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BAHAGIAN SEKOLAH BERASRAMA PENUH DAN SEKOLAH KLUSTER

ADDITIONAL MATHEMATICS

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PAGE 1 2 3 4 5 6 7 8 ABOUT THIS MODULE WE LEARN EXAMINATION FORMAT ANALYSIS TABLE OF SPM ADD MATHS QUESTIONS LIST OF FORMULAE AND NORMAL TABLE ADDITIONAL MATHEMATICS NOTES PROBLEM SOLVING STRATEGY PARTITION I II II-III IV V-VII VIII-XVI XVII XVIII

This module is 1. specially planned for students who will be sitting for SPM. 2. to provide exposure and to familiarize students with the needs of the actual SPM exam questions. 3. to prepare students with adequate knowledge prior to the examination. 4. comprises challenging questions which incorporate a variety of questioning techniques and levels of difficulty and conforms to the current SPM farmat.

That which we persist in doing becomes easier not that the nature of the task has changed, but our ability to do has increased.

I

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Key towards achieving 1A Read question carefully Follow instructions Start with your favourite question Show your working clearly Choose the correct formula to be used +(Gunakannya dengan betul !!!) Final answer must be in the simplest form The end answer should be correct to 4 S.F. (or follow the instruction given in the question) 3.142

Kunci Mencapai kecemerlangan

Proper / Correct ways of writing mathematical notations Check answers! Proper allocation of time (for each question)

Paper 1 : 3 - 7 minutes for each question Paper 2 : Sec. A : 8 - 10 minutes for each question Sec. B : 15 minutes for each question Sec. C : 15 minutes for each question

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ANALYSIS TABLE OF SPM ADDITIONAL MATHEMATICS QUESTIONS 2004-2008AMaths (3472) SPM Chapter 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 Functions Quadratic Equations Quadratic Functions Simultaneous Equations Indices and Logarithms Coordinate Geometry Statistics Circular Measure Differentiation Solution of Triangle Index Number Progressions Linear Law Integration Vectors Trigonometric Functions Permutations / Combinations Probability Probability Distributions Motion Along A Straight Line Linear Programming Total 04 3 1 2 05 3 2 1 Paper 1 06 2 1 1 07 3 1 2 08 3 1 2 1 2 2 3 1 1 1 2 1 2 3 1 1 1 3 2 2 1 1 2 2 2 1 1 2 1/2 1/2 2/3 1 1 1 1 1 1 1 1/2 1 1/3 1 1 1/3 1 1/3 1 1 4 1 1 2 1 1 1 1 3 1 1 2 1 1 1 1 2 1 2 2 1 1 1 1 3 1 1 2 1 1 1 1 3 1 1 2 1 1 1 1 1 1 1 1 1 1 1 25 25 25 25 25 6 6 6 6 6 5 5 5 5 5 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1 1 1 1 1 1/3 1 1 1 1 1 1 1 1/2 1 1 2/3 1 1 1 2/3 1 1 2/3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 04 05 Section A 06 07 1 08 04 05 Paper 2 Section B 06 07 08 04 05 Section C 06 07 08

10

IV

SULIT

3472/2

The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA 1 x=

b b 4ac 2a2

8

logab =

log c b log c a

2 am a n = a m + n 3 am an = a m - n

9 10 11 12

Tn = a + (n-1)d Sn =

4 (am) n = a nm 5 6 7 loga mn = log am + loga n loga

n [2a + (n 1)d ] 2

Tn = ar n-1 Sn =

m = log am - loga n n

log a mn = n log a m

a(r n 1) a(1 r n ) , (r 1) = r 1 1 r a , r 0 2 different (distinct) roots. b2- 4ac = 0 2 equal roots b2 - 4ac < 0 no real roots. b2 - 4ac 0 with real rootsyy y

_ _ +a

_ + _b

+ + +

4. INDICES & LOGARITHM (a) x = an Index Form loga x = n Logarithmic Form

(b) Laws of Indices

1. a n a m = a n+m 2. a n a m = a nm 3. (a n ) m = a nmLaws of Logarithm 1. logaxy = logax + logay x 2. loga = logax logay y 3. loga xn = n logax 4. loga a = 1 5. loga 1 = 0 log c b 6. loga b = log c a 1 7. loga b = log b a

0

x

0

x

0

b2 - 4ac > 0

b2 - 4ac = 0

b2 - 4ac < 0

x

(b) Completing the Squares y = a(x - p)2 + q a +ve minimum point (p, q) a ve maximum point (p, q) (c) Quadratic Inequalities (x a)(x b) 0 Range: x a, x b (x a)(x b) 0 Range: a x b

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X

5. COORDINATE GEOMETRY (a) Distance between A(x1, y1) and B(x2, y2) AB = ( x 2 x1 ) 2 + ( y 2 y1 ) 2 (b) Midpoint of AB x + x 2 y1 + y 2 M= 1 , 2 2 (c) P divides AB internally in the ratio m : nm : A(x 1 , y1 ) P n B(x2 , y2 )

6. STATISTICS Measure of Central Tendency (a) Mean x x= n for ungrouped datax=

fx f fx f

for ungrouped data with frequency.x=i

nx + mx 2 ny1 + my 2 P= 1 , n+m n+m (d) Gradient of AB y y1 m= 2 x 2 x1m=

for grouped data , xi = midpoint of each class interval (b) Median The centre value of a set of data after the data is arranged in the ascending or descending order. FormulanF C fm L = Lower boundary of the Median class n = Total frequency F = Cumulative frequency before the median class fm = Frequency of the median class C = Size of the class interval

y-intercept x-intercept

(e) Equation of a straight line (i) given m and A(x1, y1) y y1 = m(x x1) (ii) given A(x1, y1) and B(x2, y2) y y1 y 2 y1 = x x1 x 2 x1 (f) Area of polygon x 1 x1 x 2 x3 L= ......... 1 y1 2 y1 y 2 y 3 (g) Parallel lines m 1 = m2 (h) Perpendicular lines m1 m2 = -1

M=L+

1 2

From the OgiveCumulative Kekerapan Frequency longgokann

n __ 2

0

Median Sempadan atas

Median

Upper Boundary

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XI

(c) Mode Data with the highest frequency From the Histogram :KFrequency ekerapan

For ungrouped data

==

( x x)n

2

x

2

n For grouped data

x

2

=

f ( x x) f

2

=0 Mod Sempadan kelas Mode Class Boundary

fx f

2

x

2

Measure of Dispersion (a) Interquartile Range Formulae : 1 n F1 Q1 = L1 + 4 C f Q1 Q3 = L3 + Ogive:Cumulative FrequencyKekerapan longgokan3 4

7. INDEX NUMBERS (a) Price Index P I = 1 100 P0 where Po = price at the base time P1 = price at a specific time (b) Composite Index Iw I= w where I = price index or index number w = weightage 8. CIRCULAR MEASURE (a) Radian Degree 180 0 r= (b) Degree Radian rad 180 (c) Arc length s = j (d) Area of sector 1 1 L = j2 = js 2 2

n F3 C f Q3

3 __ n 4

1 __ n 4

0

Q

Interquartile Range = Q3 Q1 (b) Variance, Standard Deviation Variance = (Standard Deviation)2

1

Q 3 Sempadan atas Upper Boundary

o =

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XII

(e) Area of segment 1 L = j2( r sin o) 2 9. DIFFERENTIATION (a) Differentiation using the First Principal

10. INTEGRATION ax n +1 (a) ax n dx = +c n +1

(ax + b) n+1 (b) (ax + b) dx = +c a (n + 1)n

dy y = lim dx x0 x

(c)

f ( x) + g ( x) dxa

b

(b)

d (a) = 0, a = constant dx d n (x ) = nxn-1 (c) dx d (d) (axn) = anxn-1 dx

=b

a

b

f ( x) dx + g ( x) dxac c

b

(d) (e) (f)y

f ( x) dx + f ( x) dx = f ( x) dxa b a

af ( x) dx = a f ( x) dxa a

b

b

(e) Product Rule d dv du (uv) = u +v dx dx dx (f) Quotient Rule dv d u v du dx u dx = dx v v2 (g) Composite Function dy du d (ax+b)n = dx du dx = an(ax+b)n-1 dy =0 (h) Turning point dx Maximum point: d2y dy = 0 and 0 dx dx 2 (i) Rate of change dy dy dx = dt dx dt

f ( x) dx = f ( x) dxa b

b

a

(h) Area under the curve

A=

y dxa

b

0 y b

a

b

x

A=a

x dya

b

0

x

(i) Volume of revolutiony

0

a

b

x

V = y 2 dxa

b

(j) Small change : dy y . x dxXIII

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y b

V = x 2 dya

b

a 0

x

(a) s = 0 at the fixed point O (b) v = 0 stops momentarily maximum / minimum displacement (c) a = 0 v constant v maximum/ minimum 13. TRIGONOMETRIC FUNCTIONS (a) y y P(x, y) sin = r x cos = r r y y tan = x0 x x

11. PROGRESSIONS Arithmetic Progressions (a) Tn = a + (n - 1)d n (b) Sn = {2a + (n - 1)d} 2 n = (a + l) 2 (c) d = T2 - T1 Geometric Progressions (a) Tn = arn-1 a(1 r n ) (b) Sn = for r < 1 1 r a(r n 1) Sn = for r > 1 r 1 a (c) S = for -1 < r < 1 1 r and n T (d) r = 2 T1 General (a) S1 = T1 = a (b) Tn = Sn Sn-1 (c) Sum of terms from Ta to Tb = Sb Sa-1

(b) tan =

sin cos 1 sec = cos 1 sin

cosec = cot = (c)

1 cos = tan sin

Sin +ve

All Semua +ve Cos Kos +ve

Tan +ve

12. MOTION ALONG A STRAIGHT LINE ds dv dt dt s v a v dt a dtXIV

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(d) Special Angles 0o 30o 0 1 Sin 2 1 Cos 3 Tan 0

45 1

o

60

o

2 121

2 1 3

3 2 1 2 3

(f) sin2 + cos2 = 1 1 + tan2 = sec2 1 + cot2 = cosec2 (g) sin(A B) = sinA cos B cos Asin B cos(A B) = cosA cosB m sinA sinB tan (A B) tan A tan B = 1 m tan A tan B (h) sin2A = 2 sinA cosA cos2A = cos2A sin2A = 2 cos2A 1 = 1 2 sin2A 2 tan A tan 2A = 1 tan 2 A 14. VECTORS (a) Addition of Vectors 1. Triangle Lawb a +

Sin Cos Tan

90o 1 0

180o 270o 0 -1 -1 0 0

360o 0 1 0

(e) Trigonometric Graphs y = a sin bxy a

0

__ 90 b

180 __ b

270 __ b

__ 360 b

x

-a

y = a cos bxy aa

b

2. Parallelogram Law0__ 90 b 180 __ b 270 __ b __ 360 b

xb a +

-a

b

y = a tan bxya

3. Polygon Law0__ 90 b 180 __ b 270 __ b __ 360 b

xA

B C

uuu r uuu v uuu v uuu v D uuuv AE = AB + BC + CD + DEXV

E

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