This document contains a 2010 Additional Mathematics exam paper from the Sijil Pelajaran Malaysia (SPM). It consists of 25 multiple choice and short answer questions covering topics like:
- Relations and functions
- Quadratic equations
- Geometric and arithmetic progressions
- Trigonometry
- Probability and statistics
The questions require students to apply concepts like domain and range, inverse functions, maximum/minimum values, and normal distributions to solve problems involving graphs, equations, and word problems.
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Matematik tambahan spm tingkatan 4 geometri koordinat {add maths form 4 coord...Hafidz Sa
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Embracing GenAI - A Strategic ImperativePeter Windle
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Spm add math 2009 paper 1extra222
1. SPM ADD MATH 2010 Paper 1 1
SIJIL PELAJARAN
MALAYSIA 2009
ADDITIONAL MATHEMATICS
Paper 1
3472/1
2 hours
2. SPM ADD MATH 2010 Paper 1 2
Diagram 1 shows the relation between set X and set Y in
the graph form.
1
State
(a) the objects of q,
(b) the codomain of the relation.
( 2 , q )
( 2 , s )
( 4 , p )
( 4 , r )
( 6 , q )
( 2 , q )
( 6 , q )
(a)the objects of q
2 , 6
(b) the codomain of the relation.
{ p , q , r , s }
3. SPM ADD MATH 2010 Paper 1 3
Given the functions g:x 2x - 3and h:x 4x, find
(a) hg(x),
(b) the value of x if hg(x) = ½ g(x).
2
h g (x )
= h ( 2x - 3 )
= 4 ( 2x - 3 )
= 8x - 12
(a) (b) hg(x) = ½ g(x)
8x - 12 = ½ ( 2x - 3 )
16x - 24 = 2x -3
14x = 21
x = 3/2
4. SPM ADD MATH 2010 Paper 1 4
3 Given the function g : x 3x -1, find
(a) g(2),
(b) the value of p when g-1
(p) = 11.
g : x 3x -1
(a) g(2)
= 3( 2 ) -1
(b) g-1
(p) = 11
= 5
3(11) -1 = p
p = 33-1
= 32
5. SPM ADD MATH 2010 Paper 1 5
4 The quadratic equation x2
+ x = 2px - p2
, where p is a
constant, has two different roots.
Find the range of values of p.
x2
+ x = 2px - p2
x2
+ x -2px + p2
= 0
x2
+ x( 1 -2p) + p2
= 0
b2
- 4ac > 0
( 1-2p )2
- 4(1)(p2
) > 0
1 – 4p + 4p2
- 4p2
> 0
1 – 4p > 0
4p < 1
p < ¼
6. SPM ADD MATH 2010 Paper 1 6
5 Diagram shows the graph of a quadratic function
f(x) = - (x + p)2
+ q, where p and q are constants.
State
(a)the value of p,
(b) the equation of the axis of symmetry.
f(x) = - (x + p)2
+ q
x = -3
x + 3 = 0
x + 3 = x + p
p = 3
(a)
(b) x = -3
7. SPM ADD MATH 2010 Paper 1 7
6 The quadratic function f(x) = -x2
+ 4x + a2
, where a is a
constant, has maximum value 8.
Find the values of a.
f(x) = -x2
+ 4x + a2
= -( x2
- 4x ) + a2
= -[ x2
- 4x + ( -2 )2
- ( -2 )2
] + a2
= -( x - 2 )2
+ ( -2 )2
+ a2
( -2 )2
+ a2
= 8
a2
= 4
a = ± 2
8. SPM ADD MATH 2010 Paper 1 8
7 Given 3n - 3
x 27 n
= 243, find the value of n.
3n - 3
x 27 n
= 243
3n - 3
x ( 33
) n
= 35
3n - 3
x 33n
= 35
3n – 3 + 3n
= 35
n – 3 + 3n = 5
4n = 8
n = 2
9. SPM ADD MATH 2010 Paper 1 9
8 Given that log8 p - log2 q = 0, express p in terms of q.
log8 p - log2 q = 0
log8 p = log2 q
q
p
2
2
2
log
8log
log
=
log2 p = 3log2 q
log2 p = log2 q3
p = q3
10. SPM ADD MATH 2010 Paper 1 10
9 Given the geometric progression -5, 10/3 , - 20/9 ,...,
find the sum to infinity of the progression.
10 20
5 , , ,...
3 9
− −
a = -5
10
3
5
r =
−
2
3
r = −
5
2
1
3
S∞
−
=
− − ÷
5
5
3
−
=
3= −
11. SPM ADD MATH 2010 Paper 1 11
10 Diagram 10 shows three square cards.
The perimeters of the cards form an arithmetic
progression. The terms of the progression are in
ascending order.
(a)Write down the first three terms of the progression.
(b)Find the common difference of the progression.
(a) 4(3) , 4(5), 4(7)
= 12 , 20, 28
(b) d = 20 - 12
= 8
12. SPM ADD MATH 2010 Paper 1 12
11 The first three terms of a geometric progression are x, 6,
12. Find
(a)the value of x,
(b) the sum from the fourth term to the ninth term.
(a) x , 6 , 12
6
126
=
x
6
2
x
=
6
2
x =
= 3
(b)
a = 3 r = 2
( )
12
123 9
9
−
−
=S = 3( 29
– 1 )
= 1533
( )3
3
3 2 1
2 1
S
−
=
−
= 3( 23
– 1 )
= 21
4 9 1533 21S −> = − = 1512
13. SPM ADD MATH 2010 Paper 1 13
12 Diagram 12 shows a sector BOC of a circle with centre 0.
It is given that ∠BOC= 1.42 radians, AD = 8 cm and 0A =
AB = OD = DC = 5cm.
Find
(a) the length, in cm, of arc BC,
(b) the area, in cm2
, of the coloured region.
(a) arc BC
= 10 ( 1.42 )
= 14.2
(b) the area
= ½ (10)2
( 1.42 )
- ½ (8) (3)
= 71 - 12
= 59
14. SPM ADD MATH 2010 Paper 1 14
Given that a = 13 i + j and b = 7 i – k j, find
(a) a - b in the form x i + y j,
(b) the values of k if | a - b | = 10.
13
a = 13 i + j , b = 7 i – k j,
(a) a - b
= 13 i + j – ( 7 i – k j )
= 6 i +(1+k) j
(b) | a - b | = 10
( ) 1016
22
=++ k
36 + 1 + 2k + k2
= 100
k2
+ 2k – 63 = 0
( k +9 ) ( k -7 ) = 0
k = -9 , 7
15. SPM ADD MATH 2010 Paper 1 15
14 Diagram 14 shows a triangle PQR.
Given and point S lies on QR such
that QS : SR = 2 : 1, express in terms of a, and b
= -3 a + 6 b
SR RP= +
uur uuur
( )
1
3 6 6
3
a b b= − + −
2 6a b b= − + −
4a b= − −
16. SPM ADD MATH 2010 Paper 1 16
15 Diagram shows a straight line AC.
The point B lies on AC such that AB : BC = 3 : 1.
Find the coordinates of B.
( ) ( ) ( ) ( )3 4 1 2 3 0 1 3
,
3 1 3 1
B
+ − +
= ÷
+ +
12 2 0 3
,
4 4
− +
= ÷
10 3
,
4 4
= ÷
5 3
,
2 4
= ÷
17. SPM ADD MATH 2010 Paper 1 17
16 Solve the equation 3 sin x cos x - cos x = 0 for 00
≤ x ≤
3600
.
3 sin x cos x - cos x = 0
cos x ( 3 sin x – 1 ) = 0
cos x = 0 3 sin x – 1 = 0
x = 90o
, 270o sin x = 1/3
x = 19.47o
, 160.53o
18. SPM ADD MATH 2010 Paper 1 18
17 It is given that sin A = 5/13 and cos B = 4/5 where A is
an obtuse angle and B is an acute angle.
Find
(a) tan A,
(b) cos(A - B).
A5
13 B
5
4
12
(a) tan A
5
12
= −
3
(b) cos( A – B )
= cosA cosB – sinA sinB
12 4 5 3
13 5 13 5
= − + ÷ ÷ ÷ ÷
48 15
65 65
= − +
33
65
= −
19. SPM ADD MATH 2010 Paper 1 19
18
Given that and∫ =
m
dxxf
5
6)( [ ]
5
( ) 2 14
m
f x dx+ =∫
find the value of m.
∫ =
m
dxxf
5
6)([ ]
5
( ) 2 14
m
f x dx+ =∫
5 5
( ) 2 14
m m
f x dx dx+ =∫ ∫
5
6 2 14
m
dx+ =∫
[ ]5
2 8
m
x =
2 ( m – 5 ) = 8
m – 5 = 4
m = 9
20. SPM ADD MATH 2010 Paper 1 20
19 The gradient function of a curve is = kx - 6, where k
is a constant.
It is given that the curve has a turning point at (2, 1).
Find
(a)the value of k,
(b) the equation of the curve.
dy
dx
6
dy
kx
dx
= −(a)
( )0 2 6k= −
2 6k =
3k =
3 6
dy
x
dx
= −(b)
2
3
6
2
x
y x c= − +
( )
( )
2
3 2
1 6 2
2
c= − +
7c =
2
3
6 7
2
x
y x= − +
21. SPM ADD MATH 2010 Paper 1 21
20 A block of ice in the form of a cube with sides x cm, melts
at a rate of 972 cm3
per minute. Find the rate of change of
x at the instant when x = 12cm.
3
V x=
2
3
dV
x
dx
=
( )
2
3 12=
432=
dV dV dx
dt dx dt
= •
972 432
dx
dt
= •
972
432
dx
dt
=
2.25 /cm s=
22. SPM ADD MATH 2010 Paper 1 22
21 Diagram shows part of the curve y = f(x) which passes
through the points (h, 0) and (4, 7).
Given that the area of the coloured region is 22 unit2
,
find the value of h∫4
f(x)dx.
∫
4
)(
h
dxxf
= 4(7) - 22
= 6
23. SPM ADD MATH 2010 Paper 1 23
22 There are 4 different Science books and 3 different
Mathematics books on a shelf.
Calculate the number of different ways to arrange all the
books in a row if
(a) no condition is imposed,
(b) all the Mathematics books are next to each other.
S1 S2 S3 S4 M1 M2 M3
(a) 7
7p
7!=
5040=
(b)
S1 S2 S3 S4M1 M2 M3
5 3
5 3p p×
5! 3!= ×
120 6= ×
720=
24. SPM ADD MATH 2010 Paper 1 24
23 The probability that a student is a librarian is 0.2. Three
students are chosen at random.
Find the probability that
(a) all three are librarians,
(b) only one of them is a librarian.
P( x = 3 )(a)
= 1 × ( 0.2)3
× 1
= 0.008
X ~ Bin ( 3 , 0.2 )
= 3
C3 (0.2)3
(0.8)0
P( x = 1 )(a)
= 3 × ( 0.2) × 0.64
= 0.384
= 3
C1 (0.2)1
(0.8)2
25. SPM ADD MATH 2010 Paper 1 25
24 A set of 12 numbers x1, x2, ... , x12 , has a variance of 40
and it is given that ∑x2
= 1 080.
Find
(a)the mean,
(b) the value of ∑ x.
(a) 2
2
2
x
N
x
−=
∑σ
21080
40
12
x= −
2
40 90 x= −
2
50x =
7.071x =
(b)
12
x
x =
∑
12(7.071)x =∑
84.853=
26. SPM ADD MATH 2010 Paper 1 26
25 The masses of apples in a stall have a normal
distribution with a mean of 200 g and a standard
deviation of 30 g.
(a) Find the mass, in g, of an apple whose z-score is 0.5.
(b) If an apple is chosen at random, find the probability
that the apple has a mass of at least 194g.
X ~ N ( 200 , 302
)
(a) X
Z
µ
σ
−
=
200
0.5
30
X −
=
15 200X= −
215X =
(b) ( 194)P X >
194 200
( )
30
P Z
−
= >
( 0.2)P Z= > −
1 ( 0.2)P Z= − >
1 0.4207= −
0.5793=