4 cm

A B

CD

V

10 cm

8 cm

3 cm

9) 2 r − 3 s = 9

r + 6 s = −310) 3 v + 4 w = −6

v − 2 w = 8

LINES AND PLANES IN 3 DIMENSIONS9. 1 Angle Between Lines And Planes

9.1.1 a)Based on the diagram, calculate the angle between the line and the plane givenExample 1: Plane :EFGH Line :GC

1. a) Plane : ABCD Line : DV

b) Plane : SRLK Line : QL

P Q

RS

LK

12 cm

7 cm

5 cm

G H

EF

DA

B C

15 cm

6 cm

8 cmP Q

RS

LK

RS

U T

YX

24 cm

7 cm

4 cm

A B

CD

V

8 cm

6 cm

4 cm

J K

LM

N

12 cm

9 cmG H

EF

DA

B C6 cm

5 cm

12 cm

Angle : ∠CGH

tan ∠CGH =

CHGH

=

48

∠CGH = 26.57o / 26o 34’

Angle : Angle :

Example 2 : Plane : PSK Line : KR

Angle : ∠RKS

tan ∠RKS =

SRKS

=

125

∠RKS = 67.38o / 67o 23’

2. a) Plane : CDEH Line : FD

Angle :

b) Plane : URST Line : RX

Angle :

Example 3 : Plane : JKLM Line : NK NM = 11 cm

3. a) Plane : ABCD Line : AV

b) Plane : ABCD Line : DG

12 cm

5 cm

J K

QM

RS

P

L

20 cm

12 cm 15 cm

G H

EF

DA

B C

P Q

RS

X

W

V

U

7 cm

4 cm 6 cm

5 cm

Angle : ∠ NKM

KM = √122+92 = 15 cm

tan ∠NKM =

NMKM

=

1115

∠NKM = 36.25o / 36o 15’

9. 2 Angle Between Two Planes

a) Calculate the angle between the two planes.Example 1: Plane EFGH and plane GHDA

Angle : ∠ DHE = ∠ AGF

tan ∠DHE =

AFGF

=

96

∠DHE = 56.31o / 56o19’

1. a) Plane KLSP and plane JKLM

b) Plane PSWV and plane VUXW

8 cm

6 cm

9 cm

BA

F

E

D C

20 cm

10 cm

13 cm

P Q

RS

LK

12 cm

10 cm

7 cm

RS

U T

YX

12 cm

9 cm

5 cm

T

RS

QP

A B

CD

V

8 cm

5 cm

P Q

RS

LK

Example 2 : Plane PQLK and plane SRLK

Angle : ∠ QLR = ∠ PKS

tan ∠QLR =

QRLR

=

107

∠QLR = 55o

2. a) Plane ABCD and plane ADEF

b) Plane URST and plane XRSY

Example 3 : Plane TRQ and plane SRQP

Angle : ∠ TRS

tan ∠TRS =

TSRS

=

411

∠QLR = 19.98o / 19o59’

3. a) Plane ABCD and plane ABV

b) Plane PQSR and plane PQKL

PROBABILITYJuly 2006

6. Table below shows the number of teachers in a two-session school.

SessionNumber of teachersMen Women

5 cm

5 cm

11 cm

4 cm 4 cm

3 cm

A B2 C D E3 4 F G

2 63 Y R

Morning 6 10Afternoon

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 chosenare 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

Gender1 Alpha 1 Beta

Boys 3 6Girls 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] 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 2008Diagram 10 shows three numbered cards in box P and two cards labeled with letters in box Q.

[5 marks]

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]