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MODUL GEMILANG SMKA SHEIKH MALEK 1 by CYY
MODUL KEM GEMILANG
SMKA SHEIKH MALEK
GROUP HALUS
2013
MATEMATIK TAMBAHAN
DISEDIAKAN OLEH : CIKGU YUSRI BIN YAAHMAT
yusriyaahmat.blogspot.com
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MODUL GEMILANG SMKA SHEIKH MALEK 2 by CYY
ALGEBRA
1. x =2 4
2
b b ac
a
2. ama
n = a
m+ n----pan-pen
3. ama
n = a
mn----pan-pen
4. (am)n = a
mn
5. logamn= loga m+ loga n-----pan-pen
6. logan
m= loga mloga n -----pan-pen
7. logamn = nlogam-----kuda pan
8.log
loglog
ca
c
bb
a= ---- kuda
9. ( 1)n
T a n d = + ------- kaki atok gbai
10. [2 ( 1) ]2
n
nS a n d = +
11.1n
nT ar
=
12. nS =1
)1(
r
ran
=r
ran
1
)1(, 1r
13. S =r
a
1, | r| < 1
KALKULUS
1. y= uv,dx
dy= u
dx
dv+ v
dx
du-----sida
2. y=v
u,
dx
dy =
2v
dx
dvu
dx
duv
3.dx
dy=
du
dy
dx
du----3p/u
4. Area under a curve
=
b
a
y dx or = b
a
dyx
5. Volume generated
= b
a
dxy2 or = b
a
dyx2
STATISTIK
1. x =N
x----tiada f
2. x =
f
fx
3. =
N
xx 2)(=
22
)(x
x
N
tiada f
4. =
f
xxf 2)( =
22)(
fxx
f
5. m = CLmf
FN
+
21
6. I =0
1
Q
Q100-2 thn
7. I =
i
ii
W
IW----fungsi gubahan
8. rn P =
!)(
!
rn
n
9. rnC =
!!)(
!
rrn
n
10. P(AB) = P(A) + P(B) P(AB)
11. )( rXp = = rnrrn
qpC , p+ q = 1
12. Mean /Min = np
13. = npq
14. Z =
X
=0
> 0
Paksi-y
Paksi-x
1
1
4Q N=
3
3
4
Q N=
DaTo TaBah
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MODUL GEMILANG SMKA SHEIKH MALEK 3 by CYY
GEOMETRI
1. Distance = 2212
21 )()( yyxx +
2. Midpoint
(x,y) =
++
2,
2
2121 yyxx
3. A point dividing a segment of a line
(x,y) = 1 2 1 2,nx mx ny my
m n m n
+ +
+ +
4. Area of triangle /Luas segi tiga
)()(3123121332212
1yxyxyxyxyxyx ++++
5. r = 22 yx +
6. r =22
yx
yx
+
+ ji
TRIGONOMETRI
1. Arc length, s = r
2. Area of sector =2
1 2r
3. AA 22 cossin + = 1
4. A2sec = A2tan1+
5. A2cosec = A2cot1+
6. sin 2A = 2 sinAcosA
7. cos 2A = cos2Asin
2A
= 2 cos2A1
= 1 2 sin2A
8. )(sin BA = sinAcosB cosAsinB
9. )(cos BA = cosAcosB m sinAsinB
10. )(tan BA =BA
BA
tantan1
tantan
m
11. tan 2A =A
A2
tan1
tan2
12. A
a
sin = B
b
sin = C
c
sin
13. a2 = b
2+ c22bccosA
14. Area of triangle =2
1absin C
FORMULA TAMBAHAN
1. x2 (SOR)x+ POR = 0 2
b cx
a a
+ = 0
2. f (x) =2 2
2 4
b ba x c
a a
+ +
3.100
X YZ
=
Z
Year C
X
YearA
Y
YearB
Sila t2
Vektor unit
SiLa
Anak
Panah
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MODUL GEMILANG SMKA SHEIKH MALEK 4 by CYY
MANUAL PENGGUNAAN-14 JAM
SLOT TAJUK PAGE
1 FORM 4 BAB 1- 4
FUNCTION
QUADRATICS EQUATION
QUADRATICS FUNCTIOND
SIMULTANEOUS EQUATION
5 - 17
2 FORM 4 BAB 5 6
INDICES N LOGARITME
COORDINATE GEOMETRY 18-26
3 FORM 4 BAB 7 8
STATISTICS
CIRCULAR MEASURE27-33
4 FORM 4 BAB 10 11
SOLUTIONS OF TRIAGLES
INDEX NUMBERS
34 -42
5 FORM 5 BAB 1 2
PROGRESSIONS
LINEAR LAW
43-50
6 FORM 5 BAB 4 5
VECTORS
TRIGONOMETRIC FUNCTIONS
51-63
7 FORM 4 FORM 5 BAB 3
INTEGRATION64-70
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MODUL GEMILANG SMKA SHEIKH MALEK 5 by CYY
FUNCTIONS
EXAMPLE 1:
Given that 43: xxf and xxg 2: ,
find fg(3).
Answer : f(x) = 3x - 4 , g(x) = 2x
g(3) = 2( )
= ( )
fg(3) = f [ g(3) ]
= f ( )
= 3 ( ) - 4
= ( )
EXAMPLE 2:
Given that xxf 23: and 2: xxg , find
gf(4).
Answer : f(x) = 3 2x , g(x) = x2.
f( ) = 3 2( )
= ( )
gf(4) = g ( )
= ( )2
= ( )
EXERCISES
1. Given that 12: + xxf and xxg 3: ,
find f g(1) .
2. Given that 92: xxf and xxg 31: + ,
find gf(3).
Given that 23: xxf , find f2(2).
Answer : f(x) = 3x - 2
f(2) = 3( ) 2 = ( )
f2(2) = f [ f(2) ]
= f ( )
= 3 ( ) 2
= ( )
Given that xxg 43: , evaluate gg(1).
Answer : g(x) = 3 4x
g(1) = 3 4( ) =
gg(1) = g [g(1)]
= g ( -1)
= 3 4 ( )
= ( )
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MODUL GEMILANG SMKA SHEIKH MALEK 6 by CYY
1.5 FINDING THE VALUE OF COMPOSITE FUNCTIONS 22 , gf
Notes : f2(a) = ff (a) =f [f(a) ]
1.7 FINDING COMPOSITE FUNCTIONS
EXAMPLE 1:
Given that 43: xxf and xxg 2: ,
find fg(x).
Answer : f(x) = 3x - 4 , g(x) = 2x
fg(x) = f [ g(x) ]
= f ( 2x)
= 3 (2x) - 4
= 6x 4
OR
fg(x) = f [ g(x) ]
= 3 [g(x)] - 4
= 3 (2x) 4
= 6 x - 4
EXAMPLE 2:
Given that xxf 23: and 2: xxg , find the
composite fuction gf.
Answer: f(x) = 3 2x , g(x) = x2.
gf(x) = g[f(x)]
= g (3 2x)
= (3- 2x)2
OR
gf(x) = g[f(x)]
= [f(x)]2
= (3- 2x)2
thus 2)23(: xxgf
EXERCISES
1. Given that 32: + xxf and xxg 4: ,
find fg(x) .
2. Given that 52: xxf and xxg 5: , find
the composite function gf .
3. Given the functions 4: +xxf and
12: xxg , find
(a) fg(x) (b) gf(x)
4. Given that 43: + xxf and xxg 25: ,
find
(a) f g (b) gf
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MODUL GEMILANG SMKA SHEIKH MALEK 7 by CYY
EXAMPLE 1 :
Given that f(x) = 4x 6 , find
x + ( )
f1
(x) =4
6+x
EXAMPLE 2 :
Given that f(x) = 2x + 3 , find f1
(x).
1. Given that f : x 4 + 8x , find f1
. 2. Given that g : x 5 + 6x , find g1
.
3. Given that g : x 10 2x , find g1
. 4. Given that f : x 4 3x , find f1
.
5. Given that f : x 5 + 2x , find f1
. 6. Given that g : x 3 2x , find g1
.
7. Given that f : x 6x - 15 , find f1
.8. Given that g : x 3
4
3x , find g
1.
9. Given that f(x) = 1 2x , find f 1(x). 10. Given that g(x) = 3x + 2 , find g1 (x).
11. Given that f(x) = 5 2x , find f
(x). 12. Given that g(x) = 5x + 2 , find g
(x).
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MODUL GEMILANG SMKA SHEIKH MALEK 8 by CYY
EXERCISES-QUESTIONS BASED ON SPM FORMAT
1.
Base on the information above, a relation from P
into Q is defined by the set of ordered pairs
{ (1, 4), (1, 6), (2, 6), (2, 8) }.
State
(a) the images of 1, Ans:
(b)the object of 4, Ans:
(c) the domain, Ans:
(d)the codomain, Ans:
(e) the range, Ans:
(f) the type of relation Ans:
2.
The above diagram shows the relation between set
A and set B
State
(a) the images of b, Ans:
(b)the objects of p, Ans:
(c) the domain, Ans:
(d)the codomain, Ans:
(e) the range, Ans:
(f) the type of relation Ans:
3. Given the functions 12: + xxf and
3: 2 xxg , find
(a) f1
(5) ,
(b) gf (x) .
4. Given the functions 14: + xxg and
3: 2 xxh , find
(a) g1
(3) ,
(b) hg (x) .
5. Given the function xxf 32: and
2:2
+ xxxh , find(a) f
1(3) ,
(b) hf (x) ,
(c) f2(x) .
6. Given the functions 12: + xxf and2
2: xxh , find(a) f
1( 1) ,
(b) hf (x) ,
(c) f h(x).
P = { 1, 2, 3}
Q = {2, 4, 6, 8, 10}
P = { 1, 2}
Q = {2, 4, 6, 8, 10} a
b
c
p
q
r
s
A B
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MODUL GEMILANG SMKA SHEIKH MALEK 9 by CYY
QUADRATIC EQUATIONS/FUNCTIONS
Solve (x 3) = 1.
Ans : 2, 4
L4. Solve 1 + 2x = 5x + 4.
Ans : 1, 3/2
Solve (2x 1) = 2x 1 .
Ans : , 1
L4. Solve 5x 45 = 0.
Ans : 3 , 3
Selesaikan (x 3)(x + 3) = 16.
Ans : 5 , 5
L6. Selesaikan 3 + x 4x2= 0.
Ans : , 1
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MODUL GEMILANG SMKA SHEIKH MALEK 10 by CYY
Solve x 2x 9 = 0 by completing the square,
give your answers correct to 4 significant figures.
Ans : 2.212 , 4.162
L4. Solve x + 10x + 5 = 0 , give your
answers correct to 4 significant figures.
Ans : 0.5279, 9.472
Find the quadratic equation with roots 2 dan - 4.
x2+ 2x 8 = 0
L1. Find the quadratic equation with roots -
3 dan 5.
Ans : x2 2x 15 = 0
Given that the roots of the quadratic equation
x2+ (h 2)x + 2k = 0 are 4 and -2 . Find h and k.
(Ans : h = 0, k = -4)
L6. Given that the roots of the quadratic
equation 2x2+ (3 k)x + 8p = 0 are p and 2p
, p 0. Find k and p.
(Ans: p = 2, k = 15)
Given that the roots of the quadratic equation
2x2+ (p+1)x + q - 2 = 0 are -3 and . Find the
value of p and q.
x = -3 , x =
x + 3 = 0 or 2x 1 = 0
(x + 3) ( 2x 1) = 0
2x2+ 5x 3 = 0
Comparing with the original equation :
p + 1 = , q - 2 =
p = , q =
L4. Given that the roots of the quadratic
equation 3x2+ kx + p 2 = 0 are 4 and
- . Find k and p.
(Ans : k = -10 , p = -6)
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MODUL GEMILANG SMKA SHEIKH MALEK 11 by CYY
The roots of the quadratic equation
2x2+ px + q = 0 are - 6 and 3.
Find
(a) p and q,
(b) range of values of k such that
2x2+ px + q = k does not have real roots.
Answer :
(a) x = -6 , x = 3
(x + 6) (x 3) = 0
x2+ 3x - 18 = 0
2x2+ 6x 36 = 0
Comparing : p = , q =
(b) 2x2+ 6x 36 k = 0
a = 2, b = 6, c = -36 - k
62 4(2)(-36 k) < 0
k < 40.5
L1. The roots of the quadratic equation
2x2+ px + q = 0 are 2 and -3.
Find
(a) p and q,
(b) the range of values of k such that
2x2+ px + q = k does not have real roots.
Find the range of k if the quadratic equation
2x2 x = k has real and distinct roots.
( Ans : k > - 1/8 )
L3. The quadratic equation 9 + 4x2= px
has equal roots. Find the possible values of p.
( Ans : p = -12 atau 12)
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MODUL GEMILANG SMKA SHEIKH MALEK 12 by CYY
Find the range of p if the quadratic equation 2x +
4x + 5 + p = 0 has real roots.
(Ans : p - 3 )
L5. Find the range of p if the quadratic
equation x2+ px = 2p does not have real
roots.
( Ans : -8 < p < 0 )
The roots of the quadratic equation
2x2+ 8 = (k 3)x are real and different. Determine
the range of values of k.
( Ans : k < -5 , k > 11 )
L7. Find the range of values of k if the
quadratic equation x2+ 2kx + k + 6 = 0 has
equal roots.
( Ans : k -2 , 3 )
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MODUL GEMILANG SMKA SHEIKH MALEK 13 by CYY
(a) The equation x 6x + 7 = h(2x 3) has roots
which are equal. Find the values of h. [4]
(b) Given that and are roots of the equation
x2 2x + k = 0 , while 2 and 2 are the
roots of the equation x2 + mx + 9 = 0.
Determine the possible values of k and m [6]
( h = -1 , -2 ; k = 49
L2. One of the roots of the equation
2x2+ 6x = 2k 1 is twice the other.
Find the value of k and the roots of the
equation.
( x = -1 , x = -2 ; k = 2
3
)
(SPM 2003 , P1, S3). Solve the quadratic equation
2x(x 4) = (1 x)(x + 2). Give your answer correct to
4 significant sigures. [3]
( x = 2.591, - 0.2573)
L3. (SPM 2003, P1, S4) The quadratic
equation x (x+1) = px 4 has two distinct
roots.Find the range of values of p. [3]
( p , -3 , p > 5)
(SPM 2002) Find the range of k if the Q.E.
x2+ 3 = k (x 1), k constant, has two distinctroots. [3]
( k < -2 , k > 6)
(SPM 2001) Given 2 and m are roots of the
equation (2x 1)(x + 3) = k (x 1), with k as aconstant, find k and m. [4]
( k = 15 , m = 3 )
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MODUL GEMILANG SMKA SHEIKH MALEK 14 by CYY
FUNCTION VALUE MAX/MINAXIS OF
SYMMETRY
POINT
MAX/MINGRAPH SKETCHING
f(x) = (x 1)2+ 2 2 (x 1)
2
= 0x = 1
(1, 2)
g(x) = (x- 2)2+ 4 4
(x 2)2 = 0
x =( , )
f(x) = (x + 1) - 4
f(x) = 1 + 2 (x 3)
f(x) = 3x2
- 2
x
O
y
xO
y
(1,2)
x
O
y
x
O
x
y
xO
y
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MODUL GEMILANG SMKA SHEIKH MALEK 15 by CYY
Solve the inequality x + x - 6 0
x2 + x - 6 0
(x + 3) ( x 2) 0
Consider f(x) = 0. Then x = -3 , x = 2
Range of x is : x -3 atau x 2
L4. Solve the inequality x + 3x - 10 0.
x -5 , x 2
Solve the inequality 2x2 + x > 6.
x < -2 , x > 3/2
L6. Solve the inequality x(4 x) 0.
0 x 4
Diven y = h + 4kx 2x = q 2(x + p)
(a) Find p and q in terms of h and / or k.(b) If h = -10 and k = 3,
(i) State the equation of the axisof symmetry,
(ii) Sketch the graph of y = f(x)
(Ans : p = -k , q = 2k2+ h ; paksi simetri : x = 3)
L8. Sketch the graphs of
(a) y = x2+ 3
(b) y = 2 (x - 3)2 1
x
-3 2
y=f(x)
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MODUL GEMILANG SMKA SHEIKH MALEK 16 by CYY
SIMULTANEOUS EQUATIONS
1. Solve x + y = 3, xy = 10 .
x + y = 3 ........ (1)
xy = 10 ........ (2)
From (1), y = ( )......... (3)
Substitute (3) into (2),
x ( ) = 10
3x x2
= 10
x2
3x 10 = 0
(x ) (x ) = 0
x = ( ) atau x = ( )
From ( ), when x = ( ) , y = 3 ( ) =
x = ( ) , y = 3 ( ) =
2. Solve x + y = 5, xy = 4 .
(Ans : x = 1, y = 4 ; x = 4, y = 1)
3. Solve x + y = 2 , xy = 8 .
(Ans : x = 4 , y = 2 ; x = 2, y = 4 )
4. Solve 2x + y = 6, xy = 20 .
(Ans : x = 2 , y = 10 ; x = 5, y = 4 )
5.Solve the simultaneous equations
3x 5 = 2y
y(x + y) = x(x + y) 5
(Ans : x = 3 , y = 2 )
6. Solve the simultaneous equations :
x + y 3 = 0
x2
+ y2
xy = 21
(Ans : x = 1, y = 4 ; x = 4, y = 1 )
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MODUL GEMILANG SMKA SHEIKH MALEK 17 by CYY
6. Solve the simultaneous equations :
4x + y = 8
x2+ x y = 2
(Ans : x = 2 , y = 0 ; x = 3 , y = 4 )
7. Solve the simultaneous equations p m = 2
and p2+ 2m = 8. Give your answers correct
to three decimal places .
(m = 0.606, p = 2.606 ; m = 6.606 , p = 4.606 )
7. Solve the simultaneous equations
11
2x y+ = and y
2 10 = 2x
(Ans : x = 4 , y = 3 ; x = , y = 3 )
8. Solve the simultaneous
x + 3y 1 = x2+ 3x + 6y 4 = 0
x = 1, y = 0 ; x = 2 , y = 1]
9 Solve the simultaneous equations 2x + y = 1 and 2x + y + xy = 5. Give your answers
correct to three decimal places .
(Ans : x = 1.618, y = 2.236 , x = 0.618, y = 0.236)
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MODUL GEMILANG SMKA SHEIKH MALEK 18 by CYY
INDICES N LOGARITME
Solve the equation log2(x+1) = 3.
Answers: log2(x+1) = 3
x + 1 = 23
x =
L1. Solve the equation log2(x 3 ) = 2.
Jawapan:
Ans : x = 7
Solve the equation log10(3x 2) = 1 .
Jawapan: 3x 2 = 10-1
3x 2 = 0.1
3x = 2.1
x = 0.7
L2. Solve the equation log5(4x 1 ) = 1 .
Ans : x = 0.3
Solve the equation log3 (2x 1)+ log24 = 5 .
Ans : x = 14
L6. Solve the equation
log4 (x 2)+ 3log2 8 = 10.
Ans : x = 6
Solve the equation
log2 (x + 5) = log2(x 2) + 3.
Ans : x = 3
L8. Solve the equation
log5 (4x 7) = log5(x 2) + 1.
Ans : x = 3
Solve log3 3(2x + 3)
= 4
Ans : x = 12
L10 . Solve log2 8(7 3x)
= 5
Ans : x = 1
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MODUL GEMILANG SMKA SHEIKH MALEK 19 by CYY
Given log 2 T - log4 V = 3, express T in terms of
V. [4]
(Ans: T = 8V
)
L8. Given log 4 T + log 2 V = 2, express
T in terms of V. [4]
(Ans: 16V-2
)
Solve 4x
= 7x. [4]
( Ans: x = 1.677 )
L10. Solve 4x
= 9x. [4]
( Ans: x = 2.409 )
Solve the equation 3 3log 9 log (2 1) 1x x + = .
[3]
(Ans: x = 1 )
6. Given that log 2m p= and log 3m r= ,
express227
log16
m
m
in terms ofpand r. [4]
(Ans: 3r 4p +2 )
Solve the equation 5 3 68 32x x += .
[3]
(Ans : x = 3.9 )
8. Given that 5log 2 m= and 5log 3 p= ,
express 5log 2.7 in terms of mandp.[4]
(Ans: 3p m 1 )
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MODUL GEMILANG SMKA SHEIKH MALEK 20 by CYY
PAPER 1- BAB 1/2/3/5
1 The relation is shown in the following set of ordered pairs.
{ (2, 4), (1, 1), (0, 0), (1, 1), (2, 4) }.
State
(a) the type of relation,
(b) the range of the relation.
[2 marks]
Satu hubungan ditunjukkan oleh set pasangan tertib berikut.
{ (2, 4), (1, 1), (0, 0), (1, 1), (2, 4) }.
Nyatakan
(a) jenis hubungan itu,
(b) julat hubungan itu.
2 The functions gis defined by : , 55
x pg x x
x
+
. Find the value ofpif g
1(3) = 12.
[2 marks]
Fungsi g ditakrifkan oleh : , 55
x pg x x
x
+
. Carikan nilai p jikag
1(3) = 12.
3 Form the quadratic equation which has the roots 3 and3
2. Write your answer in the
form 02 =++ cbxax where a, band care constants.
[2 marks]Bentukkan persamaan kuadratik yang mempunyai punca-punca 3 dan
3
2. Beri jawapan
anda dalam bentuk 02 =++ cbxax dengan keadaan a, b dan c adalah pemalar.
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MODUL GEMILANG SMKA SHEIKH MALEK 21 by CYY
4 Find the range of values ofxwhich satisfies the inequality (x1)(x+ 5) > 8(x+ 2). [3 marks]
Cari julat nilai x yang memuaskan ketaksamaan (x1)(x+ 5) > 8(x+ 2).
5 Solve the equation 2 13 5x x = . [4 marks]
Selesaikan persamaan2 13 5x x = .
6 Given that 3log 5 =p and 3log 2 = q. Express 3log 75 in terms of pand q. [4 marks]
Diberi 3log 5 = p dan 3log 2 = q. Ungkapkan 3log 75dalam sebutan p dan q.
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MODUL GEMILANG SMKA SHEIKH MALEK 22 by CYY
GEOMETRY COORDINATE
Eg. 1 Given two points A(2,3) and B(4,7)
Distance of AB =2 2
(4 2) (7 3) +
= 4 16+
= 20 unit.
E1. P(4,5) and Q(3,2)
PQ =
[ 10 ]
E1. The distance between two points A(1, 3) and
B(4, k) is 5. Find the possible vales of k.
7, -1
E2. The distance between two points P(-1, 3)
and Q(k, 9) is 10. Find the possible values of k.
7, -9
Eg. P(3, 2) and Q(5, 7) E1 P(-4, 6) and Q(8, 0)
(2, 3)
Eg1. The point P internally divides the line
segment joining the point M(3,7) and N(6,2) in
the ratio 2 : 1. Find the coordinates of point P.
P =
+
++
+12
)2(2)7(1,12
)6(2)3(1
E1. The point P internally divides the line
segment joining the point M (4,5) and N(-8,-5) in
the ratio
1 : 3. Find the coordinates of point P.
51,
2
1
2
N(6, 2)
M(3, 7)
P(x, y)
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E1. P(0, 1), Q(1, 3) and R(2,7)
Area of PQR =2
1 0 1 2 0
1 3 7 1
=
= 1 unit 2
1. P(2,3), Q(5,6) and R(-4,4)
Area of PQR =
17
2unit
2
1. P(1,5), Q(4,7), R(6,6) and S(3,1).
Area of PQRS =
= 8 unit 2
2. P(2, -1), Q(3,3), R(-1, 5) and S(-4, -1).
[27]
1. Given that the points P(5, 7), Q(4, 3) and
R(-5, k) are collinear, find the value of k.
k= 33
2. Show that the points K(4, 8), L(2, 2) and M(1, -
1) are collinear.
E1. Find the equation of a straight line that
passes through the point (5,2) and has a
gradient of -2.
y = -2x + 12
E2. Find the equation of a straight line that passes
through the point (-8,3) and has a gradient of4
3.
4y = 3x + 36
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P
Q
O x
y
EXERCISES-PAPER 1
1 Diagram shows a straight line PQwith the equation 3x+ 4y 12 = 0. The point Plies on thex-
axis and the point Qlies on they-axis. Find the equation of the straight line perpendicular to PQ
and passes through the point Q. [3 marks]
Rajah 2 menunjukkan garis lurus PQ yang mempunyai persamaan 3x + 4y 12 = 0.
Titik P terletak pada paksi-x dan titik Q terletak pada paksi-y. Carikan persamaan
garis lurus yang berserenjang dengan PQ dan melalui titik Q. [3 markah]
2 Given that P(0, 1), Q(k, s),R(6,1) and S(5, 6) are the vertices of a rhombus PQRS.
Find the values of kand s. [3 marks]
Diberi bahawa P(0, 1), Q(k, s), R(6, 1) dan S(5, 6) ialah bucu-bucu bagi rombus PQRS.
Cari nilai bagi k dan s. [3 markah]
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P
Q
O x
y
EXERCISES-PAPER2-SECTION A
1 Diagram 1 shows a parallelogram PQRSon a Cartesian plane.
(a) Find the value of t.
Hence, state the equation of a straight line PQin the intercept form. [3 marks]
(b) Nis a moving point such that its distance is in the ratioRQ: QN= 2 : 3.
Find the equation of the locus ofN. [2 marks]
(c) Calculate the area of PQRS. [2 marks]
Rajah 1 menunjukkan segi empat selari PQRSpada satah Cartesan.
(a) Cari nilai t.
Seterusnya, nyatakan persamaan garis lurus PQdalam bentuk pintasan. [3 markah]
(b) Nialah titik bergerak di mana jaraknya adalah dalam keadaanRQ: QN= 2 : 3.
Cari persamaan lokusN. [2 markah]
(c) Hitung luas PQRS. [2 markah]
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EXERCISES-PAPER2-SECTION B
1 In Diagram 3, the straight lineABCintersects the line 5y+x+ 35 = 0 at the point C.
(a) Write the equation of AC in intercept form. [1 mark]
(b) Find the coordinates of C. [2 marks]
(c) Given the pointRmoves such that the ratioRA:RC = 1 : 2, find the equation
of the locusR. [3 marks]
(d) Find the points of intersection of the locusRwith they-axis. [2 marks]
(e) Find the area of ACO. [2 marks]
Dalam Rajah 3, garis lurusABCbersilang dengan garis 5y+x+ 35 = 0 pada titik C.
(a) Tuliskan persamaanACdalam bentuk pintasan. [1 markah]
(b) Cari koordinat titik C. [2 markah]
(c) Diberi titikRyang bergerak dengan keadaan nisbah RA:RC= 1 : 2, carikan
persamaan lokusR. [3 markah]
(d) Cari titik persilangan lokusRdengan paksi-y. [2 markah]
(e) Cari luas ACO. [2 markah]
x
y
C
A
B5
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STATISTICS
(a) Find the mode, median and mean for
2, 3, 1, 2, 6, 8, 9, 3, 2, 3.
[Mode = 2, Median = 3, Mean = 4]
(b) Find the mode, median and mean for the data
in the table below.
[Mode = 4, Median = 6, Mean = 7.2]
Score 2 4 6 9 12 13
Frequency 1 3 2 1 2 1
(a) Find the range and the interquartile for
5, 1, 2, 3, 4, 6, 3, 8, 2, 5, 9.
[Range = 8, Interquartile range = 4]
(b) Find the range and the interquartile range for
12, 17, 13, 19, 15, 8, 12, 11.
[Range = 11, Interquartile range = 4.5]
(c) Find the range and the interquartile for
Score 1 4 5 6 8 9Frequency 1 3 1 1 2 1
[Range = 8, Interquartile range = 4]
(d) Find the range and the interquartile range for
Points 2 4 6 8No. of person 3 5 2 2
[Range = 6, Interquartile range = 3]
(a) Find the mean, variance and the standard
deviation for the data below.
5, 12, 6, 3, 6, 10.
[Mean = 7,92= 9.333,9= 3.055]
(b) Find the mean, variance and the standard
deviation for the data below.
18, 12, 16, 11, 19, 18, 12, 14.
[Mean = 15,92= 7.5,9= 2.739]
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MODUL GEMILANG SMKA SHEIKH MALEK 28 by CYY
EXERCISES-PAPER 1
1 Table 1 shows the distribution of the scores acquired by 40 students in a quiz competition.
Determine the inter-quartile range of the distribution.
[3 marks]
Markah 0 1 2 3 4
Bilangan pelajar 2 9 16 12 1
JADUAL 1
Jadual 1 menunjukkan taburan markah yang diperolehi 40 pelajar dalam suatu pertandingan
kuiz. Tentukan julat antara kuartil bagi taburan tersebut. [3 markah]
2 The marks for students who sat for a test is shown in Table 1.
Marks 1 20 21 40 41 60 61 80 81 100
Number of students 4 k 12 9 5
TABLE 1
If the median of the marks is 505, find the value of k. [4 marks]
Jadual 1 menunjukkan markah pelajar-pelajar yang menduduki suatu ujian.
Marks 1 20 21 40 41 60 61 80 81 100
Number of students 4 k 12 9 5
JADUAL 1
Jika markah median ialah 505, cari nilai k. [4 markah]
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EXERCISES-PAPER2-SECTION A
3 (a) The mean and standard deviation of a set of numbers 1 2 3, , , ..., nx x x x are 20 and 7
respectively. Calculate the mean and standard deviation of the following set of data :
1 2 35 2, 5 2, 5 2,..., 5 2.nx x x x [4marks]
Min dan sisihan piawai bagi satu set nombor 1 2 3, , ,..., nx x x x adalah masing-masing
20 dan 7. Hitungkan min dan sisihan piawai bagi set data berikut :
1 2 35 2,5 2,5 2,...,5 2.nx x x x [4markah]
Marks Number of students
0 19 5
20 39 8
40 59 11
60 79 10
80 99 6
TABLE 1
(b) Table 1 shows the distribution of Chemistry marks obtained by 40 students.
Calculate the median of the distribution. [3marks]
Jadual 1 menunjukkan taburan markah Kimia yang diperolehi 40 orang pelajar.
Hitungkan median bagi taburan itu. [3markah]
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MODUL GEMILANG SMKA SHEIKH MALEK 30 by CYY
CIRCULAR MEASURE
Example: 25.8o
Since 180o= rad
rad4504.0
1808.258.25
=
=
(a) 30o=
[0.5237 rad]
(b) 1.54 rad =
[88.22o]
(c) =rad5
18
[206.24o]
Example:
r= 10 cm and = 1.5 rad
Since S= r
= 10 1.5
= 15 cm
(a) r= 20 cm and = 0.6 rad
[12 cm]
Example:r= 3 cm and = 1.8 rad
Since 2
2
1rA =
8.132
1 2=
= 8.1 cm2
(a) r= 5 cm and = 0.8 rad
[10 cm2]
(c)
[Perimeter = 10.19 cm, Area = 1.145 cm2]
(d)
[Perimeter = 8.599 cm, Area = 4.827 cm2]
25.8o
O
A
B
O
P
Q
1.8 rad
3cm
3cm
O
P
Q
3.8
cm
3.8cm
1 rad
OA
B
4.2cm
4.2cm
60o
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MODUL GEMILANG SMKA SHEIKH MALEK 31 by CYY
P O R
Q
EXERCISES
1 Diagram 1 shows a semicircle OPQRwith centre O. Given that the length of arc PQis
64 cm, calculate the area of the sector OQR.
(Use = 3142)[3 marks]
Rajah 1 menunjukkan sebuah semibulatan OPQR berpusat O. Diberi panjang lengkok PQ
ialah 64 cm. Hitungkan luas sektor OQR.(Gunakan = 3142)
[3markah]
DIAGRAM 2 / RAJAH 2
2 Diagram 2 shows a sector OABof a circle with centre O and radius r. If the radius is
35 cm and the perimeter of sector OABis 122 cm, find AOB in radian.
[3 marks]
Rajah 2 menunjukkan sektor OAB bagi bulatan dengan pusat O dan jejari r. Jika jejarinya
ialah 35 cm dan perimeter sektor OAB ialah 122 cm, cari AOB dalam radian.
[3 markah]
O
A
B
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MODUL GEMILANG SMKA SHEIKH MALEK 32 by CYY
3 The diagram above shows a circle of radius of 6 cm and the arcEFwhich subtends an angle of 60
at the centre, O. Calculate the perimeter of the shaded segment.
[3 marks]
Rajah di atas menunjukkan bulatan berjejari6 cm dan lengkokEFmencangkup sudut60pada pusatO. Hitungkan perimeter kawasan yang berlorek.
Diagram 3 / Rajah 3
4 Diagram 3 shows an equilateral triangle PQR and a sector PTS with centre P. Given that
QR = 18 cm and the area of the shaded region is 225 cm2. Find the radius PT. [3 marks]
Rajah3 menunjukkan segitiga sama PQR dan sektor PTS berpusatP.Diberi QR = 18 cm dan luas
kawasan berlorek ialah225 cm2. Cari panjang jejariPT.
E
F
O
60
R
P
Q
ST
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MODUL GEMILANG SMKA SHEIKH MALEK 33 by CYY
5 Diagram 4 shows a circle with centre Oand radius 10 cm. PTQ is a tangent to the circle
at T. PSO and QRO are straight lines. Given that OP = OQ and S is a mid-point of OP.
Calculate
Rajah4 menunjukkan bulatan berpusatOdengan jejari10 cm. PTQ ialah tangen kepada bulatan
itu di T. PSO dan QRO adalah garis lurus.Diberi bahawa OP = OQ dan S adalah titik
tengah OP. Hitungkan
(a) SOT, [2 marks]
(b) the length of the minor arc ST, [2 marks]panjang lengkok minorST,
(c) the area of the minor sector OST, [2 marks]
luas sektor minorOST,
(d) the area of the shaded region. [4 marks]
luas kawasan berlorek.
[Use / Gunakan = 3142 ]
S
O
P T Q
R
Diagram 4 / Rajah 4
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INDEX NUMBER
Notes :
a. Price Index,I= 1
0
100P
P , P1= price at a specific time, P0= price at the base year.
b. Composite Index ,IW
I
W
=
,I =price index, W= weightage
1. Table below shows the price indices and percentage of usage of four main
ingredients ,P,Q,Rand S, in the production of a type of cake.
Ingredients
Price index for the
year 2012
(2010=100)
Percentage of
usage
P m 20
Q 105 30
R 108 10
S 120 40
(a) Calculate
(i) the price of ingredient Qin the year 2010 if its price in the year 2012 is RM 50.00,
(ii) the price index ofRin the year 2012, based on the year 2008, given that its price
index in the year 2010, based on the year 2008 is 110.
(b) The composite index number of the cost of production of this type of cake in the year
2012,based on the year 2010 is 112.8. Calculate.
(i) the value of m,
(ii) the cost of these ingredients for the production of this type of cake in
the year 2012 if the corresponding cost in year 2010 is RM60.00.
Guided Solutions
a (i)
,
(ii)( ) ( )
( )1 0 0
=
b (i)
(ii)
5 0( ) 1 0 5 , ( ) = =x
x
( )( ) 105 ( 3 0 ) 1 08 (1 0 ) 120 ( 40 )( ) , ( )
10 0
+ + += =
mm
2012100 ( ) , ( )2012
60 = =
QQ
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MODUL GEMILANG SMKA SHEIKH MALEK 35 by CYY
2. The bar chart in Diagram shows the monthly cost of ingredients P, Q,Rand Sfor making biscuit
in year 2009. Table shows the prices and the price indices used to make the biscuit.
a) Find the values ofxandy.
b) Calculate the composite index for the cost of making the biscuit in year 2011 based on the
year 2009.
c) If the price for making biscuit in 2009 is RM 1200, find the corresponding price in year 2011.
d) The cost of these ingredients increases by 15% from the year 2011 to 2012. Find the
composite index for the year 2012 based on 2009.
Ingredient Price in 2009 Price in 2011 Price index in 2011
(2009=100)
P 0.90 1.35 150
Q x 3.00 120
R 3.20 y 125
S 1.25 1.75 140
15
20
25
40
P Q R S
Monthly cost (RM)
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MODUL GEMILANG SMKA SHEIKH MALEK 36 by CYY
1 Table 3 shows the price indices and percentage of usage of four main ingredients, P,
Q,Rand S, in the production of a type of cake.
Jadual 3 menunjukkan indeks harga dan peratus penggunaan empat bahan utama, P, Q,R
dan Sdalam penghasilan sejenis kek.
Ingredients /
Bahan
Price index for the year 2006Indeks harga pada tahun 2006
(2003 = 100)
Percentage of usage /
Peratus penggunaan
P m 20
Q 105 30
R 108 10
S 120 40
Table 3 / Jadual 3
(a) Calculate
Hitungkan
(i) the price of ingredient Qin the year 2003 if its price in the year 2006 is RM 5000,
harga bahanQpada tahun2003jika harganya pada tahun2006 ialahRM 5000,
(ii) the price index ofRin the year 2006, based on the year 2000, given that its price
index in the year 2003, based on the year 2000 is 110.indeks hargaRpada tahun2006, berasaskan tahun2000jika indeks harganya
pada tahun2003, berasaskan tahun2000 ialah 110.
(b) The composite index number of the cost of production of this type of cake in the year
2006, based on the year 2003 is 1128.
Calculate
Nombor indeks gubahan bagi kos bahan-bahan untuk menghasilkan kek tersebut pada
tahun2006 berasaskan tahun2003 ialah1128 .
Hitungkan
(i) the value of m,
nilaim,
(ii) the cost of these ingredients for the production of this type of cake in the year 2006
if the corresponding cost in year 2003 is RM 6000.
kos bahan-bahan untuk menghasilkan kek itu pada tahun2006jika kos yang
sepadan pada tahun 2003 ialahRM6000.
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Expenditure /
Barangan Keperluan
Price index /
Indeks Harga
Weightage /
Pemberat
Shoe / kasut
Bag / beg
Shirt / bajuTrouse / seluar
125
120
n108
m
2
7 m4
TABLE 3 / JADUAL 3
2 Table 3 showsthe price indices and the weightages for 4 types of goods for En. Ahmad in the
year 2004 based on the year 2002.
Jadual3 menunjukkan indeks harga dan pemberat untuk empat jenis barangan keperluan tahunan
Encik Ahmad bagi tahun2004 berdasarkan tahun2002 sebagai tahun asas.
(a) Calculate the value of nif the expenditure for a shirt in the year 2002 is RM 35 and
increases to RM 3920 in the year 2004
Hitungkan nilainjika harga sehelai baju pada tahun2002 ialahRM35 dan meningkat
menjadiRM 3920pada tahun2004. [2 markah]
(b) The composite index number for the expenditures in the year 2004 based on the year 2002
is 115. Calculate the value of m.
Nombor indeks gubahan barangan keperluan itu pada tahun2004 berasaskan tahun2002
ialah115.Hitungkan nilai m. [3 markah]
(c) Calculate En. Ahmads total yearly expenses for the expenditures in the year 2002 if the
corresponding expenses for the year 2004 is RM1380.
Hitung jumlah perbelanjaan tahunan Encik Ahmad untuk barangan keperluan tersebut
pada tahun2002jika perbelanjaan yang sepadan pada tahun2004 ialah RM 1380.[2 markah]
(d) The expenses increases 25% from the year 2004 to 2006. Find the composite index number
in the year 2006 based on the year 2002.
Kos barangan itu meningkat25% dari tahun2004 ke tahun2006.
Cari nombor indeks gubahan tahun2006 berasaskan tahun2002.
[3 markah]
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MODUL GEMILANG SMKA SHEIKH MALEK 38 by CYY
SOLUTION OF TRIANGLES
(1) Diagram 1 shows the triangle ABC.
Calculate the length of BC.
Answer :
8.2
( ) ( )
BC=
( ) ( )( )
BC =
Using the scientific calculator,
BC= ( )
(2) Diagram 2 shows the triangle PQR
Calculate the length of PQ.
( ) ( )
( ) ( )=
[ 8.794 cm ]
(3) Diagram 3 shows the triangle DEF.
Calculate the length of DE.
( ) ( )( ) ( )
=
[ 10.00 cm ]
(4) Diagram 4 shows the triangle KLM.
Calculate the length of KM.
( ) ( )
( ) ( )=
[ 11.26 cm ]
Diagram2
D
E F
600 35
016
15 cm
Diagram 3
LK
M
420
630
15 cm
Diagram 4
Diagram 1
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MODUL GEMILANG SMKA SHEIKH MALEK 39 by CYY
(1) Diagram 1 shows the triangle ABC.
Find ACB.
( ) ( )
( ) ( )=
(2) Diagram 2 shows the triangle KLM
Find KLM
( ) ( )
( ) ( )=
[ 27.360]
(3) Diagram 3 shows the triangle DEF.
Find DFE.
( ) ( )
( ) ( )=
[ 11.090]
(4) Diagram 4 shows the triangle PQR.
Find QPR.
( ) ( )
( ) ( )=
[ 36.110]
D
E F
3.5 cm
43024
12.5 cm
Diagram 3
LK
M
9 cm 500
15 cm
Diagram 2
RP
Q
10 cm
1300
13 cm
Diagram 4
A
B C60
0
15 cm
Diagram 1
10 cm
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MODUL GEMILANG SMKA SHEIKH MALEK 40 by CYY
a = b + c 2bc cos A
b2= a
2+ c
2 2ac cos B
c2= a
2+ b
2 2ab cos C
SOLUTION OF TRIANGLES
2.2 Use Cosine Rule to find the unknown sides or angles of a triangle.
(1)
Find the value ofx.
2 2 2( ) ( ) 2( )( )cos( )x = +
x=
(2)
Find the value ofx.
2 2 2( ) ( ) 2( )( )cos( )x = +
[ 7.475 ]
(3)
Find the value of x.
2 2 2( ) ( ) 2( )( )cos( )x = +
[ 9.946 ]
cm
E13 cm
430
5 cm
Diagram
12.3
P
cm
Q R
670
x cm
7PQ
R
750
5
bc
acbA
2cos
222+
=
ac
bcaB
2
cos222
+=
ab
cbaC
2cos
222+
=
Diagram
12.3
P
16.4 cm
cm
Q R67
0
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MODUL GEMILANG SMKA SHEIKH MALEK 41 by CYY
(1) In
Find BAC .
2 2 215 ( ) ( ) 2( )( )cos BAC= +
)14)(13(2
151413cos
222+
=BAC
(2)
Find BAC .
2 2 2( ) ( ) ( ) 2( )( )cos BAC= +
[ 83.17]
(3)
Calculate BAC
2 2 2( ) ( ) ( ) 2( )( )cos BAC= +
[ 73.41]
(4)
Calculate BCA
2 2 2( ) ( ) ( ) 2( )( )cos BAC= +
[39.17]
5)
Find RQP .
2 2 2( ) ( ) ( ) 2( )( )cos BAC= +
[108.07]
C
A
B
13cm 14 cm
15 cm
Diagram 1
C
A
B
11cm 13 cm
16 cm
Diagram 2
C
A
B
13cm 16 cm
17.5 cmDiagram 3
A
B
12.67cm 16.78 cm
19.97 cm
Diagram 4
C
2.23 cm6.45 cm
5.40 cmP
Q
R
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MODUL GEMILANG SMKA SHEIKH MALEK 42 by CYY
1 (a) In Diagram 6, PQRand QRS are two triangles. Given that QPR= 65, PQ= QR= 5 cm
and SR= 4 cm, calculate
Dalam Rajah 6, PQR dan QRS ialah dua buah segi tiga. Diberi QPR = 65, PQ = QR = 5cm
dan SR = 4 cm, hitungkan
(i) the length of SQ,
panjangSQ,
(ii) QSR,
(iii) the area of PQS.
luas PQS.
[8 marks]
(b) A triangle QRShas the same measurement the triangle QRSas in the Diagram 6, that is
QR= 5 cm, RS= 4 cm and RQS = RQS, but which is different in shape to triangle
QRS.
Sebuah segi tiga QRS mempunyai ukuran-ukuran yang sama sebagaimana segi tiga
QRS seperti dalam Rajah 6, iaitu QR = 5 cm, RS = 4 cmdan RQS = RQS,
tetapi bentuk yang berbeza daripada segi tiga QRS.
(i) Sketch the triangle QRS,
Lakarkan segi tigaQRS,
(ii) State the size of QSR.
Nyatakan saizQSR.
[2 marks]
P
Q
RS
5 cm
4 cm
65
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PROGRESSION
(a) State the common difference of each of the following arithmetic progressions
1. 2, 6, 10, 14,
[4]
2. 21, 18, 15, 12,
[3]
(b) Find the tenth term and the twentieth term of the following arithmetic progressions
1. 2, 6, 10, 14,
[38 ; 78]
2. 21, 18, 15, 12,
[-6 ; -36]
(c) Calculate the number of terms in each of the following arithmetic progressions
1. 2, 6, 10, , 82
[21]
2. 21, 18, 15, , -66.
[30]
(a) Find the sum of the first 20 terms of each of the following arithmetic progressions
1. 2, 6, 10, 14,
[800]
2. 21, 18, 15, 12,
[-150]
(b) Find the sum of the following arithmetic progressions
1. 2, 6, 10, 14, , 54
[1456]
2. 21, 18, 15, 12, , -30
[-81]
(c) Sum of a specific number of consecutive terms
1. Given an arithmetic progression 2, 6, 10, 14,
find the sum from fifth term to the sixteenth
term.
[504]
2. Given an arithmetic progression21, 18, 15, 12, find the sum from seventh
term to the eighteenth term.
[-180]
(d) (i) Find the sum to infinity of each of the following geometric progressions
1. 8, 4, 2, 1,
[16]
2. 27, 9, 3, 1,
[40.5]
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MODUL GEMILANG SMKA SHEIKH MALEK 44 by CYY
PAPER 1
1 The first three terms of an arithmetic progression are 2k3, 4k+ 1 and 5k+ 6.
Find
(a) the value of k, [2 marks]
(b) the eighth term. [1 mark]
Tiga sebutan pertama suatu janjang aritmetik adalah 2k 3, 4k + 1 dan 5k + 6.
Carikan
(a) nilai k, [2 markah]
(b) sebutan ke lapan. [1 markah]
2 The sum of the first nterms of the geometric progression 128, 64, 32, 16 . is 255875.
Find
(a) the common ratio of the progression, [1 mark]
(b) the value of n. [3 marks]
Hasil tambah n sebutan pertama suatu janjang geometri 128, 64, 32, 16 ialah
255875. Carikan
(a) nisbah sepunya, [1 markah]
(b) nilai n. [3 markah]
3 Given that 25, 22, 19,. are the first three terms in an arithmetic progression, find the
sixteenth term. [2 marks]
Diberi 25, 22, 19,. adalah tiga sebutan pertama bagi suatu janjang aritmetik, cari sebutan
keenam belas. [2 markah]
4 A geometric progression has a first term of 27 and a sixth term of9
1. Find
(a) the common ratio, [2marks]
(b) the sum to infinity. [2marks]
Sebutan pertama suatu janjang geometri ialah 27 dan sebutan keenam ialah9
1.
Cari
(a) nisbah sepunya, [2markah]
(b) hasil tambah sehingga ketakterhinggaan. [2markah]
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MODUL GEMILANG SMKA SHEIKH MALEK 45 by CYY
PAPER 2-SECTION A
1 Diagram 1 shows several sectors of circles, centre O, which are the results of electromagnetic
wave emission from the radio station at O.
Rajah 1menunjukkan beberapa sektor bulatan berpusat di O, yang merupakan hasil
pancaran gelombang elektromagnet dari sebuah stesen radio di O.
The radius of each sector increases by 3 cm compared to that of the previous sector.
Given that the area of the nth
sector is 512cm2, the radius of the first sector, OA = 10 cm dan
'4
AOA
= rad. Find
Panjang jejari sektor bulatan bertambah sebanyak 3 cmdaripada bulatan sebelumnya. Diberi
luas sektor bulatan ke-n ialah 512cm2, jejari sektor bulatan awal, OA = 10 cm dan
'
4AOA
= rad. Cari
(a) the radius of the nth
sector,
panjang jejari sektor bulatan ke-n, [2 marks]
(b) the value of n
nilai n, [3 marks]
(c) the sum of the arc lengths of the first 20 sectors.
hasil tambah 20panjang lengkok pertama gelombang tersebut. [2 marks]
rad4 O
A
CB
D
AB
CD
10 cm
3 cm
3 cm
Diagram 1 / Rajah 1
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MODUL GEMILANG SMKA SHEIKH MALEK 46 by CYY
LINEAR LAW
1.1 Draw the line of best fit
1 2
1.2 Write the equation for the line of best fit of the following graphs.
1
[ 53
3y x= +
]
2
[ 5 52
y x= + ]
P(0,5)
Q(2,0)
x
.
.
X
X
X
X
X
y
x2
y
x
X
X
X
X
y
x
P(0,3)
Q(6,13)
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MODUL GEMILANG SMKA SHEIKH MALEK 47 by CYY
2.2 Determine values of constants of non-linear relations given lines of best fit
1 The diagram below shows the line of best fit
for the graph of y2against x. Determine the
non-linear equation connecting y and x.
[y2=-2x+4]
2 The diagram below shows the line of best fit for the
graph of y2against
x
1 . Determine the non-linear
equation connecting y and x.
[ 21
5 2yx
= +
]
3 The diagram below shows the line of best fit
for the graph of 2x
y
against x. Determine the
non-linear equation connecting y and x.
[ 2 4 12y xx= ]
4 The diagram below shows the line of best fit for the
graph of 2
y
x against x. Determine the non-linear
equation connecting y and x.
[2
1 32
y xx
= + ]
P(0,4)
Q(2,0)
y
x
X
X P(0,2)
Q(2,12)y
x
X
P(3,0)
Q(6,12)2x
y
x
X
X
(2,4)
2x
y
(4, 5)
x0
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MODUL GEMILANG SMKA SHEIKH MALEK 48 by CYY
5 The diagram below shows the line wheny
x
against x is drawn. Determine the non-linear
equation connecting y and x
[3+
=x
xy ]
6 The diagram below shows the line of best fit
for the graph of log10y against x. Determine
the relation between y and x.
[ y = 103x
]
7 The diagram below shows part the graph oflog10y against x. Form the equation that
connecting y and x.
[ 16410 += xy ]
8 The diagram below shows the line of best fitfor the graph of log10y against log10x.
Determine the relation between y and x.
[y= 100x2]
9 The diagram below shows part the graph of
log10y against log10x. Form the equation that
connecting y and x.
[1000
xy= ]
10 The diagram below shows part the graph of
log 2y against log2x. Determine the relation
between y and x.
[16
2
xy= ]
(3,6)
(0,3)
y
x
x0(0,0)
(2,6)y10log
x
X
X
log10y
(3,4)
(4,0) x
(0,2)
(2,6)
y10log
x10og
X
X
6
-3
log10x
log10y
2
(5,6)
log2x
log2y
0
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MODUL GEMILANG SMKA SHEIKH MALEK 49 by CYY
1. Use graph paper to answer this question.
The table below records the values of an experiment for two variables x and y which are related
byx
qpxy +=
2 where p and q are constants.
x 0.8 1 1.3 1.4 1.5 1.7
y 108.75 79 45.38 36.5 26.67 8.19
(a) Plot xy against x3using scale 2 cm represents 1 unit in x-axis and 2 cm represents
10 units for y-axis.
Hence, draw the line of best fit
[5marks]
(b) From the graph, estimate the value of
(i) p and q
(ii) x when y=x
45 [5marks]
[Answer:p=-16.67, q=95, x=1.458]
2. Use graph paper to answer this question.
The table below records the values of an experiment for two variables x and y which are related
by kxx
p
x
y+= where p and k are constants.
x 3 5 6 7 8 9
y 4.7 4.0 3.6 3.0 2.5 1.8
(a)Plot the graph y against x2 [4 marks]
(b)use the graph to estimate the values of
(i) p
(ii) k.
(iii) x which satisfy the simultaneous equation kxx
p
x
y+= and y = 2 [6 marks]
[answer: p=5, k= -0.04, x= 8.60 - 8.75]
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MODUL GEMILANG SMKA SHEIKH MALEK 50 by CYY
3. Use graph paper to answer this question.
The table below records the values of an experiment for two variables x and y which are related
by y=p qx where p and q are constants.
x 3 4 5 6 7
y 5 10 20 40 80
(a) Plot the graph log 10y against x [4 marks]
(b)Use the graph to estimate the values of
(i) p
(ii) q.
(iii) ywhenx=4.8 [6marks]
[answer: 1.995, 0.6166, 17.38]
4. SPM 2003 Paper 2 Question 7
Use graph paper to answer this question.
Table below shows the value of two variables, x and y, obtained from an experiment. It is known that x
and y are related by the equation2
xy pk= ,wherep and kare constants
x 1.5 2.0 2.5 3.0 3.5 4.0
y 1.59 1.86 2.40 3.17 4.36 6.76
(a)Plot log10y against x2
Hence, draw the line of best fit. [5 marks]
(b)Use the graph in (a) to find the value of
(i) p
(ii) k [5 marks]
[Answer: p=1.259 , k =1.109 ]
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MODUL GEMILANG SMKA SHEIKH MALEK 51 by CYY
VECTOR
Task 1 : Express the following vectors in terms of x%
and y%
.
(1)
ABCD is a parallelogram.
ACuuur
=
(2)
QRuuur
=
(3)
TQuuur
=
PRuuur
=
(4)
EFGH is a parallelogram.
EG
uuur
=
(5)
FGuuur
=
(6)
BC
uuur
=
ADuuur
=
(1) PQuuur
=
6
8
magnitude of PQuuur
PQuuur
=
Unit vector in the direction
of PQuuur
PQuuur
=
10 ;
4
535
(2) STuuur
=
8
15
magnitude of STuuur
STuuur
=
Unit vector in the direction
of STuuur
STuuur
=
17 ;1517
817
(3) ) CDuuur
=
8
6
magnitude of CDuuur
CDuuur
=
Unit vector in the direction
of CDuuur
CDuuur
=
10 ;35
45
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MODUL GEMILANG SMKA SHEIKH MALEK 52 by CYY
VECTORS
3.4 3.7 Addition, Subtraction and Multiplication of Vectors
Task 1 : Given 2 5a i j= +% % %
, 4b i j= % % %
and 3 7c i j= +% % %
, find the following vectors in terms of i%
and j%
.
(1) 2b%
=
2 8i j% %
(2) 3a%
=
6 15i j+% %
(3) 4c%
=
12 28i j +% %
(4) 12
a
%
=
52
i j+
% %
(5) 3a b+
% %
=
5 7i j% %
(6) 2a b+
% %
=
5 6i j+% %
(7) b c+% %
=
2 3i j % %
(8) 12b a+% %
=
52
3i j+% %
(9) 3a b% %
=
17i j +% %
(10) 4b c% %
=
7 23i j% %
(11) 3 2c a% %
=
13 11i j +% %
(12) 12
b a% %
=
132
j%
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MODUL GEMILANG SMKA SHEIKH MALEK 53 by CYY
Task 2 : Given3
4a
=
%,
2
5b
=
%and
6
1c
=
%, express the following in the form of
y
x.
(1) 4b%
=
8
20
(2) 2c%
=
12
2
(3) 12
a%
=
32
2
(4) 2a b+% %
=
4
13
(5) 3b c+% %
=
16
2
(6) 2c a+% %
=
15
2
(7) 3a b% %
=
11
7
(8) 2b c% %
=
10
11
(9) 2c a% %
=
0
9
(10) 3 2c b% %
=
2213
(11) 3b a+% %
=
3
19
(12) 4a c% %
=
617
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MODUL GEMILANG SMKA SHEIKH MALEK 54 by CYY
Example:
Given that a%
= 13i+jand b%
= 7i kj, find
(a) a%
b%
in the formxi +yj,
(b) the value of kif | a%
b%
| = 10.
Solution:
(a) a%
b%
=
(b) | a%
b%
| = 10
2 26 (1 ) 10k+ + =
1. Given that b%
= 3i 2j, c%
= 5i+ mj, where m is a constant. Find the value of m.
(a) if b%
c%
is parallel to b%
(b) | 2 b%
c%
| = 125
2. Given u= 2i+ 3j dan v= 2i+ kj, find the values of k if 2 +u v = 10.
[3 marks]
Diberi u= 2i+ 3j dan v= 2i+ kj, cari nilai-nilai k jika 2 +u v = 10.
[3 markah]
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MODUL GEMILANG SMKA SHEIKH MALEK 55 by CYY
1. SPM 2003 P1 Q12
Diagram shows two vectors, OP andQOuuur uuur
.
Express
(a) OPuuur
in the formx
y
(b) OQuuur
in the form xi yj+% %
[2marks]
(5
3
, -8i+4j)
2. SPM2003 P1Q13
Use the information given to find the values of h
and kwhen r= 3p-2q
[3marks]
( h= 2 , k= 13)
3. SPM 2004 P1 Q16Given that O(0,0)A(-3,4) andB(2,16), find in terms
of unit vectors, i and j% %,
(a) ABuuur
(b) the unit vector in the direction ofABuuur
[4marks]
(a)
12
5(b)
12
5
13
1
4. SPM 2004 P1 Q17Given thatA(-2,6),B(4,2) and C(m,p), find the
value of mand ofpsuch that2 10 12AB BC i j+ =
uuur uuur
% %
[4 marks]
m= 6 ,p= 2
p =2a +3 b
q = 4a b
r = ha + (h-k) b, where h and k are
constants
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MODUL GEMILANG SMKA SHEIKH MALEK 56 by CYY
1 In Diagram 2,APC and BPD are straight lines.
DIAGRAM 2
Given that AB
= 2x, AD
= y and BC
= 3AD
.
(a) Express in terms of x and/or y ,
(i) AC
,
(ii) BD
.
[2 marks]
(b) Given that AP
= m AC
and BP
= n BD
.
Express AP
(i) in terms of m, x and y.
(ii) in terms of n, x and y.
Hence, show that m+ n = 1. [5 marks]
(c) If AQ
=4
3x y, prove thatAC and QB are parallel. [3 marks]
D
A B
C
P
Q
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MODUL GEMILANG SMKA SHEIKH MALEK 57 by CYY
DIAGRAM 6
2 In Diagram 6, OP
= p%
, OQ
= q%
, PR
= 2q%
. Sis the mid point of OPand QU
=1
2UR
.
Given that SUand OR intersects at T.
(a) Express in terms of p%
and / or q%
,
(i) OR
(ii) QR
(iii) QU
[3marks]
(b) Given that ST
= h SU
and OT
= kOR
. State OT
(i) in terms of h, p%
and q%
.
(ii) in terms of k, p%
and q%
.
[4marks]
(c) Hence, find the values of hand k.
[3marks]
R
P
Q
O
T
S
U
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MODUL GEMILANG SMKA SHEIKH MALEK 58 by CYY
TRIGONOMETRIC FUNCTIONS
5.2 Six Trigonometric Functions of any Angle (1)
1. Given that px=sin and 00
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Solve each of the following trigonometric equations for 00
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5.3 Graphs of Trigonometric Functions- (2) Graphs of cosine
Sketching the graphs of each of the following trigonometric functions for 20 x .
(a). xy cos= (b). xy cos2=
(c). xy cos3= (d). xy cos
2
1=
(e) xy cos= (f) 1cos += xy
(g) 1cos2 = xy (h) 1cos2 += xy
(i) xy cos= (j) 1cos2 += xy
(k) xy 2cos= (l) 12cos += xy
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MODUL GEMILANG SMKA SHEIKH MALEK 61 by CYY
Notes :
Using :y =asinpx or y =acospx or y =atanpx
STEPS TO SKETCH GRAPH
1. Shape of graph : look at basic sin, cos and tan graph.
2. Constant of angle : periodic/cycle (repeat) in 360or 2.
3. Constant of trigonometric equation : amplitude (maximum/minimum).
i. Negative shape : upside down/reverse shape.
ii. Shifting upward and downward.
iii. Modulus/absolute sign
4. Find a suitable lineby solving simultaneous equations using substitution method.5. Draw the straight line.
6. State the number of solutions.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
1. (a) Sketch the graph of y= 3 sin 2xfor 0 x2.
(b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the
equation 2x+ 3sin 2x= 0 for 0 x2.
2. (a) Sketch the graph ofy= |cos 3x|for 0 x.
(b) Hence, using the same axes, sketch a suitable straight line to solve the equation
x2|cos 3x|= 0.
Hence, state the number of solutions to the equation for 0 x.
ANSWERS
1 b) 3 solutions 2. b) 6 solutions
Period/cycle in 360
ShapeAmplitude
1
y
0
Cosinus graph
0
y
x
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MODUL GEMILANG SMKA SHEIKH MALEK 62 by CYY
Paper 1
1. Given that t=tan ,00
900
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MODUL GEMILANG SMKA SHEIKH MALEK 63 by CYY
2. (a) Sketch the graphy= cos 2xfor 00 1800 x . [ 3 marks ]
(b) Hence, by drawing a suitable straight line on the same axes, find the number of solutions
satisfying the equation180
-2sin22 x
x = for 01800 xo .
[ 3 marks ]
( SPM P2 No. 3 )
3. (a) Prove that cosec
2
x 2 sin
2
x cot
2
x= cos 2x. [ 2 marks ]
(b) (i) Sketch that graph of y = cos 2xfor 2x0 .
(ii) Hence, using the same axes, draw a suitable straight line to find the number of
solutions to the equation 3( cosec2
x 2 sin2x cot
2x) = 1-
xfor 20 x .
State the number of solutions. [ 6 marks ]
(SPM P2 No.5)
4. (a) Sketch the graph y = -2cos x for 20 x . [ 4 marks ]
(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions
to the equation 0cos2 =+ xx
for 20 x . State the number of solutions.
[ 3 marks ]( SPM P2 No. 4 )
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MODUL GEMILANG SMKA SHEIKH MALEK 64 by CYY
INTEGRATION
1. 43x dx 2. 523
x dx
3.6
2
3dx
x 4.
4
7dx
x
5. 3(2 3)x dx 6. ( )42 5 43
x dx
4.
2
1
8x dx =
=
[12]
5.4
3
2
x dx =
=
[60]
13.
2
1
(2 1)(2 1)x x dx +
=
[ 253
]
14.
3
2
1
(3 2)x dx
=
[38]
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MODUL GEMILANG SMKA SHEIKH MALEK 65 by CYY
7.
2
2
1
3x dx =
=
[ 32
]
8.
3
3
1
2( ) dxx
=
[ 8
9]
10.
3
0
(2 6 )x dx =
[-21]
11.
3
2
1
(4 3 )x x dx =
[-10]
y = 3x + 2
pintasan y=
pintasan x=
x dalam sebutan y =
y2dalam sebutan x =
y = 3x23
pintasan y=
pintasan x=
x2dalam sebutan y =
y2dalam sebutan x =
y = 3x + 2 , y = 3x 3
cari titik persilanganantara garis dan lengkung di atas
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MODUL GEMILANG SMKA SHEIKH MALEK 66 by CYY
Notes :
a. To integrate, use : nax dx =1
1
nax
n
+
++ c
b. Gradient function, m =dy
dx
c.dy
ydx
=
d. Area under the curve :
(a) atx-axis, Ax=b
a
y dx (b) aty-axis, Ay=b
a
x dy
e. Volume of revolution :
(a) aboutx-axis, Vx= 2
b
a
y dx (b) abouty-axis, Vy= 2
b
a
x dy
PAPER 1
1.5
4
(2 3)dx
x ,
Answer :
2. 2 (4 1)x x dx
Answer :
3. Given that3
1( )g x dx = 6, find
a) the value of3
1
( )
2dx
g x ,
b) the value of1
35 ( ) dxg x ,
c) the value of ksuch that3
1[ ( ) ]g x k dx+ = 10.
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MODUL GEMILANG SMKA SHEIKH MALEK 67 by CYY
Guided Solutions
a)3
1
( )
2dx
g x
=3
1
( )1
2g x dx
=1
( )2
= ( )
c)3 3
1 1( )g x dx k dx+ = 10
( ) + [ ]3
1kx
= 10
( ) ( ) = 4
k= ( )
4. Given that5
2( )f x dx = 9, find
a) the value of5
2
2 ( )
3dx
f x ,
b) the value of2
54 ( )f x dx ,
c) the value of ksuch that5
2[ ( ) ]f x kx dx+ = 30.
b)1
35 ( )g x dx
5( ) = ( )
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MODUL GEMILANG SMKA SHEIKH MALEK 68 by CYY
5. Diagram shows the curvey = 5x4and the straight linex=p.
If the area of the shaded region is 32 unit2, find the value ofp.
Guided Solutions
Area =b
ay dx =
4
0
5p
x dx= ( )
[ ] 0p
= ( )
( ) ( ) = 32
p = ( )
6. Diagram shows the shaded region bounded byy-axis, the curvey2= 4xand
a straight liney= k.
Given that the area of the shaded region is9
4unit
2, find the value of k.
Answer :
y2= 4x
y= k
0 x
y
y= 5x4
x=p
0 x
y
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MODUL GEMILANG SMKA SHEIKH MALEK 69 by CYY
PAPER 2
7. Diagram shows the straight line PQis normal to the curve2
13
xy= + atM(3, 4). The straight line
MNis parallel toy-axis.
Find
a) the value of h,
b) the area of the shaded region,
c) the volume of revolution, in terms of , when the region bounded by the curve, the
y-axis and straight liney= 4 is rotated through 360abouty-axis.
Guided Solutions
a) m1 =
dy
dx =
2
3
x
=
2( )
3 = ( )
m2 =1
1
m
= ( ) =
4 0
3 h
(m2 is gradient of PQ)
h = ( )
b) Area of regionA =23
0
( 1)3
xdx+ =
Area of regionB =
Hence, the area of the shaded region =A+B=
c) The volume of revolution = 4
2
1
x dy = 4
1
( ) dy =
.
BA
N Q(h, 0)Ox
y
y=
2
13
x +
M (3, 4)
P
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AO
B
y
P
C
xk
Q
32 +=xy
9=+xy
8. Diagram shows the curve 32 +=xy intersects the straight lineAC at point B.
It is given that the equation of straight lineACisy + x = 9and the gradient of the curve at pointBis
4. Find
a) the value of k,
b) the area of the shaded region P,
c) the volume of revolution, in terms of , when the shaded region Q is rotated through
360 about the xaxis .
9. Diagram shows the curve 22 1y x= + intersects the straight line 9 9y x= at point (2, k).
Find,
(2,k)
y= 2x2+ 1
y= 9x9
x