MOHD NAZAN BIN KAMARUL ZAMAN SMK. KOTA KLIAS, BEAUFORT

TEKNIK MENJAWAB MATEMATIK SPM 2010

SETS3 marks

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

On the diagrams in the answer space, shade

RQ P

RQ)(P')QPa) b

PR

QP

R

Q

QPa)

PR

QFirstly – label each part at the diagram with numbers or letters

2 53I 4 P = 1, 2, 3, 4

Q = 3, 4, 5

P ∩ Q = 3, 4

RQ)(P') b

PR

Q

Firstly – mark all the area at the diagram with numbers or letters

2 53I 4

P = 1, 2, 3, 4P’= 5Q = 3, 4, 5P’ ∩ Q = 5R = 2, 3 5 3, 2, RQ)(P'

Elimination method

SIMULTANEOUS LINEAR SIMULTANEOUS LINEAR EQUATIONSEQUATIONS

Substitution methodSubstitution method

Matrix methodMatrix method

4 MARKS

2 Calculate the value of d and of e that satisfy the following simultaneous linear equations: 8d − 9e = 5 2d − 3e = −1

(i)(ii)

(ii) x 4 2d(4) – 3e(4) = -1(4)

8d – 12e = - 4 (iii)

8d = 32

0 – 3e = -9

(iii) – (i)

339

e

e

Substitute e = 3 to (i)

8d – 9(3) = 5

8d – 27 = 5

8d = 5 + 27

8d – 9e = 5 (i)

4832

d

d

1 mark

1 mark

2 marks3 e and 4 d

Using matrices

8d − 9e = 52d − 3e = −1

1-5

ed

3 - 29- 8

3415

8293

61

15

8293

)9(2)3(81

1

eandded

ed

CAB

CBA

1 mark

2 marks

1 mark

2 Calculate the value of d and of e that satisfy the following simultaneous linear equations:

2723

331

ed

ed

Using matrices

273

23311

ed

65

273

13312

31

273

13312

)3(31)2(1

1

1

eandd

ed

ed

CAB

CBA

1 mark

2 marks

1 mark

QUADRATIC EQUATIONS- General Form- Factorisation

4 Marks

3. Solve the quadratic equation x7

32x2

Change to general form

0 3x72x

x73x2

7x32x

x7

32x

2

2

2

2

21 x and 3 x

1 2x 0 1 -2x 0 3 - x

01 -2x )3-(x

1 mark

2 marks

1 mark

MATRICESNOTES1. When the matrix has no inverse

2. MATRIX FORM

3. Formula of the inverse matrix

4. State the value of x and of y

4. The inverse matrix of

2 m3- 7

k1 is

7 43 2

a) Find the value of m and of k

b) Write the following simultaneous linear equations as matrix equation :

2x + 3y = - 14x + 7y = 5

Hence, using matrix method, calculate the value of x and of y

7432

21

2321

7432

12141

7432

)4(3)7(211 a)

mk

acbd

bcad

k = 2 and m = - 4

b) Write the following simultaneous linear equations as matrix equation :

2x + 3y = - 14x + 7y = 5

Hence, using matrix method, calculate the value of x and of y

711711

51

2437

21

51

7432

1

yandxyx

yx

CAB

CBAyx 1 mark

2 marks

1 mark

THE STRAIGHT LINE – 6 MARKS

REMEMBER :

1. Gradient

2. Equation of a line21

21

xxyym

cmxy 1

by

ax

3. Parallel lines , same gradient

4. Perpendicular lines , the product of their gradients = - 1

21 mm

121 mm

y

● R(4,12)

●Q

● P(3, -6)

0

Diagram 3

5. In Diagram 3,OPQR is parallelogram and O is the origin.

Find(a) the equation of the straight line PQ,(b) the y-intercept of the straight line QR

x

a) mPQ = mRO

mRO =

34

1204012

xxyy

12

12

mPQ = mRO = 3

m = 3 and P(3, -6) y = mx + c-6 = 3(3) + c-6 = 9 + c-6 – 9 = c- 15 = c

m = 3 and c = -15y = mx + cy = 3x - 15

b) mQR = mOP =

236

0306

xxyy

12

12

m = - 2 and R(4, 12)

y = mx + c12 = - 2(4) + c12 = - 8 + c12 + 8 = c20 = c

y-intercept of the straight line QR = 20

GRADIENT AND AREA UNDER A GRAPH

6. In the diagram, OPQ is the distance-time graph of a car traveling from town A to town B. The straight line RPS represents the distance-time graph of a van traveling from town B to town A

0 t 5 6

144

250

Distance from A (km)

Time(hrs)

P

Q R

S

Calculate thea) average speed, in km h-1

, of the car from town A to Bb) value of t if the van travelled at uniform speed.

a) Average speed = timedistancetotal

4240

= 60 km h-1

b)

t144

80t = 144 t = 1.2

= 80

LINES AND PLANES IN 3 DIMENSION

a) Line and Plane

b) Plane and plane

7. Diagram 10 shows a right prism. Right angled triangle SUT is the uniform cross-section of the prism

U

Q

ST

P

R

5 cm12 cm

20 cm

Identify and calculate the angle between the plane PSR and the plane PUTR.

Using open & close method

S

P

U

T

R

5 cm

i. Identify the plane PSR and the plane PUTR.

ii. open the plane PSR and the plane PUTR.

P R

TU

S

Identify three points when we joint together become a straight line.

The straight line is SPU or UPS ,

so the angle between the plane PRS and the plane PUTR is SPU or UPS

S U

P

12

20 SPU

'00 5730@96.302012 tan

D F

E

G

722

8. The diagram shows a solid formed by combining a right pyramid with a half cylinder on the rectangular plane DEFG.

DE = 7 cm, EF = 10 cm and the height of the pyramid is 9 cm. Clculate the volume, in cm3, of the solid. [ using =

Volume of pyramid + volume of half cylinder

210

9 x 10 x 7 x 31

height x base of Area x 31 pyramid of volume

192.5

10x 27 x

722 x

21

height x r x 21 cylinder half of volume

2

2

Volume of the combine solid = 402.5

CIRCLES : Perimeter and Area

1. Use the correct formulae

2. Substitute with the correct values.

9. In Diagram, BC and AD are archs of two different circle which have the same cente O

It is given that

a)The perimeter, in cm of the whole diagramb)The area, in cm of the shaded region

calculate,722 Using

14cmOB,30OBCand90OAED 00

O

C

BA

E

D

7cm

3242

73177

31714

777222

36060

7147222

3603014

)

DOEDCEBCOBperimetera

6164

6512

3151

3225

7722

36030

14722

360307

722

36060

)

2

22

OAEOBCODEareab

PROBABILITY

)()()(SnAnAP

10. Diagram 9 shows two boxes , P and Q . Box P contains four cards labeled with letters and box Q contains three cards labeled with numbers.

TSEB 764

Two cards are picked at random, a card from box P and another card from box Q .

a)List the sample space and the outcomes of the events .

b) Hence , find the probability that(i) a card labeled with letter E and a card labelled with an even number are picked

(ii) a card lebelled with letter E or a card labelled with an even number arepicked

P Q

a) {(B, 4), (B, 6), (B, 7), (E, 4), (E, 6), (E, 7), (S, 4), (S, 6), (S, 7), (T, 4), (T, 6), (T, 7)} Notes : 1. Accept 8 correct listings for 1 mark

b) i) {(E, 4), (E, 6)}

ii) {(E, 4), (E, 6), (E, 7), (B, 4), (B, 6), (S, 4), (S, 6), (T, 4), (T, 6)}

61@

122

43@

129

1(m)

1(m)

2(m)

1(m)1(m)

MATHEMATICAL REASORNING

a) State whether each of the following statement is true or false 12 > 5 and 1472

It is a false statement

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

6 is greater than 0

c) Make a general conclusion by induction for the sequence of number 7, 14, 27,….which follows the following pattern.

.................3)2(327

2)2(314

1)2(37

3

2

1

3

(2) n

+ n n = 1, 2, 3, …