Teknik Menjawab Matematik Spm 2010
Transcript of Teknik Menjawab Matematik Spm 2010
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,