TEKNIK MENJAWAB MATEMATIK SPM 2010

MOHD NAZAN BIN KAMARUL ZAMAN SMK. KOTA KLIAS, BEAUFORT

SETS3 marks

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

\ ! P QROn the diagrams in the answer space, shade

a) P Q

b ) (P' Q) R

Q P R P R

Q

a) P QQ P R I2 3 4 5

Firstly label each part at the diagram with numbers or letters P = 1, 2, 3, 4 Q = 3, 4, 5 P Q = 3, 4

b ) (P' Q) R

Q P R I2 3 4 5

Firstly mark all the area at the diagram with numbers or letters P = 1, 2, 3, 4 P=5 Q = 3, 4, 5 P Q=5 R = 2, 3

(P' Q) R ! 2, 3, 5

SIMULTANEOUS LINEAR EQUATIONSElimination method Substitution method Matrix method

4 MARKS

2

Calculate the value of d and of e that satisfy the following simultaneous linear equations: 8d 2d 9e = 5 3e = 1 1 mark (iii) (i) 1 mark (i) (ii)

(ii) x 4

2d(4) 3e(4) = -1(4) 8d 12e = - 4 8d 9e = 5

Substitute e = 3 to (i) 8d 8d 9(3) = 5 27 = 5 8d = 5 + 27 8d = 3232 8 d !4 d!

(iii) (i)

0 3e = -9

9 e! 3 e!3

@ d ! 4 and e ! 3

2 marks

Using matrices 8d 2d 9e = 5 3e = 1 1 mark

8 - 9 d 5 2 - 3 e ! -1 A v B ! C B ! A1 C3 d 1 ! e 8(3) 2(9) 2 d 1 3 9 5 ! e 6 2 8 1 @d ! 4 and e ! 3

9 8

5 1

1 mark

2 marks

2

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

1 d e!3 3 3d 2e ! 27

Using matrices

1 1 d 3 3 ! e 27 3 2 v ! !1

1 mark

1 d 1 2 ! 3 e 1( 2) 1 (3) 3 1 3 1 d 1 2 3 ! 3 e 3 3 1 27 @d ! 5 and e ! 6

3 1 mark 27

2 marks

QUADRATIC EQUATIONS- General Form - Factorisation

4 Marks

3. Solve the quadratic equationChange to general form

2x 2 3 !x 7

2x 2 3 !x 7 2x 2 3 ! x 7 2x 2 3 ! 7 x 2x 2 7 x 3 ! 01 mark

(x - 3 ) 2x - 1! 0x -3!0

1 mark

2x - 1 ! 0 2x ! 1

x !3

and

1 x! 2

2 marks

MATRICESNOTES 1. 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 3 is 1 7 - 3 4 7 k m 2 a) Find the value of m and of k

b) Write the following simultaneous linear equations as matrix equation : 2x + 3y = - 1 4x + 7y = 5 Hence, using matrix method, calculate the value of x and of y

a)

2 3 1 d b 1 ! c a 2(7) 3(4) 4 7 ad bc 3 1 2 ! 4 7 14 12 1 2 3 1 2 3 ! m 2 4 7 2 k

k = 2 and m = - 4

b) Write the following simultaneous linear equations as matrix equation : 2x + 3y = - 1 4x + 7y = 5 Hence, using matrix method, calculate the value of x and of y2 4

3 x 1 ! 7 y 5 v !!1

1 mark

3 1 x 1 7 ! y 2 4 2 5 x 11 ! y 7 @ x ! 11 and y ! 7

1 mark

2 marks

THE STRAIGHT LINE 6 MARKSREMEMBER :

y1 y2 1. Gradient m ! x1 x22. Equation of a linex y ! 1 a b 3. Parallel lines , same gradient

y ! mx c

m1 ! m2

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

m1m2 ! 1

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

y R(4,12) Q

0 P(3, -6) Diagram 3

x

Find (a) (b)

the equation of the straight line PQ, the y-intercept of the straight line QR

a) mPQ = mRO y 2 y1 mRO = x 2 x1 ! ! 12 0 40

y 2 y1 b) mQR = mOP = x 2 x1 ! ! 60 30

12 4 !3 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 = -15 y = mx + c y = 3x - 15

6 3 ! 2 m = - 2 and R(4, 12) y = mx + c 12 = - 2(4) + c 12 = - 8 + c 12 + 8 = c 20 = 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 ADistance from A (km)

R 250

Q

P 144

S Time(hrs) 0 t 5 6

Calculate the a) average speed, in km h-1 , of the car from town A to B b) value of t if the van travelled at uniform speed.

a) Average speed = total distance

time

!

240 4

= 60 km h-1

b)

144 = 80 t80t = 144 t = 1.2

LINES AND PLANES IN 3 DIMENSIONa) b) Line and Plane Plane and plane

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

P 12 cm Q 5 cm R

20 cm U T S

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

Using open & close methodi. Identify the plane PSR and the plane PUTR.P 5 cm R

ii. open the plane PSR and the plane PUTR. S

P

R

U T S

U

T

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

P

20

S

12

U

SPU 12 tan U ! 20 ! 30.960 @ 300 57 '

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

G D EDE = 7 cm, EF = 10 cm and the height of the pyramid is 9 cm. Clculate the volume, in cm3, of the solid. 22 [ using T = 7

F

Volume of pyramid + volume of half cylindervolume of pyramid ! 1 x Area of base x height 3 1 ! x 7 x 10 x 9 3 ! 210

volume o hal cylinder !

1 x T r 2 x height 22

1 22 7 x x 10 ! x 2 7 2 ! 192.5

Volume of the combine solid = 402.5

CIRCLES : Perimeter and Area1. 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 D C E

O

7cm

A

B

It is given that O ED ! 90 0 and OBC ! 30 0 , OB ! 14cm

22 Using T ! , calculate 7a) The perimeter, in cm of the whole diagram b) The area, in cm of the shaded region

a ) perimeter ! OB BC CE ED DO 22 30 !14 v 2 v v14 7 7 360 22 60 v 2v v 7 7 7 360 1 1 ! 14 7 7 7 7 3 3 2 ! 42 3

b) area ! ODE OBC OAE

60 22 2 30 22 2 v v14 ! v v7 360 7 360 7 30 22 2 v v7 360 7 2 1 5 ! 25 51 12 3 3 6 1 ! 64 6

PROBABILITY

n( ) P( ) ! n( S )

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.

B

E P

S

T

4

6 Q

7

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 are picked

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

2 1 @ 12 6ii) {(E, 4), (E, 6), (E, 7), (B, 4), (B, 6), (S, 4), (S, 6), (T, 4), (T, 6)}

1(m) 1(m)

9 3 @ 12 4

1(m) 1(m)

MATHEMATICAL REASORNING

a) State whether each of the following statement is true or false 12 > 5 and 7 2 ! 14 It is a false statement

b) Write down Premise 2 to complete the following argument Premise 1 Premise 2 : If x is greater than zero, then x is a positive number 6 is greater than 0 : __________________________________________ 6 is a positive number

Conclusion :

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

7 ! 3 ( 2)1 1 14 ! 3 ( 2) 2 2 27 ! 3 (2) 3 3 .... ! .......... ...3 (2)n

+n

n = 1, 2, 3,