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1. Modul ENRICHMENT Matematik SPM 2014
Tajuk : Variations ( Ubahan )
Masa: 30 minutes / minit
1. If p q2
and q = 3 when p = 36, the value of q when p = 64 is
Jika p q2
dan q = 3 apabila p = 36, nilai q apabila p = 64 ialah
A ½
B 3
C 4
D 6
2. If R varies directly as S and R = 10 when S = 25, find the value of R
when S = 9.
Jika R berubah secara langsung dengan S dan R=10 apabila S= 25, cari nilai R apabila
S=9.
A 2
B 3
C 5
D 6
3. Given that M 2
T
P
and M = 4 when P = 6 and T = 3, express M in terms of P and T.
Diberi M 2
T
P
dan M=4 apabila P=6 dan T=3, ungkapkan M dalam sebutan P dan T.
A M = 2
6
T
P
B M = 2
3
T
P
C M = 3PT2
D M =
4
2
PT
4. Given that E
GF
1
and E =
3
1
when F = 12 and G = 16, find the value of G when E =
4 and F = 8.
Diberi E
GF
1
dan E =
3
1
apabila F = 12 dan G = 16, cari nilai G apabila E = 4 dan F
= 8.
A
4
1
2. B
2
1
C 2
D 4
5. It is given that y varies directly as the square root of x and y = 6 when x = 9. Calculate the
value of x when y = 10.
Diberi y berubah secara langsung dengan punca kuasa dua x dan y=6 apabila x=9.
Hitungkan nilai x apabila y=10.
A 5
B 10
C 15
D 25
6. P varies directly as the cube of q and inversely as r. The relationship between P, q and r is
P berubah secara langsung dengan kuasa tiga q dan secara songsang dengan r. Hubungan
P, q dan r ialah
A P
r
q3
B P
2
1
3
1
r
q
C P
r
q3
D P 2
3
r
q
7. Table 1 shows some values of variables m, n and h such that m varies directly as the square
root of n and inversely as h.
Jadual 1 menunjukkan beberapa nilai pembolehubah m, n dan h dimana m berubah secara
lansung dengan punca kuasa dua n dan secara songsang dengan h.
m 3 3
n 36 81
h 10 15
Table 1/ Jadual 1
Find the relationship between m, n and h.
Carikan hubungan di antara m, n dan h.
A m =
h
n5
3. B m =
h
n
C m =
h
n3
D m =
h
n23
8. It is given that y varies directly as xm
and y = 8 when x =1. Calculate the value of m when
y = 72 and x = 3.
Diberi y berubah secara langsung dengan xm
dan y = 8 apabila x =1. Hitung nilai m
apabila y = 72 dan x = 3.
A 1
B 2
C 3
D 4
9. Table 2 shows the relationship between three variables w, t and h.
Jadual 2 menunjukkan hubungan antara tiga pembolehubah w, t dan h.
w t h
2 2 4
4 m 9
Table 2 / Jadual 2
Given that w
h
t , calculate the value of m.
Diberi w
h
t , hitung nilai m.
A 3
B 6
C 12
D 18
10.
p 1 2
q 12 3
Table 3 /Jadual 3
Based on Table 3, if q varies inversely as the square of p. Find the relation between q
and p.
Berdasarkan Jadual 3, jika q berubah secara songsang dengan kuasa dua p. Cari
hubungan antara p dan q.
4. A q = 12p2
B q =
4
3
p2
C q = 2
12
p
D q = 2
4
3
p
ANSWER VARIATION
1 C 6 C
2 D 7 A
3 A 8 B
4 A 9 B
5 D 10 C
5. Modul ENRICHMENT Matematik SPM 2014
Tajuk : Earth as a sphere ( Bumi sebagai Sfera )
Masa: 200 minutes / minit
1. F(50ºS, 70ºE), G, H and K are four points on the earth’s surface. F, G and H are on the
same latitude such that FG is the diameter. The longitude of H is 45ºW.
F(50ºS, 70ºT), G, H and K merupakan empat titik pada permukaan bumi. F, G dan H
adalah pada latitud yang sama dan FG ialah diameter. Longitud H ialah 45ºB.
(a) Find the longitude of G. [2m]
Cari longitud bagi G.
(b) An aeroplane flew due west from F to H. It then flew 4800 nautical miles due
north to K. Given that its average speed for the whole journey was 680 knots,
calculate
Sebuah kapal terbang berlepas ke arah barat dari F ke H. Selepas itu, ia
menuju ke arah utara ke K sejauh 4800 batu nautika. Diberi purata laju
sepanjang perjalanan ialah 680 knot , hitung
(i) the latitude of K, [3m]
Latitud bagi K,
(ii) the distance, in nautical miles, from F to H, [3m]
Jarak, dalam batu nautika, dari F ke H,
(iii) the time taken to complete the journey. [4m]
Masa diambil untuk seluruh perjalanan.
Answer:
Jawapan:
(a)
(b) (i)
(ii)
(iii)
6. 2. G (60ºS, 20ºW) and H are two points on the surface of the earth where GH is a
diameter of the common parallel of latitude.
G (60ºS, 20ºB) dan H ialah dua titik di permukaan bumi di mana GH merupakan
diameter selarian latitud.
(a) Find the latitude of H. [2m]
Cari latitud bagi H
(b) Given GL is a diameter of the earth. On the diagram in the answer space, mark
the locations of H and L. Hence, state the location of L. [3m]
Diberi GL ialah diameter bumi, pada rajah di ruang jawapan, tandakan
kedudukan H dan L. Seterusnya, nyatakan kedudukan L.
(c) Calculate the shortest distance, in nautical miles, from H to the South Pole.[2m]
Hitungkan jarak terpendek dalam batu nautika dari H ke Kutub Selatan.
(d) An aeroplane took off from G and flew due east along the common parallel of
latitude at an average speed of 450 knots. The aeroplane took 8 hours to
reach a point P. Calculate
Sebuah kapal terbang berlepas dari G dan menuju ke timur melalui selarian
latitud dengan purata laju 450 knot. Ia mengambil masa 8 jam untuk sampai ke
titik P. Hitungkan
(i) the distance, in nautical miles, from G to P, [2m]
Jarak dari G ke P dalam batu nautika
(ii) the longitude of P. [3m]
Longitud bagi P
Answer:
Jawapan:
(a)
(b)
(c)
Equator/ khatulistiwa
S/S
G
N/U
7. (d)
3. P (51ºN, 20ºW) and Q are two points on the Earth’s surface. PQ is a diameter of the
latitude.
P (51ºU, 20ºB) dan Q merupakan dua titik di permukaan bumi. PQ ialah diameter latitud.
(a) Find the longitude Q. [2m]
Cari longitud bagi Q
(b) Given that PR is the Earth’s diameter, mark the positions of Q and R on the
diagram provided below. Hence, state the position of R. [2m]
Diberi PR ialah diameter Bumi, label kedudukan bagi Q dan R pada rajah
diberikan. Seterusnya nyatakan kedudukan bagi R
(c) Calculate the shortest distance, in nautical miles, from Q to the North Pole.[3m]
Hitung terpendek dari Q ke Kutub Utara dalam batu nautika
(d) An aeroplane flew due west from P along the latitude with an average speed of
500 knots.The aeroplane took 9 hours to reach a point M. Calculate
Sebuah kapal terbang berlepas dari P ke arah barat mengikut selarian latitud
dengan purata laju 500 knot. Kapal terbang tersebut mengambil masa 9 jam
untuk sampai ke titik M. Hitung
(i) the distance, in nautical miles, from P to M, [2m]
Jarak dari P ke M dalam batu nautika
(ii) the longitude of M. [3m]
Longitud bagi M
Answer:
Jawapan:
(a)
(b)
(c)
8. (d) (i)
(ii)
4. R (40ºN, 80ºW) , S and T are three points on the surface of the earth. RS is the
diameter of a parallel of latitude 40ºN. T is 3600 nautical miles to the south of R.
R (40ºU, 80ºB) , S dan T merupakan tiga titik di permukaan bumi. RS ialah diameter
selarian latitud 40ºU. T terletak 3600 batu nautika ke selatan R.
(a) State the longitude of S. [2m]
Nyatakan longitud bagi S
(b) Find the latitude of T. [4m]
Cari latitud bagi S
(c) Calculate the shortest distance, in nautical miles, from R to S measured along
the surface of the earth. [2m]
Hitung jarak terpendek dari R ke S pada permukaan bumi dalam batu nautika d
(d) A ship sailed from S to R along the common parallel of latitude and then due
south to T. The total time taken for the journey was 20 hours. Calculate the
average speed of the ship for the whole journey. [4m]
Sebuah kapal berlayar dari S ke R mengikut selarian latitud dan menuju selatan
ke T. Jumlah masa untuk perjalanan ialah 20jam. Hitungkan purata laju kapal
tersebut untuk jumlah perjalanan
Answer:
Jawapan:
(a)
(b)
9. (c)
(d)
5. G (65ºN, 10ºW) and H (65ºN, ºE) are two points on the surface of the earth. GH
is a diameter of the parallel of latitude 65ºN. A yacht sailed with an average
speed of 375 knots from G to H using the shortest distance, as measured along
the surface of the earth.
G (65ºU, 10ºB) dan H (65ºU, ºT) merupakan dua titik di permukaan bumi. GH
ialah diameter selarian latitud 65ºU. Sebuah kapal berlayar dengan purata laju 375
knot dari G ke H meenggunakan jarak terpendek diukur pada permukaan bumi.
(a) State the value of . [2m]
Nyatakan nilai
(b) Mark the positions of points G and H on the diagram. [4m]
Labelkan kedudukan bagi G dan H pada rajah diberi
(c) Calculate the shortest distance, in nautical miles, from point G to point H. [3m]
Hitungkan jarak terpendek dari G ke H dalam batu nautika
(d) Calculate the total time, in hours, taken for the yacht to sail from point G to
point H. [3m]
Hitungkan jumlah masa bagi kapal berlayar dari titik G ke titik P dalam jam
Answer:
Jawapan:
(a)
N/U
10. (b)
(c)
(d)
6. P (50ºs,40ºw), Q (50ºS, 90ºE), R and T are four points on the earth’s surface. PR
is the diameter of the common latitude.
P (50ºs,40ºB), Q (50ºS, 90ºT), R danT merupakan empat titik pada permukaan bumi. PR
ialah diameter latitud.
(a) (i) State the longitude of R. [2m]
Nyatakan longitud bagi R
(ii) Calculate the ratio of the distance from P to R through the South Pole
to its distance through North Pole. [3m]
Hitungkan nisbah jarak dari P ke R melalui kutub Selatan dan melalui
kutub Utara
(iii) Calculate the distance, in nautical miles, from P to Q due east as
measured along the common latitude. [4m]
Hitung jarak dalam batu nautika dari P ke arah Timur ke Q diukur
sepanjang laltitud.
(b) Find the latitude of T which is situated 3900 nautical miles due North
of Q. [3m]
Cari latitud T yang mana terletak 3900 batu nautika ke Utara Q
Answer / Jawapan:
S/S
O.0º
11. (a) (i)
(ii)
(iii)
(b)
7. G (45°N, 108°W ), H, J and K are four points on the surface of earth. GJ is a diameter of the
earth.
G (45°U, 108°B ), H, J dan K ialah empat titik pada permukaan bumi. GJ ialah diameter
bumi.
(a) State the loacation of point J.
Nyatakan kedudukan titik J. [3 marks/3 markah]
(b) Calculate the shortest distance, in nautical miles, from G to the North Pole, measured
along the surface of the earth.
Hitung jarak terpendek, dalam batu nautika, dari G ke Kutub Utara, diukur sepanjang
permukaan bumi. [2 marks/2 markah]
(c) K is 1410 nautical miles due south of J measured along the surface of the earth.
Calculate the latitude of K.
K ialah 1410 batu nautika dari J arah ke selatan diukur sepanjang permukaan bumi.
Hitung latitud bagi K. [3 marks/3 markah]
12. (d) An aeroplane took off from G and flew due east to H. The average speed of the
aeroplane was 647 knots and the time taken for the flight was 4 hours.
Sebuah kapal terbang berlepas dari G dan terbang arah ke timur ke H. Purata laju kapal
terbang itu ialah 647 knot dan masa yang diambil bagi penerbangan itu ialah 4 jam.
Calculate
Hitung
(i) the distance, in nautical miles, from G to H, measured along the common parallel
of latitude,
jarak, dalam batu nautika, dari G ke H, diukur sepanjang selarian latitud sepunya,
(ii) the longitude of H.
longitud bagi H. [4 marks/4 markah]
Answer/ Jawapan :
(a) (b)
(c) (d) (i)
(ii)
8. K(50°S, 20°W ) and L are two points on the surface of the earth such that KL is a diameter
of the parallel of latitude.
K(50°S, 20°B ) dan L ialah dua titik pada permukaan bumi dengan keadaan KL ialah
diameter selarian latitud.
(a) Find the longitude of L.
Cari longitud bagi L. [1 marks/1 markah]
(b) KM is a diameter of the earth. On Diagram 8 in the answer space, mark the locations of
L and M. Hence, state the location of M.
KM ialah diameter bumi. Pada Rajah 8 di ruang jawapan, tanda kedudukan L dan M.
Seterusnya, nyatakan kedudukan M. [4 marks/4 markah]
(c) Calculate the shortest distance, in nautical miles, from L to the South Pole.
Hitung jarak terpendek, dalam batu nautika, dari L ke Kutub Selatan. [2 marks/2
markah]
(d) An earoplane took off from K and flew due east along the parallel of latitude at an
average speed of 550 knots. The aeroplane took 8 hours to reach a point Q.
Sebuah kapal terbang berlepas dari K dan terbang arah ke timur di sepanjang selarian
latitud dengan purata laju 550 knot. Kapal terbang itu mengambil masa 8 jam untuk
tiba di Q.
(i) Calculate the distance, in nautical miles, from K to Q.
Hitung jarak, dalam batu nautika, dari K ke Q.
(ii) Find the longitude of Q.
Cari longitud bagi Q.
[5 marks/5 markah]
Answer/ Jawapan :
(a) (d) (i)
13. (b)
(c) (ii)
9. Table 9 shows the latitude and longitude of points F, G and H on the surface of the earth.
Jadual 9 menunjukkan latitud dan longitud bagi titik F, G dan H pada permukaan bumi.
Point
Titik
Latitude
Latitud
Longitude
Longitud
F 60°S / 60°S 50°W / 50°B
G 60°S / 60°S y° E / y° T
H x° N / x°U 50°W / 50°B
(a) P is a point on the surface of the earth such that FP is a diameter of the earth. State the
location of P.
P ialah satu titik pada permukaan bumi dengan keadaan FP ialah diameter bumi.
Nyatakan kedudukan P. [2 marks/2 markah]
(b) The distance from F to H measured along the meridian is 4020 nautical miles. Calculate
the value of x.
Jarak dari F ke H diukur sepanjang meridian ialah 4020 batu nautika. Hitung nilai x.
[3 marks/3 markah]
(c) G is due east of F and the distance of FG measured along the common parallel of
latitude is 2250 nautical miles. Calculate the value of y.
G terletak arah ke timur F dan jarak FG diukur sepanjang selarian latitud sepunya ialah
2250 batu nautika. Hitung nilai y. [4 marks/4 markah]
(d) Q(45°N, y°E ) is another point on the surface of the earth. An aeroplane took off from F
and flew due east to G along the common parallel of latitude and then due north to Q.
The aeroplane took 18 hours for the whole journey. Calculate the average speed, in
knots, for the whole journey.
Q(45°U, y°T ) ialah satu titik lain pada permukaan bumi. Sebuah kapal terbang berlepas
dari F dan terbang arah ke timur ke G di sepanjang selarian latitud sepunya dan
kemudian arah ke utara ke Q. Kapal terbang itu mengambil masa 18 jam bagi seluruh
perjalanan. Hitung purata laju, dalam knot, bagi seluruh penerbangan itu.
[3 marks/3 markah]
Diagram 8/Rajah 8
14. Answer/ Jawapan:
(a) (b)
(c) (d)
10. J (50°N, 85°E ), K (50°N, 120°E ), L and M are four point on the surface of the earth. JL is
a diameter of the earth.
J (50°U, 85°T ), K (50°U, 120°T ), L dan M ialah empat titik di permukaan bumi. JL ialah
diameter bumi.
(a) State the location of point L.
Nyatakan kedudukan bagi titik L. [3 marks/3 markah]
(b) Calculate the shortest distance, in nautical miles, from J to the South Pole measured
along the surface of the earth.
Hitung jarak terpendek, dalam batu nautika, dari J ke Kutub Selatan diukur sepanjang
permukaan bumi. [2 marks/2 markah]
(c) M is 4380 nautical miles due south of K measured along the surface of the earth.
Calculate the latitude of M.
M adalah 4380 batu nautika arah ke selatan dari K diukur sepanjang permukaan bumi.
Hitung latitud M. [3 marks/3 markah]
(d) An aeroplane took off from J and flew due east to K along the common parallel of
latitude and then flew due south to M. The average speed of the whole flight was 550
knots. Calculate the total time, in hours, taken for the whole flight.
Sebuah kapal terbang berlepas dari J dan terbang arah ke timur ke K di sepanjang
selarian latitud sepunya dan kemudian terbang arah ke selatan ke M. Purata laju kapal
terbang bagi keseluruhan penerbangan itu ialah 550 Knot. Hitung jumlah masa, dalam
jam, yang diambil bagi keseluruhan penerbangan itu. [4 marks/4 markah]
Answer/ Jawapan:
(a) (b)
(c) (d)
15. ANSWERS
TOPIC: EARTH AS A SPHERE
1. (a) 110 W [2m]
(b)(i) HOK = [1m]
= 80 [1m]
Latitude of K = 80 – 50
= 30N [1m]
(ii) Distance F to H
= ( 70 + 45 ) 60 cos 50 [2m]
= 4435.23 n.m [1m]
(iii) Time = [3m]
= 13.58 hrs [1m]
2. (a) 60S [2m]
(b) Point H [1m]
Point L [1m]
L ( 60N, 160E ) [1m]
(c) Distance H to south pole
= 30 X 60 [1m]
= 1800 n.m. [1m]
(d) (i) Distance = 450 x 8 [1m]
= 3600 n.m. [1m]
(ii) GOP =
= 120 [2m]
Longitude = 120 - 20 = 100E [1m]
60
4800
680
480023.4435
60cos60
3600
16. 3. (a) Longitude of Q = 160E [2m]
(b) Position R (510
N,1600
E) [2m]
(c) Shortest Distance Q to North Pole= ( 90 – 51 ) 60 [2m]
= 2340 n.m. [1m]
(d) (i) Distance PM = 500( 9) = 4500 n.m. [2m]
(ii) GOP =
= 11911' [2m]
Longitude = 11911' + 20 = 11911'W [1m]
4. (a) Longitude of S = 100 E [2m]
(b) ROT = [1m]
= 60 [1m]
Latitude of T = 60 – 40 [1m]
= 20S [1m]
(c) Distance R to S
= 100 x 60 [1m]
= 6000 n.m. [1m]
(d) Average speed =
= [3m]
= 593.66 n.m. [1m]
5. (a) =170E [2m]
(b) Point G [2m]
Point H [2m]
60
3600
20
360040cos60180
20
360030.8273
51cos60
4500
17. (c) Distance G to H
= 50 x 60 [2m]
= 3000 n.m [1m]
(d) Time = [2m]
= 8 hrs [1m]
6. (a) (i) Longitude of R = 140E [2m]
(ii) Distance PR via S : Distance PR via N
80x 60 : 280 x 60 [2m]
2 : 7 [1m]
(iii) Distance P to Q
= ( 40 + 90 ) 60 x cos 50 [3m]
= 5013.74 n.m [1m]
(b) QOT = [1m]
= 65 [1m]
Latitude of T = 65º – 50º
= 15N [1m]
7. (a) Latitude of J = 45°S [1m]
Longitude of J = (180 – 108)°E
= 72°E [1m]
Location of J = (45°S , 72°E ) [1m]
(b) Shortest distance from G to the North Pole
= (90 – 45) x 60 [1m]
= 2700 nautical miles [1m]
(c) = 1410 / 60 = 23.5 ° [1m]
Location of K = (45 + 23.5) °S
= 68.5°S [2m]
(d) (i) Distance of GH
= 647 X 4
= 2588 nautical miles [1m]
(ii) Distance of GH = 2588 nautical miles
(60 x cos 45°) = 2588 [1m]
= 61° [1m]
375
3000
60
3900
18. Location of H = (108 – 61)°W
= 47°W [1m]
8. (a) Longitude of L = (180 – 20)°E
= 160°E [1m]
(b)
* Point L [1m]
* Point M [1m]
* Location of M [2m]
(c) Shortest distance from L to the South Pole
= (90 – 50) x 60 [1m]
= 2400 nautical miles [1m]
(d) (i) Distance from K to Q along the parallel of latitude
= 550 X 8 [1m]
= 4400 nautical miles [1m]
(ii) (60 x cos 50°) = 4400 [1m]
= 114 ° 5 ' [1m]
Location of H = (114 ° 5' – 20°) E
= 94 ° 5' E [1m]
9. (a) Latitude of P = 60°N [1m]
Longitude of P = (180 – 50)°E
= 130°E [1m]
Location of P = (60°N , 130°E )
(b) (60) = 4020 [1m]
= 67°
x + 60 = 67 [1m]
x = 7 [1m]
(c) (60 x cos 60°) = 2250 [1m]
= 75 ° [1m]
50 + y = 75 [1m]
y = 25 [1m]
Location of M = (50°N , 160°E )
19. (d) Distance from F to G along the parallel of latitude 60° S
= 2250 nautical miles [1m]
Distance from G to Q along the meridian 25° E
= (45 + 60) x 60
= 6300 nautical miles [1m]
Average speed =
18
63002250
= 475 knots [1m]
10. (a) Latitude of L = 50°S [1m]
Longitude of L = (180 – 85)°W
= 95°W [1m]
Location of L = (50°S , 95°W ) [1m]
(b) Shortest distance from J to the South Pole
= (50 + 90) x 60 [1m]
= 8400 nautical miles [1m]
(c) Difference latitude between M and K = 73° [1m]
Latitude of M = (73 – 50)°S [1m]
= 23°S [1m]
(d) Total time =
550
438085.1349
[3m]
= 10.42 hours [1m]